# 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]]