Use the trigonometric substitution to select the algebraic expression as a trigonomet- ric function of $\theta$ , where $- \frac { \pi } { 2 }

Question

Use the trigonometric substitution to select the algebraic expression as a trigonomet- ric function of $\theta$ , where $- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 } .$ . Then find $\sin \theta \text { and } \cos \theta$ $2 = \sqrt { 4 - x ^ { 2 } } , x = 2 \sin \theta$
A) $2 \cos \theta = 2 ; \sin \theta = 0 ; \cos \theta = 1$
B) $2 \sin \theta = 2 ; \sin \theta = 0 ; \cos \theta = 1$
C) $2 \cos \theta = 2 ; \sin \theta = 1 ; \cos \theta = 0$
D) $4 \sin \theta = 2 ; \sin \theta = 0 ; \cos \theta = 1$
E) $4 \cos \theta = 2 ; \sin \theta = 0 ; \cos \theta = 1$

Quizplus · Accepted Answer

Given $x = 2\sin\theta$, substituting $x = 2$ gives $2 = 2\sin\theta$, leading to $\sin\theta = 1$. Using the Pythagorean identity, $\cos\theta = \sqrt{1 - \sin^2\theta} = 0$. The correct expression is $2\cos\theta = 2$, but the values for $\sin\theta$ and $\cos\theta$ are mistakenly swapped in the explanation.

Use the Trigonometric Substitution to Select the Algebraic Expression as a Trigonomet-