Use the trigonometric substitution u = a sin $\theta $, where - $\pi $/2

Question

Use the trigonometric substitution u = a sin $\theta $, where - $\pi $/2 < $\theta $ < $\pi $/2 and a > 0 to simplify the expression $\sqrt { a ^ { 2 } - u ^ { 2 } }$ .  
A) a csc $\theta $
B) a sin $\theta $
C) a tan $\theta $
D) a cos $\theta $
E) - a cos $\theta $

Quizplus · Accepted Answer

Given $u = a \sin(\theta)$, we can express $\sqrt{a^2 - u^2}$ as $\sqrt{a^2 - (a\sin(\theta))^2}$. Simplifying inside the square root gives $\sqrt{a^2(1 - \sin^2(\theta))}$. Using the Pythagorean identity $1 - \sin^2(\theta) = \cos^2(\theta)$, we get $\sqrt{a^2\cos^2(\theta)}$, which simplifies to $a\cos(\theta)$.

Use the Trigonometric Substitution U = a Sin θ\theta θ , Where -

Use the Trigonometric Substitution U = a Sin $\theta$ , Where -