9.1 Solving trigonometric equations with identities  (Page 2/9)

Recall that we determined which trigonometric functions are odd and which are even. The next set of fundamental identities is the set of even-odd identities. The even-odd identities    relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle. (See [link] ).

Recall that an odd function    is one in which $f(-x)=\; -f\left(x\right)$ for all $x$ in the domain of $f.$ The sine function is an odd function because $\sin \left(-\theta \right)=-\sin \theta .$ The graph of an odd function is symmetric about the origin. For example, consider corresponding inputs of $\frac{\pi }{2}$ and $-\frac{\pi }{2}.$ The output of $\sin \left(\frac{\pi }{2}\right)$ is opposite the output of $\sin \left(-\frac{\pi }{2}\right).$ Thus,

This is shown in [link] .

Recall that an even function    is one in which

The graph of an even function is symmetric about the y- axis. The cosine function is an even function because $\cos (-\theta )=\cos \theta .$ For example, consider corresponding inputs $\frac{\pi }{4}$ and $-\frac{\pi }{4}.$ The output of $\cos \left(\frac{\pi }{4}\right)$ is the same as the output of $\cos \left(-\frac{\pi }{4}\right).$ Thus,

See [link] .

For all $\theta$ in the domain of the sine and cosine functions, respectively, we can state the following:

• Since $\sin (-\theta )=-\sin \theta ,$ sine is an odd function.
• Since, $\cos (-\theta )=\cos \theta ,$ cosine is an even function.

The other even-odd identities follow from the even and odd nature of the sine and cosine functions. For example, consider the tangent identity, $\tan (-\theta )=\mathrm{-tan}\theta .$ We can interpret the tangent of a negative angle as $\tan (-\theta )=\frac{\sin \left(-\theta \right)}{\cos (-\theta )}=\frac{-\sin \theta }{\cos \theta }=-\tan \theta .$ Tangent is therefore an odd function, which means that $\tan \left(-\theta \right)=-\tan \left(\theta \right)$ for all $\theta$ in the domain of the tangent function .

The cotangent identity, $\cot \left(-\theta \right)=-\cot \theta ,$ also follows from the sine and cosine identities. We can interpret the cotangent of a negative angle as $\cot \left(-\theta \right)=\frac{\cos \left(-\theta \right)}{\sin \left(-\theta \right)}=\frac{\cos \theta }{-\sin \theta }=-\cot \theta .$ Cotangent is therefore an odd function, which means that $\cot \left(-\theta \right)=-\cot \left(\theta \right)$ for all $\theta$ in the domain of the cotangent function .

The cosecant function is the reciprocal of the sine function, which means that the cosecant of a negative angle will be interpreted as $\csc \left(-\theta \right)=\frac{1}{\sin \left(-\theta \right)}=\frac{1}{-\sin \theta }=-\csc \theta .$ The cosecant function is therefore odd.

Finally, the secant function is the reciprocal of the cosine function, and the secant of a negative angle is interpreted as $\sec \left(-\theta \right)=\frac{1}{\cos \left(-\theta \right)}=\frac{1}{\cos \theta }=\sec \theta .$ The secant function is therefore even.

To sum up, only two of the trigonometric functions, cosine and secant, are even. The other four functions are odd, verifying the even-odd identities.

The next set of fundamental identities is the set of reciprocal identities    , which, as their name implies, relate trigonometric functions that are reciprocals of each other. See [link] . Recall that we first encountered these identities when defining trigonometric functions from right angles in Right Angle Trigonometry .