If $x = 8 \cos \theta$, use trigonometric substitution to write $\sqrt { 64 - x ^ { 2 } }$ as a trigonometric function of $\theta$, where $0

Question

If $x = 8 \cos \theta$, use trigonometric substitution to write $\sqrt { 64 - x ^ { 2 } }$ as a trigonometric function of $\theta$, where $0 < \theta < \pi$.
A) $8 \sec \theta$
B) $8 \sin \theta$
C) $8 \cot \theta$
D) $8 \cos \theta$
E) $8 \csc \theta$

Quizplus · Accepted Answer

Given $x = 8 \cos \theta$, we can substitute $x$ into the expression $\sqrt{64 - x^2}$ to get $\sqrt{64 - (8 \cos \theta)^2}$. Simplifying this gives $\sqrt{64 - 64\cos^2 \theta} = \sqrt{64(1 - \cos^2 \theta)}$. Since $1 - \cos^2 \theta = \sin^2 \theta$, the expression simplifies further to $\sqrt{64\sin^2 \theta} = 8\sin \theta$, which corresponds to choice B.

If x=8cos⁡θx = 8 \cos \thetax=8cosθ , Use Trigonometric Substitution to Write

If $x = 8 \cos \theta$ , Use Trigonometric Substitution to Write