Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ,where $- \frac { \pi } { 2 }

Question

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ,where $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$ .Then find sin θ and cos θ. $4 = \sqrt { 16 - x ^ { 2 } } , x = 4 \sin \theta$
A) $16 \cos \theta = 4 ; \sin \theta = 0 ; \cos \theta = 1$
B)​ $4 \cos \theta = 4 ; \sin \theta = 0 ; \cos \theta = 1$
C)​ $16 \sin \theta = 4 ; \sin \theta = 0 ; \cos \theta = 1$
D)​ $4 \cos \theta = 4 ; \sin \theta = 1 ; \cos \theta = 0$
E)​ $4 \sin \theta = 4 ; \sin \theta = 0 ; \cos \theta = 1$

Quizplus · Accepted Answer

Given $x = 4\sin\theta$, substituting $x$ into the equation $4 = \sqrt{16 - x^2}$ gives $4 = \sqrt{16 - (4\sin\theta)^2}$. Simplifying inside the square root gives $4 = \sqrt{16 - 16\sin^2\theta}$, which simplifies further to $4 = \sqrt{16(1 - \sin^2\theta)}$, or $4 = 4\sqrt{\cos^2\theta}$, leading to $4 = 4\cos\theta$. Since $x = 4\sin\theta$, and given the domain for $\theta$, $\sin\theta = 0$ when $\theta = 0$, which also makes $\cos\theta = 1$.

Use the Trigonometric Substitution to Rewrite the Algebraic Expression as a Trigonometric