Chapter 5.2, Problem 92E

To determine

To simplify the expression:

  $\sqrt{a^2-u^2}$

Using substitution $u=a\cos \theta$ . Where $0<\theta <\frac{\pi }{2},a>0$

Answer to Problem 92E

  $\sqrt{a^2-u^2}=a\sin \theta$

Explanation of Solution

Given:

  $\sqrt{a^2-u^2}$

Concept Used:

The trigonometric identity:

• ${\sin }^2\theta +{\cos }^2\theta =1$

Calculation:

In order to simplify the expression

  $\sqrt{a^2-u^2}$

Substitute the value of $u=a\cos \theta$ in the expression as,

  $\begin{array}{c}\sqrt{a^2-u^2}=\sqrt{a^2-{\left(a\cos \theta \right)}^2}\\ =\sqrt{a^2-a^2{\cos }^2\theta }\\ =\sqrt{a^2\left(1-{\cos }^2\theta \right)}\\ =\sqrt{a^2{\sin }^2\theta },\text{ using }\left({\sin }^2\theta +{\cos }^2\theta =1\right)\\ =asin\theta \end{array}$

Thus, putting $u=a\cos \theta$ ,gives

  $\sqrt{a^2-u^2}=a\sin \theta$