# 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} $ ## 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 ![[Angle Sum Identities of Trigonometric Functions#^64yfg7]] 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 $\cos^2 \alpha = \frac 1 2 (1 + \cos (2\alpha))$ $\sin^2 \alpha = \frac 1 2 (1 - \cos (2 \alpha))$ $\tan^2 \alpha = \frac {1- \cos(2\alpha)} {1+ \cos(2\alpha)}$ ## Derivatives ![[Derivatives of Sine and Cosine#^qei2uq]] ![[Derivatives of Tangent and Cotangent#^0ytid3]] ![[Derivatives of Secant and Cosecant#^fc25n3]] ![[Derivatives of Inverse Trigonometric Functions#^3wzecb]] ## Antiderivatives 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 ## Expressing Trigonometric Functions Using The Exponential Function ![[Expressing Trigonometric Functions Using The Exponential Function#^s1jedx]] ## The Unit Circle With Values ![[edeea030d1fe459b2773bd33f79733712c40e136.png|400]]