Trigonometry

Sin3x : Formula, Derivation, Proof, and Examples

Sin3x equals 3sin x - 4sin³ x, the triple-angle identity that writes the sine of a tripled angle in terms of the sine of the single angle alone. This article covers the formula, its step-by-step derivation, the rearranged sin³ x form, six worked examples, the mistakes students make, and where the triple-angle identity is used.

Tan Pi : Exact Value 0, and How to Find It

The value of tan π is exactly 0. This article shows why, using tanθ = (sinθ)/(cosθ) with sinπ = 0, reads the same answer off the unit-circle point (-1, 0), gives a tangent table in degrees and radians, and works through examples, including why tanπ is a clean zero rather than an undefined value.

Sin 47 Degrees — Value of sin(47°) and How to Find It

The value of sin 47 degrees is approximately 0.7314 — it is not a special-angle exact value, so there is no clean surd for it. This article shows how to find sin 47° honestly (calculator, the cofunction cos 43°, and table interpolation), gives the radian form, and places it on the unit circle.

Cot2x Formula : Proof, Graph, Properties, and Examples

The cot2x formula is cot 2x = (cot² x - 1)/(2cot x), the double-angle identity for cotangent. This article covers its proof from the cotangent angle-sum rule, three equivalent forms (in terms of tan x, in terms of sin and cos, and as 1/2(cot x - tan x)), the graph with its asymptotes, six worked examples, and the mistakes to avoid.

Law of Cosines : Formula, Proof, and Examples

The law of cosines states c² = a² + b² - 2abcos C, relating the three sides of any triangle to the cosine of one angle. This article covers when to use it (the SAS and SSS cases), its derivation, six worked examples, how it generalises the Pythagorean theorem, and the errors students make with obtuse angles.

Inverse Sine (Arcsin): Domain, Range & Graph

The inverse sine function, written sin⁻¹x or x, takes a ratio between -1 and 1 and returns the unique angle in [-(π)/2, (π)/2] whose sine is that ratio. This article covers its definition, why the domain is [-1, 1], the principal branch, the arcsin graph, and worked examples.

Reciprocal of Sine : Cosecant Relationship & Graph

The reciprocal of sine is the cosecant function, defined by cscθ = 1/(sinθ), so multiplying the two always gives 1. This article covers how cosecant is derived from sine, why the reciprocal is undefined wherever sinθ = 0, its range of (-∞, -1] ∪ [1, ∞), its graph, and worked examples, while keeping it clearly apart from inverse sine.

Unit Circle With Tangent : Values, Chart, and the Tangent Segment

On the unit circle, the tangent of an angle is the y-coordinate divided by the x-coordinate of the point where the terminal ray meets the circle: tanθ = (sinθ)/(cosθ) = y/x. This article covers how to read tangent off the unit circle, the full tangent values chart, the geometric tangent-line segment, quadrant signs, and where tangent is undefined.

Tan Pi/6 : Exact Value 1/√3, Rationalized, and How to Find It

The value of tan pi/6 is exactly 1/(√3), which rationalizes to (√3)/3 and rounds to about 0.5774. This article proves it on the unit circle, ties (π)/6 to its degree twin 30^°, and covers rationalizing the denominator, a reference table, worked examples, and the usual mistakes.

Tan Pi/4 : Exact Value, Why It Equals 1, and How to Find It

The value of tan pi/4 is exactly 1, because at (π)/4 radians the sine and cosine are equal, so their ratio is one. This article proves it from the unit circle, ties the radian angle to its degree twin, and walks through a reference table, worked examples, and the mistakes students hit.

Tan Pi/3 : Exact Value √3, and How to Find It

The value of tan π/3 is exactly √3, which is about 1.732. This article shows where that value comes from using the 30-60-90 triangle and the unit-circle point (1/2, (√3)/2), connects (π)/3 to 60^°, gives a tangent reference table in degrees and radians, and works through examples and the mistakes students make.

Tan Pi/2 : Why It Is Undefined, From the Unit Circle

The value of tan π/2 is undefined: it is not a number, because cos(π)/2 = 0 forces a division by zero. This article works from the radian angle and the unit-circle point (0, 1), shows the vertical asymptote and the one-sided limits, notes why the odd-function rule does not rescue a value, and clears the "tan π/2 equals infinity" mix-up with examples.

Tanh : Hyperbolic Tangent Formula, Graph & Properties

Tanh, the hyperbolic tangent, is defined by x = (e x - e -x)/(e x + e -x) and produces a smooth S-curve squashing every real number into the open interval (-1, 1). This article covers its definition, the tanh graph with horizontal asymptotes, its properties, derivative, and how it differs from the circular tangent function.

Tangent Function : Graph, Period, Asymptotes & Properties

The tangent function, tanθ = (sinθ)/(cosθ), gives the slope of the line from the origin to a point on the unit circle. This article covers its definition, the tangent graph with vertical asymptotes, why the period is π (not 2π), its domain, range, odd symmetry, and worked examples.

Tan(A−B) Formula: Proof, Derivation & Examples

The tan(A−B) formula is tan(A-B) = (tan A - tan B)/(1 + tan A tan B), the tangent subtraction identity. This article derives it from the sine and cosine difference formulas, works six examples including tan 15^°, and shows where the identity earns its keep — most sharply in finding the angle between two lines.

Tan 120 Degrees : Value −√3 and How to Find It

The value of tan 120 degrees is exactly -√3, about -1.7321, because 120^° sits in the second quadrant where tangent is negative and its reference angle is 60^°. This article proves it with the reference-angle rule and the unit circle, gives a standard-angle table, and walks through the sign mistakes students make.

Tan 90 Degrees : Why It Is Undefined (Not Infinity

The value of tan 90 degrees is undefined: it is not a number, because computing it forces a division by zero. This article shows why using tanθ = (sinθ)/(cosθ) and cos 90^° = 0, reads the answer off the unit circle, explains the vertical asymptote and the one-sided limits, and clears up the common "tan 90 equals infinity" mix-up with worked examples.

Tan 60 Degrees : Exact Value, √3, and How to Find It

The value of tan 60 degrees is exactly √3, which is about 1.7321. This article shows where that value comes from using the 30-60-90 triangle and the unit circle, gives a standard-angle table in both degrees and radians, and works through examples plus the mistakes students make.

Tan 45 Degrees : Exact Value, 1, and How to Find It

The value of tan 45 degrees is exactly 1. This article shows why that value falls straight out of the 45-45-90 triangle and the unit circle, gives a standard-angle table in both degrees and radians, and works through examples plus the mistakes students make.

Tan 30 Degrees : Exact Value, 1/√3, and How to Find It

The value of tan 30 degrees is exactly 1/(√3), which rationalises to (√3)/3 and is about 0.5774. This article shows where that value comes from using the 30-60-90 triangle and the unit circle, gives a standard-angle table in degrees and radians, and works through examples plus common mistakes.

Tan 20 Degrees : Value 0.3640 and How to Find It

The value of tan 20 degrees is approximately 0.3640 (more precisely 0.36397023), and unlike 30^° or 60^° it has no clean surd form. This article shows why 20^° is a calculator value rather than a memorised one, where it sits among the special angles, and how to find and use it reliably.

Tan 7pi/6 : Exact Value 1/√3 Explained

The value of tan 7π/6 is exactly 1/(√3), or in rationalised form (√3)/3, which is about 0.5774. This article shows why the answer is positive (the angle lands in the third quadrant), how the reference angle (π)/6 produces the value, and how the unit circle confirms it, with a tangent table, worked examples, and the mistakes to sidestep.

Tan 3pi/4 : Exact Value, −1, and How to Find It

The value of tan 3pi/4 is exactly -1. This article works from the unit circle and the reference angle (π)/4, explains why the sign is negative in the second quadrant, gives a radian table, and works through examples plus common mistakes.

Tan 2pi/3 : Exact Value, −√3, and How to Find It

The value of tan 2pi/3 is exactly -√3, which is about -1.7321. This article works from the unit circle and the reference angle (π)/3, explains why the sign is negative in the second quadrant, gives a radian table, and works through examples plus common mistakes.

Tan 2pi : Value, Unit Circle Proof, and Why It's 0

The value of tan 2π is exactly 0, because 2π radians is one full turn around the circle back to the point (1, 0), where tangent reads 0/1. This article proves it from the unit circle, gives a tangent table in radians and degrees, and shows why periodicity guarantees the answer.

Tan 1 Degrees : Value 0.0175 and How to Find It

The value of tan 1 degree is approximately 0.0175 (more precisely 0.01745506), and like most non-standard angles it has no clean surd form. This article shows why 1^° is a calculator value, how the small-angle rule tanθ ≈ θ gets it almost exactly, and where such a tiny tangent matters.

Tan 0 Degrees : Value, Proof, and Why It Equals 0

The value of tan 0 degrees is exactly 0, because tangent is (sinθ)/(cosθ) and at 0^° that is 0/1. This article proves it from the unit circle, gives a standard-angle tangent table in degrees and radians, and walks through worked examples and the mistakes students make.

Sin Pi/6 : Exact Value 1/2 on the Unit Circle

The value of sin pi/6 is exactly 1/2, or 0.5, taken as the height of the (π)/6 point on the unit circle. This article proves it with the 30-60-90 triangle, gives a standard-angle table, links the degree twin sin 30^°, and works through examples and the errors students make.

Sin Pi/4 : Exact Value √2/2 Explained

The value of sin pi/4 is exactly (√2)/2, about 0.7071, taken as the height of the (π)/4 point on the unit circle. This article proves it with the 45-45-90 triangle, gives a standard-angle table, links the degree twin sin 45^°, and works through examples and the errors students make.

Sin Pi/3 : Exact Value √3/2 on the Unit Circle

The value of sin pi/3 is exactly (√3)/2, about 0.8660, taken as the height of the (π)/3 point on the unit circle. This article proves it with the 30-60-90 triangle, gives a standard-angle table, links the degree twin sin 60^°, and works through examples and the errors students make.

Sin Pi/2 : Exact Value on the Unit Circle (= 1)

The value of sin pi/2 is exactly 1, read straight off the unit circle as the height of the quarter-turn point. This article explains the radian angle (π)/2, proves the value on the circle, links the degree twin sin 90^°, and works through examples and the errors students make.

Sin of Sin Inverse x : Value, Domain, and Examples

Sin of sin inverse x equals x, written sin(sin⁻¹ x) = x, but only when x lies in [-1, 1]. This article covers the value, why the domain restriction exists, how sin(sin⁻¹ x) differs from sin⁻¹(sin x), six worked examples, and the mistakes students make when the input steps outside [-1, 1].

Sine Function : Graph, Period, Amplitude & Properties

The sine function gives the y-coordinate of a point on the unit circle, or the ratio (opposite)/(hypotenuse) in a right triangle. This article covers the definition, the sine graph (period 2π, range [-1, 1]), why sine is an odd function, its quadrant signs, key values, and worked examples.

Sin Double Angle Formula : Derivation, Identities, and Examples

The sin double angle formula is sin 2x = 2sin x cos x, which rewrites the sine of a doubled angle using the sine and cosine of the original angle. This article covers the formula, its derivation from the angle-sum identity, the tangent form, six worked examples, the mistakes to avoid, and where doubling an angle actually matters.

Sin A Sin B Formula: Product-to-Sum Proof & Examples

The sin A sin B formula is sin A sin B = 1/2[cos(A-B) - cos(A+B)], a product-to-sum identity that rewrites a product of two sines as a difference of two cosines. This article proves it from the cosine sum and difference formulas, works six examples, and shows where turning a product into a sum actually pays off.

Sin(a+b) Formula : Proof, Uses, and Examples

The sin(a+b) formula is sin(a+b) = sin a cos b + cos a sin b, the angle-sum identity for sine. This article covers its geometric proof, why it is not equal to sin a + sin b, how it builds the double-angle and sin(a-b) formulas, six worked examples, and the mistakes students make with it.

Sin A Cos B Formula : Derivation & Examples

Sin A cos B equals 1/2[sin(A+B) + sin(A-B)], a product-to-sum identity that turns a product of a sine and a cosine into a sum of two sines. This article covers the formula, its derivation from the angle-sum and angle-difference identities, six worked examples, the mistakes students make, and why the identity matters for integration.

Sin A Cos A Formula : Derivation & Examples

Sin A cos A equals 1/2sin 2A, the single-angle identity that rewrites the product of the sine and cosine of the same angle as half of its double-angle sine. This article covers the formula, its one-line derivation, the tangent form, six worked examples, the mistakes to avoid, and how it differs from 2sin Acos A and sin Acos B.

Sin 180 Degrees : Exact Value 0 and How to Find It

The value of sin 180 degrees is exactly 0. This article shows why the sine hits zero at a straight angle, derives it from the unit circle and the sin(180^° - θ) identity, and works through examples and the mistakes students make. Its radian twin is sin π.

Sin 150 Degrees : Exact Value 1/2 and How to Find It

The value of sin 150 degrees is exactly 1/2, or 0.5. This article shows why a second-quadrant angle keeps a positive sine, derives the value from the 30^° reference angle and the unit circle, and works through examples and the mistakes students make.

Sin 120 Degrees : Exact Value √3/2 and How to Find It

The value of sin 120 degrees is exactly (√3)/2, about 0.8660. This article shows why an obtuse angle in the second quadrant keeps a positive sine, derives the value from the 60^° reference angle and the unit circle, and works through examples and the mistakes students make.

Sin 90 Degrees : Value, Proof, and Why It Equals 1

The value of sin 90 degrees is exactly 1, the highest number the sine function ever reaches. This article proves it from the unit circle, gives a standard-angle table in degrees and radians, links the radian twin sin(π)/2, and walks through worked examples and the errors students make.

Sin 60 Degrees : Value √3/2 and How to Find It

The value of sin 60 degrees is exactly (√3)/2, about 0.8660. This article proves that value from the 30-60-90 triangle and the unit circle, gives a standard-angle table in degrees and radians, and works through examples and the mistakes students make most.

Sin 50 Degrees : Value 0.7660 and How to Find It

The value of sin 50 degrees is about 0.7660; unlike 45^° or 60^°, it has no simple radical form, so it is a calculator value. This article gives the decimal, the radian form (5π)/18, the cofunction link to cos 40^°, why the small-angle shortcut fails here, and worked examples.

Sin 45 Degrees : Value √2/2 Explained

The value of sin 45 degrees is exactly (√2)/2, about 0.7071, and it is the one angle where sine and cosine are equal. This article proves the value from the 45-45-90 triangle, reads it off the unit circle, gives a standard-angle table in degrees and radians, and walks through worked examples and mistakes.

Sin 40 Degrees : Value 0.6428 Explained

The value of sin 40 degrees is approximately 0.6428, and unlike 30^° or 45^° it has no clean radical form, 40^° is not a standard angle. This article explains why, shows how to find the value, gives a sine table for nearby angles, and walks through worked examples and the common mistakes.

Sin 35 Degrees : Value, Decimal 0.5736 & How to Find

The value of sin 35 degrees is approximately 0.5736, with no clean radical form because 35^° is not a standard angle. This article gives the decimal, the radian form (7π)/36, the cofunction link sin 35^° = cos 55^°, how to place it on the unit circle, and worked examples.

Sin 30 Degrees : Exact Value 1/2 and How to Find I

The value of sin 30 degrees is exactly 1/2, or 0.5, one of the cleanest results in trigonometry. This article proves it from the 30-60-90 triangle and the unit circle, gives the radian form sin(π)/6, a standard-angle reference table, and worked examples with the mistakes students make.

Sin 25 Degrees : Value, Decimal 0.4226 & How to Find

The value of sin 25 degrees is approximately 0.4226, with no clean radical form because 25^° is not a standard angle. This article gives the decimal, the radian form (5π)/36, the cofunction link sin 25^° = cos 65^°, how to place it on the unit circle, and worked examples.

Sin 20 Degrees : Value ≈ 0.342 and How to Find It

The value of sin 20 degrees is approximately 0.342. It is not a special angle, so it has no clean fraction or surd; this article shows how to read it from a calculator, why the triple-angle equation ties it to sin 60^°, and the mistakes students make.

Sin 10 Degrees : Value 0.1736 and How to Find It

The value of sin 10 degrees is approximately 0.1736 — a non-special angle with no clean radical like 1/2. This article shows why, where the small-angle rule starts to drift at 10^°, gives a reference table, and works through examples plus the cofunction link sin 10^° = cos 80^°.

Sin 5pi/6 : Exact Value 1/2 and How to Find It

The value of sin 5pi/6 is exactly 1/2, or 0.5. This article shows why a second-quadrant angle keeps its positive sine, finds the reference angle (π)/6 on the unit circle, gives a radian reference table, and works through examples and common mistakes.

Sin 5 Degrees : Value 0.0872 and How to Find It

The value of sin 5 degrees is about 0.0872; unlike 30^° or 60^°, it has no simple radical form, so it is a calculator value. This article gives the decimal, the radian form (π)/36, the small-angle approximation and its limits, the cofunction link to cos 85^°, and worked examples.

Sin 4pi/3 : Exact Value -√3/2 and How to Find It

The value of sin 4pi/3 is exactly -(√3)/2, about -0.8660. This article shows why the third-quadrant angle carries a negative sign, finds the reference angle (π)/3 on the unit circle, gives a radian reference table, and works through examples and common mistakes.

Sin 3pi/4 : Exact Value √2/2 and How to Find It

The value of sin 3pi/4 is exactly (√2)/2, about 0.7071, because (3π)/4 sits in Quadrant II with a reference angle of (π)/4 where sine is positive. This article shows the reference-angle method, the unit-circle reading, a standard-angle table, worked examples, and the slips to avoid.

Sin 3pi/2 : Exact Value -1 and How to Find I

The value of sin 3pi/2 is exactly -1, the lowest point sine ever reaches, because the angle (3π)/2 points straight down the negative y-axis. This article shows the unit-circle reading, the degree twin 270^°, a standard-angle table, worked examples, and where students slip.

Sin 3pi : Exact Value 0 and How to Find It

The value of sin 3pi is exactly 0, because 3π lands on the same spot as π once you use the sine function's period of 2π. This article shows the periodicity shortcut, the unit-circle reading, a standard-angle table in radians and degrees, worked examples, and the mistakes to sidestep.

Sin 2π/3 : Exact Value √3/2 and How to Find It

The value of sin 2π/3 is exactly (√3)/2, about 0.8660, because (2π)/3 is 120^° in the second quadrant where its reference angle is (π)/3 and sine stays positive. This article shows the quadrant-and-reference-angle reason, the unit-circle point, and worked examples.

Sin 2π : Exact Value, Why It Equals 0, and How to Find It

The value of sin 2π is exactly 0, because 2π radians is one complete turn around the circle, landing back at the starting point (1, 0) where the height is zero. This article shows the unit-circle reason, the periodicity and double-angle proofs, the degree link (2π = 360^°), and worked examples.

Sin 2 Degrees : Value ≈ 0.0349 and How to Find It

The value of sin 2 degrees is approximately 0.0349. It is not a special angle, so it has no clean fraction or surd; this article shows how to read it from a calculator, why the small-angle approximation sinθ ≈ θ nails it, and the mistakes students make.

Sin 1 Degrees : Value 0.0175 and How to Find It

The value of sin 1 degrees is approximately 0.01745 - a non-special angle with no clean radical form like (√3)/2. This article explains why, shows the small-angle approximation sinθ ≈ θ in radians, gives a reference table for small angles, and works through examples.

Sin 0 Degrees : Value, Why It Equals 0, and How to Find It

The value of sin 0 degrees is exactly 0. This article shows why the sine of a zero angle vanishes using the unit circle and the right triangle, gives a standard-angle table in degrees and radians, and works through examples plus the mistakes students make with 0.

Sec pi/4 : Exact Value √2 and How to Find It

The value of sec π/4 is exactly √2, about 1.4142, because secθ = 1/(cosθ) and cos(π)/4 = (√2)/2, whose reciprocal simplifies to √2. This article derives it from the 45-45-90 triangle and the unit circle, gives a standard-angle table, links the degree twin sec 45°, and works through examples.

Sec pi/3 : Exact Value 2, Unit Circle & Examples

The value of sec π/3 is exactly 2, because secθ = 1/(cosθ) and cos(π)/3 = 1/2, so its reciprocal is 2. This article derives that value from the 30-60-90 triangle and the unit circle, gives a standard-angle table, links the degree twin sec 60°, and works through examples.

Sec pi/2 : Value and Why It Is Undefined

The value of sec π/2 is undefined, because secθ = 1/(cosθ) and cos(π)/2 = 0, so sec π/2 becomes 1/0. This article shows the unit-circle reason, the vertical asymptote on the secant graph, the degree twin sec 90°, and worked examples.

Sec Pi : Exact Value −1, Why, and How to Find It

The value of sec pi is exactly -1, because secθ = 1/(cosθ) and cosπ = -1. This article shows the unit-circle proof, a secant value table, five worked examples, and the sign mistake that turns the answer into a wrong +1.

Sec Inverse x (Arcsec) : Domain, Range & Graph

Sec inverse x, written sec⁻¹x or arcsec(x), is the inverse secant function: it takes a number with |x| ≥ 1 and returns the angle whose secant is x, using the identity arcsec(x) = (1/x). This article covers its definition, why the domain is (-∞, -1] ∪ [1, ∞), its range [0, (π)/2) ∪ ((π)/2, π], the graph, and worked examples.

Sec 0 Degrees : Exact Value 1, Why, and How to Find It

The value of sec 0 degrees is exactly 1, because secθ = 1/(cosθ) and cos 0^° = 1. This article shows the unit-circle proof, a secant value table, five worked examples, and the mistake that confuses sec 0^° with the undefined csc 0^°.

Law of Sines : Formula, Proof, and Examples

The law of sines states a/(sin A) = b/(sin B) = c/(sin C), so each side of a triangle is proportional to the sine of its opposite angle. This article covers when to use it (the AAS, ASA, and SSA cases), its proof, the ambiguous case where two triangles fit the data, six worked examples, and the mistakes to avoid.

Inverse Tangent (Arctan): Domain, Range & Graph

The inverse tangent function, written tan⁻¹x or x, takes any real number and returns the unique angle in the open interval (-(π)/2, (π)/2) whose tangent is that number. This article covers its definition, why the domain is all real numbers, the principal branch, the two horizontal asymptotes, the graph, and worked examples.

Inverse Cosine (Arccos): Domain, Range & Graph

The inverse cosine, written cos⁻¹ x or x, reverses the cosine function: it takes a ratio between -1 and 1 and returns the unique angle in [0, π] whose cosine is that ratio. This article covers the definition, why the range is restricted to the principal branch [0, π], the domain and range, the graph, and six worked examples.

Derivative of Tan Inverse x : Formula, Proof & Examples

The derivative of tan inverse x is d/dx(tan⁻¹x) = 1/(1+x²). This article derives that result by implicit differentiation and by the first principle, works six examples from bare tan⁻¹x to composite arguments, and clears up the chain-rule and inverse-versus-reciprocal mistakes that trip students up.

Derivative of Cos Inverse : Formula, Proof & Examples

The derivative of cos⁻¹x (arccos x) is d/dxcos⁻¹x = -1/(√(1 - x²)), valid for -1 < x < 1. This article covers the formula, the implicit-differentiation proof, why the sign is negative, the role of the principal branch, worked examples, and the mistakes students make most.

Derivative of Cosec x : Formula, Proof & Examples

The derivative of cosec x is d/dx(csc x) = -csc x cot x. This article proves that result three ways - the quotient rule on 1/(sin x), the chain rule on (sin x)⁻¹, and the first principle - then works six examples and clears up where the sign and the chain rule go wrong.

Derivative of Cos 2x : Formula, Chain Rule & Proof

The derivative of cos 2x is -2sin 2x. The extra factor of 2 comes from the chain rule, because 2x is an inner function whose own derivative is 2. This article covers the formula, the chain-rule proof, a first-principle derivation, the anti-derivative, worked examples, and the mistakes students make most.

Csc Sec Cot - The Reciprocal Trigonometric Functions Explained

Csc, sec, and cot are the three reciprocal trigonometric functions: cscθ = 1/(sinθ), secθ = 1/(cosθ), and cotθ = 1/(tanθ). This article defines all three, gives their domains, ranges, periods, and graphs, connects them through the Pythagorean identities, and works six examples. For the single-function deep dive, the cosecant page goes further on csc alone.

Cot Pi/6 : Exact Value Root 3 (Unit Circle Proof)

The value of cot pi/6 is exactly √3, because cotθ = (cosθ)/(sinθ) and at (π)/6 that ratio is (√3/2)/(1/2). This article shows the 30-60-90 triangle proof, the unit circle, a standard-angle table, five worked examples, and the mistake that swaps it with tangent.

Cot Pi/4 : Exact Value 1 Explained (Unit Circle)

The value of cot pi/4 is exactly 1, because cotθ = (cosθ)/(sinθ) and at (π)/4 the sine and cosine are equal. This article shows the 45-45-90 triangle proof, the unit circle, a standard-angle table, five worked examples, and where students slip.

Cot Pi/2 : Exact Value 0 Explained (Unit Circle)

The value of cot pi/2 is exactly 0, because cotθ = (cosθ)/(sinθ) and at (π)/2 the top of that fraction is 0. This article shows the unit-circle proof, a standard-angle table, five worked examples, and the mistake that makes students call it undefined.

Cot Pi : Why the Value Is Undefined (Cot π)

The value of cot pi is undefined, because cotθ = (cosθ)/(sinθ) and at π that becomes (-1)/0, a division by zero. This article explains why it is undefined rather than zero or a clean infinity, ties π radians to 180^°, and covers the unit-circle picture, a reference table, examples, and the mistakes students make.

Cot 570 Degrees : Exact Value Is √3 Explained

The value of cot 570 degrees is exactly √3, about 1.7321, because 570^° is coterminal with 210^°, whose reference angle is 30^° and whose quadrant keeps cotangent positive. This article shows the coterminal reduction, the reference-angle logic, a reference table, worked examples, and the mistakes students make.

Cot 0 Degrees : Why the Value Is Undefined

The value of cot 0 degrees is undefined, because cotθ = (cosθ)/(sinθ) and at 0^° that becomes 1/0, a division by zero. This article explains why it is undefined rather than zero or a clean infinity, shows the unit-circle picture, and covers a reference table, worked examples, and the mistakes students make.

Cos Pi/6 : Exact Value √3/2 and How to Find It

The value of cos pi/6 is exactly (√3)/2, about 0.8660. This article reads (π)/6 as a radian angle, proves the value from the 30-60-90 triangle and the unit circle, gives a radian reference table, and works through examples and the mistakes students make.

Cos Pi/4 : Exact Value √2/2 and How to Find It

The value of cos pi/4 is exactly (√2)/2, about 0.7071. This article reads (π)/4 as a radian angle, proves the value from the 45-45-90 triangle and the unit circle, gives a radian reference table, and works through examples and common mistakes.

Cos pi/3 : Exact Value, 1/2, and the Unit Circle

The value of cos pi/3 is exactly 1/2, or 0.5. This article reads that value straight off the unit circle, explains what the radian (π)/3 means, links its degree twin cos 60^°, and works through examples and the mistakes students make with radian angles.

Cos pi/2 : Exact Value: 0, and the Unit Circle

The value of cos pi/2 is exactly 0. This article reads that value off the unit circle, where the radian (π)/2 lands on the point (0, 1) with a zero x-coordinate, links the degree twin cos 90^°, and works through examples and the mistakes students make with radian angles.

Cosecant Function: Graph, Domain, Range & Examples

The cosecant function is the reciprocal of sine, written csc x = 1/(sin x), so its graph is a series of U-shaped branches that never enter the band between -1 and 1. This article covers the definition, domain and range, the cosecant graph and its vertical asymptotes, period, key values, and six worked examples.

Cos A - Cos B : Difference-to-Product Formula, Proof, and Examples

The cos A − cos B formula rewrites a difference of two cosines as a product: cos A - cos B = -2sin(A + B)/2sin(A - B)/2. This article proves it from the cosine compound-angle identities, shows where the product form pays off, and works through six examples plus the mistakes students make.

Cos A Cos B : Product-to-Sum Formula, Proof, and Examples

The cos A cos B formula rewrites a product of two cosines as a sum: cos A cos B = 1/2[cos(A - B) + cos(A + B)]. This article proves it from the cosine addition and subtraction formulas, shows where it earns its keep, and works through six examples plus the mistakes students make.

90 Degrees : Exact Value: 0, and Why It Is Zero

The value of cos 90 degrees is exactly 0. This article shows why from the unit circle, where the 90^° point sits at (0, 1) with a zero x-coordinate, gives a standard-angle table in degrees and radians, links the radian twin cos((π)/2), and works through examples and mistakes.

Cos 60 Degrees : Exact Value, 1/2, and How to Find It

The value of cos 60 degrees is exactly 1/2, which is 0.5. This article proves that value from the 30-60-90 triangle and the unit circle, gives a standard-angle table in degrees and radians, then works through examples and the mistakes students make most.

Cos 50 Degrees : Value 0.6428 and How to Find It

The value of cos 50 degrees is approximately 0.6428. Because 50^° is not one of the standard angles, it has no clean radical form, so this article shows how to find it by calculator, why cos 50^° = sin 40^°, where it sits between the special angles, and the mistakes students make.

Cos 45 Degrees - Exact Value √2/2 and How to Find It

The value of cos 45 degrees is exactly (√2)/2, about 0.7071, and it equals sin 45^°. This article proves it from the 45-45-90 triangle and the unit circle, gives a standard-angle table in degrees and radians, and walks through examples and the mistakes students make.

Cos 40 Degrees : Value 0.766 and Why It Has No Simple Exact Form

The value of cos 40 degrees is approximately 0.7660, and unlike 30^° or 45^° it has no simple exact radical form. This article shows where cos 40^° sits among the special angles, why it resists a clean closed form, and how to work with its decimal value.

Cos 7pi/4 : Exact Value, root 2 over 2, on the Unit Circle

The value of cos 7pi/4 is exactly (√2)/2, about 0.7071, and it is positive. This article finds it with the reference angle (π)/4 in the fourth quadrant, reads it off the unit circle, gives a standard-angle table, and works through examples and mistakes.

Cos 5pi/6 - Exact Value -√3/2 and How to Find It

The value of cos 5pi/6 is exactly -(√3)/2, about -0.8660. The angle (5π)/6 lands in Quadrant II with a reference angle of (π)/6, where cosine is negative, so this article shows the reference-angle method, the unit circle proof, a standard-angle table, worked examples, and the common mistakes.

Cos 5pi/4 - Exact Value -√2/2 and How to Find It

The value of cos 5pi/4 is exactly -(√2)/2, about -0.7071. The angle (5π)/4 lands in Quadrant III with a reference angle of (π)/4, where cosine is negative, so this article shows the reference-angle method, the unit circle proof, a standard-angle table, worked examples, and the common mistakes.

Cos 5pi/3 - Exact Value 1/2 and How to Find It

The value of cos 5pi/3 is exactly 1/2, or 0.5. The angle (5π)/3 lands in Quadrant IV with a reference angle of (π)/3, where cosine is positive, so this article shows the reference-angle method, the unit circle proof, a standard-angle table, worked examples, and the common mistakes.

Cos 4pi - Exact Value, Why It Equals 1, and How to Find It

The value of cos 4pi is exactly 1, because 4π radians is two complete turns around the unit circle and lands back at the starting point (1, 0). This article shows the periodicity proof, the coterminal-angle logic, a standard-angle table, worked examples, and the mistakes students make.

Cos3x Formula : Derivation, Graph & Examples

The cos3x formula is cos 3x = 4cos³ x - 3cos x, a triple-angle identity built from the angle-sum and double-angle rules. This article covers its derivation, its graph (period (2π)/3, range [-1, 1]), how it differs from cos³ x, six worked examples, and the errors students make most.

Cos 3pi/4 - Exact Value -√2/2 and the Reference Angle

The value of cos 3pi/4 is exactly -(√2)/2, about -0.7071. This article shows how the reference angle of (π)/4 and the Quadrant II sign combine to give the answer, with a standard-angle table, worked examples, and the mistakes students make.

Cos 3pi/2 - Value 0, Unit Circle, and How to Find It

The value of cos 3pi/2 is exactly 0. This article shows why the angle (3π)/2 lands at the bottom of the unit circle at (0, -1), so its x-coordinate is zero, and works through a reference table, examples, and the errors students make at 270^°.

Cos 3pi - Value, Unit Circle Proof, and Why It Equals -1

The value of cos 3pi is exactly -1. This article shows why cos 3π = cos π using the 2π periodicity of cosine, proves it on the unit circle, and works through a reference table, examples, and the errors students make with large radian angles.

Trigonometric Ratios of Specific Angles — Table & Values

The trigonometric ratios of specific angles are the exact values of sin, cos, tan, csc, sec, and cot at 0°, 30°, 45°, 60°, and 90° — five angles whose ratios come out as clean surds, not decimals. This article gives the full values table, shows where each number comes from (the 30-60-90 and 45-45-90 triangles plus the unit circle), explains why entries like tan 90° are undefined, and works through six examples.

Trigonometric Ratios of Complementary Angles

Two angles are complementary when they add to 90°, and the trigonometric ratio of an angle equals the co-ratio of its complement — so sin(90° - θ) = cosθ, tan(90° - θ) = cotθ, and sec(90° - θ) = cscθ. This article gives all six complementary-angle identities, proves them from a right triangle, explains why the "co-" in cosine means complement, and works through six examples — including the classic tan 1° · tan 2° … tan 89° problem.

Trigonometric Ratios in Radians - Values & Examples

Trigonometric ratios in radians are the same six ratios — sin, cos, tan, csc, sec, cot — measured with the angle written in radians instead of degrees, where a full circle is 2π radians rather than 360°. This article gives the standard-angle table (0, (π)/6, (π)/4, (π)/3, (π)/2), how to convert, why radians exist, six worked examples, and the mistakes that cost marks.

Trigonometric Equations - General Solution & Examples

A trigonometric equation is solved by finding every angle that satisfies it, written as a general solution like x = nπ + (-1)ⁿ,α. This article gives the general-solution formulas for sin, cos, and tan, the difference between principal and general solutions, six worked examples, and the sign and periodicity errors that trip students up.

Trigonometric Chart - Values Table & How to Remember

A trigonometric chart is a reference table listing the values of the six trigonometric ratios — sin, cos, tan, csc, sec, cot — at the standard angles 0°, 30°, 45°, 60°, 90°. This article gives the full chart in degrees and radians, the square-root pattern that lets you rebuild it from memory, the signs across all four quadrants, and six worked examples.

Tan 270 Degrees — Undefined, and Why

Tan 270 degrees is undefined, because tangent is sine over cosine and cos 270° = 0 — and dividing by zero has no value. This article shows the reason on the unit circle, the sine-over-cosine method, and what "undefined" means as opposed to zero.

Tan 55 Degrees — Value of tan(55°) and How to Find It

The value of tan 55 degrees is approximately 1.4281 — it is not a special-angle exact value, so there is no clean surd for it. This article shows how to find tan 55° honestly (calculator, the cofunction cot 35°, and table interpolation), gives the radian form, and explains why the value is greater than 1.

Tan 12 Degrees — Value of tan(12°) and How to Find It

The value of tan 12 degrees is approximately 0.2126 — it is not a special-angle exact value, so there is no clean surd for it. This article shows how to find tan 12° honestly (calculator, sine over cosine, the cofunction cot 78°, and interpolation), gives the radian form, and places it on the unit circle.

Tan 5pi/6 — Exact Value, Unit Circle, Methods

Tan 5pi/6 is −1/√3, which rationalises to −√3/3 (about −0.5774), because 5π/6 lands at 150° in the second quadrant where tangent is negative. This article finds the value through the degree conversion, the π/6 reference angle, and the sine-over-cosine quotient.

Tan3x - Formula, Proof & Examples (Triple Angle)

The tan3x formula is tan 3x = (3tan x - tan³ x)/(1 - 3tan² x), the triple-angle identity for tangent. This article derives it from the angle-addition and double-angle formulas, gives its derivative (3sec² 3x) and integration, works through examples, and separates tan3x (triple angle) from tan³x (tangent cubed).

Sin (a - b) Formula - Proof & Examples

The sin(a − b) formula is sin(a - b) = sin a cos b - cos a sin b, the difference identity for sine. This article proves it, shows how to use it to find values like sin 15°, and clears up the single biggest trap: sin(a - b) is not sin a - sin b.

Sin 75 Degrees - Exact Value (√6+√2)/4 Explained

The value of sin 75 degrees is exactly (√6+√2)/4, about 0.9659. This article derives it by writing 75° as 45° + 30°, shows why sin 75° = cos 15°, places the angle on the unit circle, and works through examples and common mistake.

Sin 210 Degrees - Exact Value −1/2 Explained

The value of sin 210 degrees is exactly -1/2, or −0.5. This article finds it with the reference angle of 30°, explains why the sign is negative (210° lies in Quadrant III), gives a standard-angle table, and works through examples and common mistakes.

Sin 240 Degrees - Exact Value −√3/2 Explained

The value of sin 240 degrees is exactly -(√3)/2, about −0.8660. This article finds it with the reference angle of 60°, explains why the sign is negative (240° lies in Quadrant III), gives a standard-angle table, and works through examples and common mistakes.

Sec 7pi/4 — Exact Value, Unit Circle, Methods

Sec 7pi/4 is √2 (about 1.4142), because secant is the reciprocal of cosine and cos(7π)/4 = (√2)/2 at 315° in the fourth quadrant, where cosine is positive. This article finds the value through the reciprocal identity, the degree conversion, and the π/4 reference angle.

Sin 15 Degrees - Exact Value (√6−√2)/4 Explained

The value of sin 15 degrees is exactly (√6-√2)/4, about 0.2588. This article derives it by writing 15° as 45° − 30°, places the angle on the unit circle, gives a standard-angle table, and works through examples and the mistakes that cost marks.

Sin 5pi/4 — Exact Value, Unit Circle, Methods

Sin 5pi/4 is −√2/2 (about −0.7071), because the angle 5π/4 lands at 225° in the third quadrant where sine is negative. This article shows the value through the unit circle, the radian-to-degree conversion, and the π/4 reference-angle method.

Principal Value of Trigonometric Functions - Explained

The principal value of a trigonometric function is the single, agreed-on angle returned by its inverse — chosen from one restricted interval so the answer is unique even though infinitely many angles share the same sine, cosine, or tangent. This article defines the principal value, lists the principal-value branch for each inverse function, shows the quadrant method to find it, and works six examples.

Inverse Trigonometric Ratios - Definition & Examples

Inverse trigonometric ratios run the ordinary ratios backwards: you give them a ratio of sides and they return the angle that produced it. Written sin⁻¹, cos⁻¹, tan⁻¹ (or arcsin, arccos, arctan), they answer "what angle has this sine?" This article defines all six, gives their domain and range, untangles the inverse-versus-reciprocal trap, and works six examples.

Derivative of Tan 2x - Formula, Proof, and Examples

The derivative of tan 2x is d/dx(tan 2x) = 2sec² 2x. This article proves that result three ways — chain rule, first principle, and quotient rule — works through examples, and clears up the most common confusion: tan 2x (double angle) versus tan² x (tangent squared), which have different derivatives.

Cot 7pi/4 — Exact Value, Unit Circle, Methods

Cot 7pi/4 is −1, because 7π/4 lands at 315° in the fourth quadrant where cotangent is negative and the reference angle is π/4. This article finds the value through the degree conversion, the reference angle, and the cosine-over-sine quotient.

Basic Properties of Trigonometric Ratios With Examples

The basic properties of trigonometric ratios are the fixed relationships that tie the six ratios together — the reciprocal relations, the quotient relations, the Pythagorean identities, the sign each ratio takes in each quadrant, and the range of values each can hold. This article explains all five property groups, why sinθ never exceeds 1 while tanθ runs unbounded, and works through six examples.

Applications of Trigonometry - Real-Life Uses & Examples

The applications of trigonometry are the real-world problems solved using sine, cosine, and tangent — chiefly finding unknown heights, distances, and angles you cannot measure directly. This article covers heights and distances, navigation, astronomy, engineering, sound and light waves, plus the angle-of-elevation method, six worked examples, and the mistakes that wreck a setup.

Conversion Relations of Trigonometric Ratios — Table

Conversion relations let you write any one of the six trigonometric ratios in terms of any other — for example, expressing sinθ, secθ, and tanθ all in terms of cotθ. The method chains three engines: the reciprocal relations, the quotient relations, and the Pythagorean identities. This article gives the full conversion table, the step-by-step method, the sign caveat, and six worked examples — including the standard textbook questions.

Cos 4 Degrees — Value of cos(4°) and How to Find It

The value of cos 4 degrees is approximately 0.9976 — it is not a special-angle exact value, so there is no clean surd for it. This article shows how to find cos 4° honestly (calculator and the small-angle approximation, with its accuracy bound), gives the radian form, and places it on the unit circle.

Cos 15 Degrees - Exact Value (√6+√2)/4 Explained

The value of cos 15 degrees is exactly (√6+√2)/4, about 0.9659. This article derives it by writing 15° as 45° − 30°, locates the angle on the unit circle, gives a standard-angle table, and works through examples and the mistakes that catch students out.

Cosine Function - Graph, Properties & Examples

The cosine function gives the x-coordinate of a point on the unit circle, or the ratio (adjacent)/(hypotenuse) in a right triangle. This article covers the definition, the cosine graph (period 2π, range [-1, 1]), why cosine is an even function, its quadrant signs, key values, and worked examples.

Cos 65 Degrees — Value of cos(65°) and How to Find It

The value of cos 65 degrees is approximately 0.4226 — it is not a special-angle exact value, but it equals sin 25° by the cofunction identity. This article shows how to find cos 65° honestly (calculator, cofunction, and interpolation), gives the radian form, and places it on the unit circle.

Heights and Distances — Trigonometry Formulas & Examples

Heights and distances is the branch of trigonometry that finds the height of an object or the distance to it using a measured angle and one known length — without ever climbing or pacing it out. This article covers the angle of elevation and angle of depression, the line-of-sight idea, the three-step method, the formulas, and six worked examples on towers, poles, and buildings.

Cos 2pi/3 = −1/2 — Value of cos(2π/3) on Unit Circle

The value of cos 2pi/3 is exactly -1/2, which is -0.5. In degrees, (2π)/3 is 120°, an angle in the second quadrant where cosine is negative. This article shows why cos((2π)/3) = -1/2 on the unit circle, gives a standard-angle reference table in radians and degrees, and clears up the sign slip students hit most.

Cos 2pi = 1 — Value of cos(2π) on the Unit Circle

The value of cos 2pi is exactly 1. A full rotation of 2π radians (that is 360°) returns to the starting point, so cos(2π) = cos 0 = 1. This article shows why one complete trip around the unit circle brings cosine back to 1, gives a standard-angle reference table in radians and degrees, and clears up the slips students hit most.

Cos 270 Degrees = 0 — Value, Unit Circle, Radians

The value of cos 270 degrees is exactly 0. In radians, 270° is (3π)/2, so cos(270°) = cos((3π)/2) = 0. This article shows why three-quarters of a turn lands on the bottom of the unit circle where the x-coordinate vanishes, gives a standard-angle reference table in degrees and radians, and clears up the common mix-ups.

Cos 180 Degrees = −1 — Value, Unit Circle, Radians

The value of cos 180 degrees is exactly -1. In radians, 180° is π, so cos(180°) = cosπ = -1. This article shows why that single half-turn lands on the leftmost point of the unit circle, gives a standard-angle reference table in degrees and radians, and clears up the slips students hit most.

Cos 135 Degrees - Value −√2/2 Explained

The value of cos 135 degrees is exactly -(√2)/2, about -0.7071. This article shows why the value is negative (135° sits in Quadrant II), how the reference angle of 45° supplies the magnitude, a standard-angle table in degrees and radians, plus worked examples and the common mistakes.

Cos 120 Degrees - Value −1/2 Explained (2026)

The value of cos 120 degrees is exactly -1/2, or -0.5. This article explains why the value is negative (120° sits in Quadrant II), how the reference angle of 60° gives the magnitude, a standard-angle table in degrees and radians, plus worked examples and common mistakes.

Cos 35 Degrees - Value 0.8192 Explained

The value of cos 35 degrees is approximately 0.8192. This article explains why 35° is a non-standard angle with no simple radical form, how to find its value using the unit circle, the reference angle, and a calculator, with a reference table in degrees and radians plus worked examples.

Cos 30 Degrees - Value √3/2 Explained

The value of cos 30 degrees is exactly (√3)/2, which is about 0.8660. This article shows where that value comes from using the 30-60-90 triangle and the unit circle, gives a standard-angle reference table in both degrees and radians, and walks through worked examples and the mistakes students make.

Cos 25 Degrees - Value 0.9063 Explained (2026)

The value of cos 25 degrees is approximately 0.9063. This article explains why 25° is a non-standard angle with no simple radical form, how to find its value using the unit circle, the reference angle, and a calculator, with a reference table in degrees and radians plus worked examples.

Cos 20 Degrees — Value of cos(20°) and How to Find It

The value of cos 20 degrees is approximately 0.9397 — and 20° is not a special angle, so there is no clean exact surd for it. This article shows how to find cos 20° honestly, gives the decimal and radian form, and places it among the standard angles.

Cos 2 Degrees — Value of cos(2°) and How to Find It

The value of cos 2 degrees is approximately 0.9994 — and 2° is not a special angle, so there is no clean exact surd for it. This article shows how to find cos 2° honestly (calculator and small-angle approximation), gives the decimal and radian form, and places it among the standard angles.

Cos 1 Degree — Value of cos(1°) and How to Find It

The value of cos 1 degree is approximately 0.9998 — it is not a special-angle exact value, so there is no clean surd for it. This article shows how to find cos 1° honestly (calculator and small-angle approximation), gives the decimal and radian form, and places it among the standard angles.

Cos 0 Degrees — Value of cos(0°) with the Unit Circle

The value of cos 0 degrees is exactly 1 (1.0000 as a decimal). This article shows why with both the unit circle and the right triangle, gives a full standard-angle cosine table in degrees and radians, and clears up the mistakes that trip students on cos 0°.

Arctan 0 — Value in Degrees and Radians

Arctan 0 equals 0°, or 0 radians — the angle in (-(π)/2, (π)/2) whose tangent is 0. This article gives the value in both units, the unit-circle reason it is 0 and not π, a tan-inverse reference table, two worked methods, the mistakes around the restricted range, and FAQs.

Arcsin 1 — Value in Degrees and Radians

Arcsin 1 equals 90°, or (π)/2 radians — the angle in [-(π)/2, (π)/2] whose sine is exactly 1. This article gives the exact value in both units, the unit-circle reason it has to be 90°, a sin-inverse reference table, two worked methods, the mistakes around the restricted range, and FAQs.

Cos(A - B) Formula — Proof, Examples, Identity

The cos(A - B) formula states that cos(A - B) = cos A cos B + sin A sin B, the cosine difference identity. This article gives the formula, its unit-circle proof, why the sign is a plus (the opposite of cos(A+B)), six worked examples in degrees and radians, the most common sign mistake, and FAQs.

2 Sin A Cos A Formula — Sin 2A Proof, Examples

The 2 sin A cos A formula states that 2sin A cos A = sin 2A, the double-angle identity for sine. This article gives the formula, its proof from the sine sum formula, its tangent form, six worked examples in degrees and radians, the unit-circle picture, the most common sign and "sin 2A = 2sin A" mistakes, and FAQs.

1 Radian to Degrees — Value, Formula, Examples

1 radian to degrees equals 180/(π) ≈ 57.2958° — the fixed angle you get when an arc length equals the circle's radius. This article gives the exact and decimal value, the conversion formula, a radian–degree reference table, two worked methods in degrees and radians, the mistakes that flip the formula, and FAQs.

Arctan 2 — Value, Radians, Degrees, Worked Examples

Arctan 2 — the angle whose tangent equals 2 — is approximately 1.1071 radians or 63.435°. This article covers the exact-value status (irrational, non-terminating), three computation methods (calculator, series expansion, right-triangle reading), the unit-circle position, the common composition mistakes, and a quick reference for arctan at nearby integer inputs.

Trigonometric Identities — Formulas, Proofs, Examples

Trigonometric identities are equations involving sine, cosine, tangent and their reciprocals that hold for every angle in their domain — the algebraic glue between the six trig functions. This article covers the eight identity families (reciprocal, quotient, Pythagorean, co-function, even-odd, sum-difference, double-angle, half-angle, product-to-sum), the unit-circle proof behind each, three worked examples in both degrees and radians, and the sign-flip mistakes that cost the most marks.

Sum to Product Formulas — Trig Identities, Proof

The sum to product formula family converts a sum or difference of two sines (or two cosines) into a product of one sine and one cosine — four identities that turn sin 75° + sin 15° into a single product expression solvable in one step. This article gives the four formulas, the proof via sum-and-difference identities, three worked examples in degrees and radians, and the common mistake of mixing up the half-sum and half-difference angles.

Secant Function — Formula, Graph, Properties, Examples

The secant function secθ = 1/cosθ is the reciprocal of cosine — defined wherever cosine is non-zero, with vertical asymptotes at θ = (2n+1)π/2 and range (-∞, -1] ∪ [1, ∞). This article gives the formula, the graph paired with cosine, the table of values at special angles, the even-function symmetry, three worked examples in degrees and radians, and the common mistakes.

Reciprocal Identities — Formulas, Proof, Examples

The reciprocal identities of trigonometry are three pairings — sine with cosecant, cosine with secant, tangent with cotangent — that say each trig function equals 1 divided by its reciprocal partner. This article gives the three identities, the unit-circle proof, the related Pythagorean-style identities 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ, three worked examples in degrees and radians, and the common mistakes around domain restrictions.

Cofunction Identities — Formula, Proof, Examples

The cofunction identities state that any trig function of θ equals the corresponding co-function of the complementary angle π/2 - θ — six pairings that turn sin(60°) into cos(30°) without computation. This article gives the six identities, the right-triangle and unit-circle proof, three worked examples in degrees and radians, the application to simplifying expressions, and the common mistakes around the "co" prefix.

Sum and Difference Formulas — Sin, Cos, Tan

The sum and difference formulas of trigonometry give the sine, cosine, and tangent of A ± B in terms of the trig functions of A and B separately — six identities that let you compute exact values for non-standard angles like 15° or 75°. This article gives the six formulas, the unit-circle proof, three worked examples in degrees and radians, the sign-flip mnemonic for cosine, and the common mistakes around tangent's denominator.

Domain and Range of Trigonometric Functions

The domain and range of trigonometric functions describes which angles each function accepts and which output values it produces — sine and cosine accept all real angles and output values in [-1, 1], while tangent, cotangent, secant, and cosecant have angles where they are undefined. This article gives the full domain–range table, the graph of each function in degrees and radians, the unit-circle anchor for each definition, three worked examples, and the most common mistakes students make.

Derivative of Arccos x — Formula, Proof, Examples

The derivative of arccos x is d/dx x = -1/(√(1-x²)) on the open interval (-1, 1) — a negative quantity that reflects the fact arccosine is a strictly decreasing function. This article gives the implicit-differentiation derivation, the first-principles approach, the chain-rule version d/dx(u) = -(u')/(√(1-u²)), three worked examples, and the common mistakes around sign and domain.

Angle of Depression — Definition, Formula, Examples

The angle of depression is the angle measured downward from a horizontal line at the observer's eye to the line of sight pointing at an object below. Its formula is tanθ = h/d, where h is the vertical drop and d is the horizontal distance — this article gives the definition, the alternate-interior-angle link to the angle of elevation, three worked examples in degrees and radians, and the common mistakes pilots and students both run into.

Arccosine — Definition, Graph, Examples, Identities

Arccosine — written x or cos⁻¹ x — is the inverse of cosine restricted to [0, π]; it takes an input in [-1, 1] and returns the unique angle in [0, π] whose cosine equals the input. This article covers the definition, the principal-value branch, the graph, the derivative and integral, three worked examples in both degrees and radians, the identity sin⁻¹ x + cos⁻¹ x = π/2, and the common mistakes around restricted-domain reasoning.

Angle of Elevation — Formula, Diagram, Examples

The angle of elevation is the upward angle between a horizontal line at the observer's eye and the line of sight to an object above. Its formula is θ = tan⁻¹(height / distance) — this article gives the definition, the right-triangle and unit-circle anchors, three worked examples in both degrees and radians, the common mistakes, and where surveyors and astronomers use it daily.

Arcsin — Formula, Graph, Domain and Range

Arcsin (written sin⁻¹ x or x) is the inverse sine function. It takes a number in [-1, 1] and returns the angle in [-π/2, π/2] whose sine equals that number. Its graph is a smooth, strictly-increasing S-curve passing through the origin, with endpoints (-1, -π/2) and (1, π/2).

Sin A + Sin B Formula — Proof and Examples

The sin A + sin B formula is the sum-to-product identity: sin A + sin B = 2 sin!((A+B)/2) cos!((A-B)/2) It converts the sum of two sines into the product of a sine and a cosine. The proof uses the angle-sum identities sin(α + β) = sinαcosβ + cosαsinβ — adding two of them and substituting α + β = A, α - β = B collapses the algebra into the product form.

Trigonometric Ratios — Definition, Formulas, Examples

Trigonometric ratios are ratios of side lengths in a right triangle, indexed by one of its acute angles. The three primary ratios — sine, cosine, tangent — give the ratios of opposite/hypotenuse, adjacent/hypotenuse, and opposite/adjacent.

Inverse Trigonometric Functions — Formulas, Domain, Range

The inverse trigonometric functions —,,,,, — undo the standard trig functions. Each takes a ratio and returns an angle. The trick is that sine, cosine, and tangent each map many angles to the same ratio, so their inverses only work on restricted "principal" intervals: on [-π/2, π/2], on [0, π], on (-π/2, π/2).

Cos pi - Find the Value of cos(π) and Why It Equals −1

The value of cos pi is -1. In radians, π corresponds to 180° — the angle that points along the negative x-axis on the unit circle. The x-coordinate of that point is -1, and since cosine reads the x-coordinate on the unit circle, cosπ = -1.

Differentiation of Trigonometric Functions — Formulas & Rules

The differentiation of trigonometric functions gives the six core rules: d/dxsin x = cos x, d/dxcos x = -sin x, d/dxtan x = sec² x, d/dxcot x = -csc² x, d/dxsec x = sec x tan x, and d/dxcsc x = -csc x cot x. All six follow from the sine and cosine derivatives via the quotient rule. This article proves them from first principles and shows where students slip.

Arctan — Formula, Graph, Identities, Domain and Range

Arctan is the inverse tangent — it takes a real number and returns the angle whose tangent is that number. Its domain is every real number (-∞, ∞), its range is the open interval (-(π)/2, (π)/2), and its graph is a smooth S-curve with horizontal asymptotes at y = ±(π)/2.

Trigonometry — Complete Guide to Formulas & Identities

This trigonometry complete guide covers the six trig functions (sin, cos, tan, csc, sec, cot), the unit circle, the four families of identities (Pythagorean, reciprocal, sum-and-difference, double-angle), the six inverse trig functions and the standard derivatives.

Radian - Definition, Formula, Conversion

A radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. By definition, θ = s/r (arc length over radius). A full circle is 2π radians, so 360° = 2π rad, 180° = π rad, and 1 rad ≈ 57.296°.

Cos2x - Formula, Identity, Examples, Proof

The **cos2x identity** is the double-angle formula for cosine, with three equivalent forms: cos 2x = cos² x - sin² x = 2cos² x - 1 = 1 - 2sin² x The proof from the angle-sum identity, when to use each form, the related cos 2x in terms of tan x, worked examples, and the most common student mistakes.

Trigonometry Formulas - Full list

The complete list of **trigonometry formulas** covers seven categories: basic ratios (sin, cos, tan), reciprocal identities (csc, sec, cot), Pythagorean identities (sin² + cos² = 1), angle-sum and angle-difference formulas, double-angle formulas, half-angle formulas, and sum-to-product formulas.

Sin Cos Tan - Trigonometric Ratios and Formulas

Trigonometric Table - Sin Cos Tan Values 0-90°

The trigonometric table gives the values of sine, cosine, tangent, cosecant, secant, and cotangent at the five standard angles: 0°, 30°, 45°, 60°, and 90°. The values come from two special right triangles - the 30-60-90 and the 45-45-90.