Chapter 9.7, Problem 39E

To determine

To show: $\sin \left(\frac{\pi }{2}-\theta \right)=\cos \theta$ using graph transformation.

Explanation of Solution

Given information:

  $\sin \left(\frac{\pi }{2}-\theta \right)=\cos \theta$

Calculation:

Let us consider a parent function $f(x)=\sin x$ .

Using graph transformation rule reflection across $y-$ axis,

  $f(-x)=\sin \left(-x\right)$

Using graph transformation rule a horizontal left ward shift of $\frac{\pi }{2}$ ,

  $f\left(-x+\frac{\pi }{2}\right)=\sin \left(-x+\frac{\pi }{2}\right)$ .

This is same graph as of $\cos x$ .

Therefore, $\sin \left(\frac{\pi }{2}-\theta \right)=\cos \theta$