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Subsection B.2.2 Trig 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 \(\sin^2\theta\text{.}\)