# Trigonometric Functions **Trigonometric functions (or trig functions) output comparisons of lengths of sides of right triangles.** In other words, they are functions that, given an angle of a right angled triangle, output the ratio of two of its sides. ![[CFE29162-6C21-4E77-85CD-FD98372A076D.png|350]] The basic trigonometric functions are _sine_ and _cosine_, all other trigonometric functions can be derived from those two. Also, because of the [[#Pythagorean Identity]], one doesn't even need both sine and cosine, one is enough. By their nature, trigonometric functions are [[Periodic Functions|periodic]]. For a proof of the relationships in the image above see [[Geometric Definitions of Derived Trigonometric Functions]]. For applications of trigonometric functions on general triangles see [[The Law of Sines]] and [[The Law of Cosines]]. ## The 'Soh Cah Toa' Mnemonic The 'Soh Cah Toa' mnemonic is an easy way of remembering the definitions of the three most basic trigonometric functions. $ \text{{\color {red}s}in} \, \alpha = \frac {\text {{\color {red} o}pposite}} {\text {{\color {red}h}ypotenuse}} $ $ \text{{\color {red}c}os} \, \alpha = \frac {\text {{\color {red} a}djacent}} {\text {{\color {red}h}ypotenuse}} $ $ \text{{\color {red}t}an} \, \alpha = \frac {\text {{\color {red} o}pposite}} {\text {{\color {red} a}djacent}} = \frac {\sin \alpha}{\cos \alpha} $ ## Derivatives ![[Derivatives of Sine and Cosine#^qei2uq]] $ \begin{gather} \frac d {d\theta} \bigg [ \tan \theta \bigg] = \sec^2 \theta \\ \frac d {d\theta} \bigg [ \cot \theta \bigg] = -\csc^2 \theta \\ \frac d {dx} \bigg[\arcsin x\bigg] = \frac 1 {\sqrt {1-x^2}} \\ \frac d {dx} \bigg[\arccos x\bigg] = - \frac 1 {\sqrt {1-x^2}} \\ \frac d {dx} \bigg[ \arctan x\bigg] = \frac 1 {x^2+1} \end{gather} $ ^fc25n3 ### Proofs $ \begin{align} \frac d {d\theta} \bigg [ \tan \theta \bigg] &= \frac d {d\theta} \bigg [ \frac {\sin \theta} {\cos{\theta}}\bigg ] \\ &= \frac {\cos \theta \cos \theta + \sin \theta \sin \theta} {\cos^2 \theta} \\ &= \frac {\cos^2 \theta + 1-\cos^2\theta } {\cos^2 \theta} \\ &= \frac 1 {\cos^2 \theta} = \sec^2 \theta \end{align} $ $\cot$ is similar. For $\arcsin$ we can use implicit differentiation: Start by applying the [[The Derivative Chain Rule]] to both sides of $\arcsin(\sin \theta) = \theta$ $ \begin{align} \arcsin(\sin\theta) &= \theta \\ \frac {d (\arcsin (\sin \theta))}{d\theta} &= 1 \\ \frac {d (\arcsin (\sin \theta))}{d (\sin \theta)} \frac {d(\sin \theta)}{d\theta} &= 1 \\ \frac {d (\arcsin (\sin \theta))}{d (\sin \theta)} \cos \theta &= 1 \\ \frac {d (\arcsin (\sin \theta))}{d (\sin \theta)} &= \frac 1 {\cos \theta} \end{align} $ $\cos \theta$ can also be expressed in terms of $\sin \theta$: $\cos \theta = \sqrt {\cos^2 \theta} = \sqrt {1-\sin^2 \theta}$ $\frac {d (\arcsin (\sin \theta))}{d (\sin \theta)} = \frac 1 {\sqrt {1-\sin^2 \theta}}$ If we substitute $x = \sin \theta$ we then get $\frac {d (\arcsin x)}{dx} = \frac 1 {\sqrt {1 - x^2}}$ Proof for $\arccos x$ is similar. ## Common Antiderivatives of Trigonometric Functions For many more see [The Integral Table](https://www.integral-table.com/). $\int \sin (ax) dx = -\frac 1 a \cos(ax) + C$ $\int\cos(ax)dx = \frac 1 a \sin (ax) + C$ $\int \cos^2 x \, dx = \frac 1 2 x + \frac 1 4\sin(2x) + C$ $\int \sin^2x \, dx = \frac 1 2x - \frac 1 4 \sin (2x) + C$ ^12sj15 ## Reciprocal Functions Cosecant, secant, and cotangent are reciprocal functions of the three most basic trigonometric functions. They are defined as: $ \begin {align} \text {cosecant:}& \; \csc \alpha = \frac 1 {\sin \alpha} \\\\ \text {secant:}& \; \sec \alpha = \frac 1 {\cos \alpha} \\\\ \text {cotangent:}& \; \cot \alpha = \frac 1 {\tan \alpha} \end {align} $ ## Identities The following are true by definition of cosine and sine: - $\cos (\alpha) = \cos (-\alpha)$ - $-\sin (\alpha) = \sin(-\alpha)$ - $-\tan (\alpha) = \tan(-\alpha)$ - $-\cot(\alpha)=\cot (-\alpha)$ It can be said that the sine is an odd[^graphSymmetry] function because $-\sin \theta = \sin {-\theta}$, and that the cosine is an even function because $\cos \theta = \cos {-\theta}$. [^graphSymmetry]: For a definition of what odd and even functions are see [[Function (in Algebra)#Graph Symmetry]] ### Co-function Identities These come from looking at the definition of sine and cosine, and that in a right triangle the sum of two non-right angles is 90 degrees. - $\sin \alpha = \cos ({\frac \pi 2 - \alpha})$ - $\cos \alpha = \sin ({\frac \pi 2 - \alpha})$ - $\tan \alpha = \cot ({\frac \pi 2 - \alpha})$ - $\cot \alpha = \tan ({\frac \pi 2 - \alpha})$ - $\csc \alpha = \sec ({\frac \pi 2 - \alpha})$ - $\sec \alpha = \csc ({\frac \pi 2 - \alpha})$ ### Pythagorean Identity Comes from the application of the Pythagorean Theorem on the right triangle whose hypothenuse is the radius of the unit [[circle]]. $\sin^2 \alpha + \cos^2 \alpha = 1$ ^m8vwo0 If you take the above and divide both sides by $\sin^2 \alpha$, you get $ 1+\cot^2 \alpha = \csc \alpha $ And if you do the same with $\cos^2$ you get $ 1+\tan^2\alpha = \sec \alpha $ ### Angle Addition and Subtraction Identities $\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ $\sin (\alpha + \beta)=\sin \alpha \cos \beta +\cos \alpha \sin \beta$ $\tan (\alpha + \beta) = \frac {\tan \alpha + \tan \beta} {1- \tan \alpha \tan \beta}$ Angle subtraction identities are derived by substituting $\beta$ for $-\beta$: $\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$ $\sin (\alpha - \beta)=\sin \alpha \cos \beta-\cos \alpha \sin \beta$ $\tan (\alpha - \beta) = \frac {\tan \alpha - \tan \beta} {1+ \tan \alpha \tan \beta}$ ### Double Angle Identities $\cos (2\alpha) = 2\cos^2\alpha - 1$ $\cos (2\alpha) = \cos^2\alpha - \sin^2 \alpha$ $\cos (2\alpha) = 1-2\sin^2\alpha$ $\sin (2\alpha) = 2\cos \alpha \sin \alpha$ $\tan (2\alpha) = \frac {2\tan \alpha}{1- \tan^2 \alpha}$ ### Square or Half-Angle Identities $\cos^2 \alpha = \frac 1 2 (1 + \cos (2\alpha))$ $\sin^2 \alpha = \frac 1 2 (1 - \cos (2 \alpha))$ $\tan^2 \alpha = \frac {1- \cos(2\alpha)} {1+ \cos(2\alpha)}$ ## Expressing Trigonometric Functions Using The Exponential Function ![[Expressing Trigonometric Functions Using The Exponential Function#^s1jedx]] ## Proving Identities Using Geometry The angle addition identities can proven by analyzing the following two diagrams. The double angle and subtraction identities are derived from the addition identities (by replacing $\beta$ with $\alpha$ or $-\alpha$, see below). ![[216E731B-3AF1-4FE2-9661-B0A06C784028.png|400]] ![[82A0185D-CFF9-4C8B-9D13-DC07A51D3996.png|400]] If we substitute $\beta$ for $\alpha$, we get the double angle identities: $ \begin{align} \cos (\alpha + \beta) &= \cos \alpha \cos \beta - \sin \alpha \sin \beta \\ \cos (2\alpha) &=\cos^2 \alpha - \sin^2 \alpha \\ &= \cos^2 \alpha - (1- \cos ^2\alpha) && \text{(Pythagorean Identity)} \\ &= 2\cos^2 \alpha - 1 \end{align} $ $ \begin{align} \sin (\alpha + \beta) &= \sin \alpha \cos \beta +\cos \alpha \sin \beta \\ \sin (2\alpha) &=\sin \alpha \cos \alpha +\cos \alpha \sin \alpha \\ &= 2\cos \alpha\sin \alpha \end{align} $ ## The Unit Circle With Values ![[edeea030d1fe459b2773bd33f79733712c40e136.png|400]]