Solve exactly for $x$ for $720^{\circ} \leq x \leq 900^{\circ}$.
\[
\cos ^{3} x+\cos x \sin ^{2} x=\sin x
\]
Answer
Final Answer: The solution to the equation \(\cos ^{3} x+\cos x \sin ^{2} x=\sin x\) for \(4\pi \leq x \leq 5\pi\) is \(x = \boxed{\frac{17\pi}{4}}\).
Step 1 ：The given equation is in terms of trigonometric functions. We can simplify the equation by using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\).
Step 2 ：We can also convert the given range of \(x\) from degrees to radians because trigonometric functions use radians. The range \(720^{\circ} \leq x \leq 900^{\circ}\) is equivalent to \(4\pi \leq x \leq 5\pi\) in radians.
Step 3 ：Let's solve the equation \(\sin(x)^2\cos(x) - \sin(x) + \cos(x)^3 = 0\) for \(x\).
Step 4 ：The solution to the equation is \(x = \frac{17\pi}{4}\).
Step 5 ：Final Answer: The solution to the equation \(\cos ^{3} x+\cos x \sin ^{2} x=\sin x\) for \(4\pi \leq x \leq 5\pi\) is \(x = \boxed{\frac{17\pi}{4}}\).
