If $x = 10 \sin \theta$, use trigonometric substitution to write $\sqrt { 100 - x ^ { 2 } }$ as a trigonometric function of $\theta$, where $- \frac { \pi } { 2 }

Question

If $x = 10 \sin \theta$, use trigonometric substitution to write $\sqrt { 100 - x ^ { 2 } }$ as a trigonometric function of $\theta$, where $- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$.
A) $10 \sin \theta$
B) $10 \cos \theta$
C) $10 \tan \theta$
D) $10 \csc \theta$
E) $10 \sec \theta$

Quizplus · Accepted Answer

Given $x = 10 \sin \theta$, substituting $x$ into the expression $\sqrt{100 - x^2}$ yields $\sqrt{100 - (10\sin\theta)^2} = \sqrt{100 - 100\sin^2\theta} = \sqrt{100(1 - \sin^2\theta)} = \sqrt{100\cos^2\theta} = 10\cos\theta$, using the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$.

If x=10sin⁡θx = 10 \sin \thetax=10sinθ , Use Trigonometric Substitution to Write

If $x = 10 \sin \theta$ , Use Trigonometric Substitution to Write