for f(x)= sin x, - π/2 ≤ x ≤ π/2, g(x)= cos x, 0 ≤ x ≤ π, and h(x)= tan x, - π/2 < x < π/2, find the exact…

for f(x)= sin x, - π/2 ≤ x ≤ π/2, g(x)= cos x, 0 ≤ x ≤ π, and h(x)= tan x, - π/2 < x < π/2, find the exact value of the composite function. g^(-1)(f(-π/4)) g^(-1)(f(-π/4))=□ (simplify your answer. type an exact answer, using radicals as needed. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.)

for f(x)= sin x, - π/2 ≤ x ≤ π/2, g(x)= cos x, 0 ≤ x ≤ π, and h(x)= tan x, - π/2 < x < π/2, find the exact value of the composite function. g^(-1)(f(-π/4)) g^(-1)(f(-π/4))=□ (simplify your answer. type an exact answer, using radicals as needed. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.)

Answer

Explanation:

Step1: Calculate $f(-\frac{\pi}{4})$

Given $f(x)=\sin x$ and $x = -\frac{\pi}{4}$, then $f(-\frac{\pi}{4})=\sin(-\frac{\pi}{4})=-\frac{\sqrt{2}}{2}$.

Step2: Calculate $g^{-1}(-\frac{\sqrt{2}}{2})$

Given $g(x)=\cos x,0\leq x\leq\pi$. We want to find $x$ such that $\cos x = -\frac{\sqrt{2}}{2}$ and $0\leq x\leq\pi$. The inverse - cosine function $y = g^{-1}(u)$ gives the angle $y$ in the domain $[0,\pi]$ for which $\cos y=u$. Since $\cos(\frac{3\pi}{4})=-\frac{\sqrt{2}}{2}$ and $\frac{3\pi}{4}\in[0,\pi]$, then $g^{-1}(-\frac{\sqrt{2}}{2})=\frac{3\pi}{4}$.

Answer:

$\frac{3\pi}{4}$