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

Question

A) 10sin?
B) 10cos?
C) 10 tan?
D) 10csc?
E) 10sec?

Quizplus · Accepted Answer

Given $x = 10\sin\theta$, we can substitute this into the expression $\sqrt{100-x^2}$ to get $\sqrt{100-(10\sin\theta)^2}$. Simplifying inside the square root gives $\sqrt{100-100\sin^2\theta}$, which simplifies further to $\sqrt{100(1-\sin^2\theta)}$. Using the Pythagorean identity $1-\sin^2\theta = \cos^2\theta$, we get $\sqrt{100\cos^2\theta}$, which simplifies to $10|\cos\theta|$. Since $-\frac{\pi}{2}<\theta<\frac{\pi}{2}$, $\cos\theta$ is positive, so $10|\cos\theta| = 10\cos\theta$.

If X =10sin?, Use Trigonometric Substitution to Write 100−x2\sqrt{100-x^{2}}100−x2​ As a Trigonometric Function of ?, Where

If X =10sin?, Use Trigonometric Substitution to Write $\sqrt{100-x^{2}}$ As a Trigonometric Function of ?, Where