Trigonometry

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

The value of cos 2pi/3 is exactly $-\frac{1}{2}$, which is $-0.5$. In degrees, $\frac{2\pi}{3}$ is $120°$, an angle in the second quadrant where cosine is negative. This article shows why $\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}$ 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 $2\pi$ radians (that is $360°$) returns to the starting point, so $\cos(2\pi) = \cos 0 = 1$. 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 $\frac{3\pi}{2}$, so $\cos(270°) = \cos\left(\frac{3\pi}{2}\right) = 0$. 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 $\pi$, so $\cos(180°) = \cos\pi = -1$. 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 $-\frac{\sqrt{2}}{2}$, 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 $-\frac{1}{2}$, 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 $\frac{\sqrt{3}}{2}$, 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 $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ whose tangent is $0$. This article gives the value in both units, the unit-circle reason it is $0$ and not $\pi$, 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 $\frac{\pi}{2}$ radians — the angle in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ 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 $2\sin A \cos A = \sin 2A$, 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 "$\sin 2A = 2\sin A$" mistakes, and FAQs.

1 Radian to Degrees — Value, Formula, Examples

1 radian to degrees equals $\frac{180}{\pi} \approx 57.2958°$ — 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 $\sec\theta = 1/\cos\theta$ is the reciprocal of cosine — defined wherever cosine is non-zero, with vertical asymptotes at $\theta = (2n+1)\pi/2$ and range $(-\infty, -1] \cup [1, \infty)$. 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 $1 + \tan^2\theta = \sec^2\theta$ and $1 + \cot^2\theta = \csc^2\theta$, 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 $\theta$ equals the corresponding co-function of the complementary angle $\pi/2 - \theta$ — 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 $A \pm B$ 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 $\dfrac{d}{dx}\arccos x = -\dfrac{1}{\sqrt{1-x^2}}$ 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 $\dfrac{d}{dx}\arccos(u) = -\dfrac{u'}{\sqrt{1-u^2}}$, 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 $\tan\theta = h/d$, 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 $\arccos x$ or $\cos^{-1} x$ — is the inverse of cosine restricted to $[0, \pi]$; it takes an input in $[-1, 1]$ and returns the unique angle in $[0, \pi]$ 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 $\sin^{-1} x + \cos^{-1} x = \pi/2$, 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 $\theta = \tan^{-1}(\text{height} / \text{distance})$ — 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 $\arcsin x$) is the inverse sine function. It takes a number in $[-1, 1]$ and returns the angle in $[-\pi/2, \pi/2]$ whose sine equals that number. Its graph is a smooth, strictly-increasing S-curve passing through the origin, with endpoints $(-1, -\pi/2)$ and $(1, \pi/2)$.

Sin A + Sin B Formula — Proof and Examples

The sin A + sin B formula is the sum-to-product identity: $$\sin A + \sin B = 2 \sin!\left(\frac{A+B}{2}\right) \cos!\left(\frac{A-B}{2}\right)$$ It converts the sum of two sines into the product of a sine and a cosine. The proof uses the angle-sum identities $\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$ — adding two of them and substituting $\alpha + \beta = A$, $\alpha - \beta = B$ 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 — $\arcsin$, $\arccos$, $\arctan$, $\arccsc$, $\arcsec$, $\arccot$ — 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: $\arcsin$ on $[-\pi/2, \pi/2]$, $\arccos$ on $[0, \pi]$, $\arctan$ on $(-\pi/2, \pi/2)$.

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

The value of cos pi is $-1$. In radians, $\pi$ 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, $\cos\pi = -1$.

Differentiation of Trigonometric Functions — Formulas & Rules

The differentiation of trigonometric functions gives the six core rules: $\frac{d}{dx}\sin x = \cos x$, $\frac{d}{dx}\cos x = -\sin x$, $\frac{d}{dx}\tan x = \sec^2 x$, $\frac{d}{dx}\cot x = -\csc^2 x$, $\frac{d}{dx}\sec x = \sec x \tan x$, and $\frac{d}{dx}\csc x = -\csc x \cot x$. 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 $(-\infty, \infty)$, its range is the open interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, and its graph is a smooth S-curve with horizontal asymptotes at $y = \pm\frac{\pi}{2}$.

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, $\theta = s/r$ (arc length over radius). A full circle is $2\pi$ radians, so $360° = 2\pi$ rad, $180° = \pi$ rad, and $1\text{ rad} \approx 57.296°$.

Cos2x - Formula, Identity, Examples, Proof

The **cos2x identity** is the double-angle formula for cosine, with three equivalent forms: $$\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$$ 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 ($\sin^2 + \cos^2 = 1$), 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.