Trigonometry

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.