Chapter 4.2, Problem 56E

To determine

To prove: That the cosine and secant functions are even and that the sine, cosecant, tangent and cotangent functions are odd using unit circle.

Answer to Problem 56E

Explanation of Solution

Used Formula:

A function $f\left(x\right)$ is even if $f\left(-x\right)=f\left(x\right)$ and odd if $f\left(-x\right)=-f\left(x\right)$

The unit circle can be drawn as,

  

Form the diagram,

  $\cos \left(\theta \right)=x$ and $\cos \left(-\theta \right)=x$ this implies that $\cos \left(\theta \right)=\cos \left(-\theta \right)$

Hence, cosine function is even function.

Similarly, $\sec \left(\theta \right)=\frac{1}{\cos \left(\theta \right)}=\frac{1}{x}$ and $\sec \left(-\theta \right)=\frac{1}{\cos \left(-\theta \right)}=\frac{1}{x}$

Then, secant function is also even function.

Now, $\sin \theta =y$ and $\sin \left(-\theta \right)=-y$ this implies that $\sin \left(-\theta \right)=-\sin \left(\theta \right)$

Hence, sine function is an odd function.

Since, $\csc \theta =\frac{1}{\sin \theta }=\frac{1}{y}$ and $\csc \left(-\theta \right)=\frac{1}{\sin \left(-\theta \right)}=-\frac{1}{y}$ this implies that $\csc \left(-\theta \right)=-\csc \left(\theta \right)$

Hence, cosecant function is an odd function.

Now, $\tan \theta =\frac{y}{x}$ and $\tan \left(-\theta \right)=-\frac{y}{x}$ this implies that $\tan \left(-\theta \right)=-\tan \theta$

Hence, tangent function is an odd function.

Since, $\cot \theta =\frac{x}{y}$ and $\cot \left(-\theta \right)=-\frac{x}{y}$ this implies that $\cot \left(-\theta \right)=-\cot \theta$

Hence, cotangent function is an odd function.