Chapter 7.3, Problem 9AYU

To determine

Whether the statement “most trigonometric equation have unique solution” is true or false.

Answer to Problem 9AYU

The statement “most trigonometric equation have unique solution” is false.

Explanation of Solution

Given information:

The statement “most trigonometric equation have unique solution.”

Consider the provided statement “most trigonometric equation have unique solution.”

Recall that the trigonometric equation $\sin \theta =\frac{1}{2}$ has the solution $\langle \theta =2n\pi +\frac{\pi }{6},\theta =2n\pi +\frac{5\pi }{6}\rangle$ .

For all value of trigonometric function $\sin \theta =\frac{1}{2}$ is $\sin \theta =\sin \frac{\pi }{6}\Rightarrow \theta =2n\pi +\frac{\pi }{6}$ where n belongs to integer.

Also $\sin \left(\pi -\frac{\pi }{6}\right)=\sin \frac{5\pi }{6}$ .

Therefore $\sin \theta =\sin \frac{5\pi }{6}\Rightarrow \sin \theta =\sin \left(2\pi +\frac{5\pi }{6}\right)$ where n is any integer.

So the trigonometric equation $\sin \theta =\frac{1}{2}$ does not unique solutions.

Thus, the statement “most trigonometric equation have unique solution.” is false.