Chapter 5.4, Problem 91E

To determine

To write: The trigonometric expression $2\sin \left(\theta +\frac{\pi }{2}\right)$ in the form to $a\sin B\theta +b\cos B\theta$ .

Answer to Problem 91E

Thetrigonometric expression $2\sin \left(\theta +\frac{\pi }{2}\right)$ can be represented as $2\cos \theta$ .

Explanation of Solution

Given information:

The given equationis $2\sin \left(\theta +\frac{\pi }{2}\right)$ .

Calculation:

Compare the expression $2\sin \left(\theta +\frac{\pi }{2}\right)$ with $\sqrt{a^2+b^2}\sin \left(B\theta +C\right)$ .

  $B=1,\sqrt{a^2+b^2}=2,C=\frac{\pi }{2}$

Calculate the value of $a$ .

  $\begin{array}{c}C=\arctan \left(\frac{b}{a}\right)\\ \tan \left(\frac{\pi }{2}\right)=\frac{b}{a}\\ a=0\end{array}$

Calculate the value of $b$ .

  $\begin{array}{c}\sqrt{a^2+b^2}=2\\ \sqrt{0^2+b^2}=2\\ b=2\end{array}$

Substitute the values in the formulae $\sqrt{a^2+b^2}\sin \left(B\theta +C\right)=a\sin B\theta +b\cos B\theta$ .

  $\begin{array}{c}2\sin \left(\theta +\frac{\pi }{2}\right)=0\sin \theta +2\cos \theta \\ =2\cos \theta \end{array}$

Therefore, the trigonometric expression $2\sin \left(\theta +\frac{\pi }{2}\right)$ can be represented as $2\cos \theta$ .