### SubsectionB.2.2Trig Function Definitions

The trigonometric functions are defined as ratios of the lengths of the sides of a right-angle triangle as shown in the left of the diagram below . These ratios depend only on the angle $\theta\text{.}$

The trigonometric functions sine, cosine and tangent are defined as ratios of the lengths of the sides

\begin{align*} \sin\theta &= \frac{\text{opposite}}{\text{hypotenuse}} & \cos\theta &= \frac{\text{adjacent}}{\text{hypotenuse}} & \tan\theta &= \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin \theta}{\cos \theta}.\\ \end{align*}

These are frequently abbreviated as

\begin{align*} \sin\theta &= \frac{\text{o}}{\text{h}} & \cos\theta &= \frac{\text{a}}{\text{h}} & \tan\theta &= \frac{\text{o}}{\text{a}} \end{align*}

which gives rise to the mnemonic

\begin{align*} \text{SOH} && \text{CAH} && \text{TOA} \end{align*}

If we scale the triangle so that they hypotenuse has length $1$ then we obtain the diagram on the right. In that case, $\sin \theta$ is the height of the triangle, $\cos \theta$ the length of its base and $\tan \theta$ is the length of the line tangent to the circle of radius 1 as shown.

Since the angle $2\pi$ sweeps out a full circle, the angles $\theta$ and $\theta+2\pi$ are really the same.

Hence all the trigonometric functions are periodic with period $2\pi\text{.}$ That is

\begin{align*} \sin(\theta+2\pi) &= \sin(\theta) & \cos(\theta+2\pi) &= \cos(\theta) & \tan(\theta+2\pi) &= \tan(\theta) \end{align*}

The plots of these functions are shown below

\begin{equation*} \sin \theta \end{equation*}
\begin{equation*} \cos \theta \end{equation*}
\begin{equation*} \tan \theta \end{equation*}

The reciprocals (cosecant, secant and cotangent) of these functions also play important roles in trigonometry and calculus:

\begin{align*} \csc \theta &= \frac{1}{\sin\theta} = \frac{\text{h}}{\text{o}} & \sec\theta &= \frac{1}{\cos \theta} = \frac{\text{h}}{\text{a}} & \cot\theta &= \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin\theta} = \frac{\text{a}}{\text{o}} \end{align*}

The plots of these functions are shown below

\begin{equation*} \csc \theta \end{equation*}
\begin{equation*} \sec \theta \end{equation*}
\begin{equation*} \cot \theta \end{equation*}

These reciprocal functions also have geometric interpretations:

Since these are all right-angled triangles we can use Pythagoras to obtain the following identities:

\begin{align*} \sin^2\theta + \cos^2 \theta &=1 & \tan^2\theta + 1 &= \sec^2\theta & 1 + \cot^2 \theta &=\csc^2\theta \end{align*}

Of these it is only necessary to remember the first

\begin{align*} \sin^2\theta + \cos^2 \theta &=1 \end{align*}

The second can then be obtained by dividing this by $\cos^2\theta$ and the third by dividing by $\sin^2\theta\text{.}$