For f, of, x, equals, sine, xf(x)=sinx , minus, start fraction, pi, divided by, 2, end fraction, is less than or equal to, x, is less than or equal to, start fraction, pi, divided by, 2, end fraction− 
2
π
​
 ≤x≤ 
2
π
​
  , g, of, x, equals, cosine, xg(x)=cosx , 0, is less than or equal to, x, is less than or equal to, pi0≤x≤π , and h, of, x, equals, tangent, xh(x)=tanx , minus, start fraction, pi, divided by, 2, end fraction, is less than, x, is less than, start fraction, pi, divided by, 2, end fraction− 
2
π
​
 <x< 
2
π
​
  , find the exact value of the composite function.

g, to the power minus 1 , left parenthesis, f, of, minus, start fraction, pi, divided by, 4, end fraction, right parenthesisg 
−1
 (f(− 
4
π
​
 ))

**Simplify your answer. Type an exact answer, using radicals as needed. Type an exact answer, using piπ as needed. Use integers or fractions for any numbers in the expression.**

We are tasked with finding the exact value of the composite function \( g^{-1}(f(-\frac{\pi}{4})) \).

### Step 1: Evaluate \( f(-\frac{\pi}{4}) \)
We are given the function:
\[
f(x) = \sin(x)
\]
Now, substitute \( x = -\frac{\pi}{4} \) into the sine function:
\[
f\left(-\frac{\pi}{4}\right) = \sin\left(-\frac{\pi}{4}\right)
\]
Using the identity \( \sin(-\theta) = -\sin(\theta) \), we know that:
\[
\sin\left(-\frac{\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\]
Thus:
\[
f\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\]

### Step 2: Solve for \( g^{-1}(f(-\frac{\pi}{4})) = g^{-1}\left(-\frac{\sqrt{2}}{2}\right) \)
We are given the function:
\[
g(x) = \cos(x)
\]
Now, we need to find the inverse function, \( g^{-1}(y) \), such that \( g(g^{-1}(y)) = y \). This means we are looking for \( x \) such that:
\[
\cos(x) = -\frac{\sqrt{2}}{2}
\]
We know that \( \cos(x) = -\frac{\sqrt{2}}{2} \) at \( x = \frac{3\pi}{4} \), within the interval \( 0 \leq x \leq \pi \).

Thus:
\[
g^{-1}\left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4}
\]

### Final Answer:
The exact value of \( g^{-1}(f(-\frac{\pi}{4})) \) is:
\[
\frac{3\pi}{4}
\]

Would you like more details on any steps?

Here are five related questions:
1. How do we determine the inverse of a trigonometric function?
2. What is the domain and range of \( \cos(x) \) within \( 0 \leq x \leq \pi \)?
3. How do we find the exact value of trigonometric functions at specific angles?
4. What is the significance of the interval restrictions for \( x \) in trigonometric functions?
5. How do we use trigonometric identities to simplify expressions?

**Tip**: Always check the domain of the function and its inverse to ensure you're using the correct range for inverse trigonometric values.

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)). Simplify your answer using radicals as needed and use π as needed.

Learn how to solve the composite function g⁻¹(f(−π/4)) step-by-step, involving sine, cosine, and inverse trigonometric functions. This trigonometry problem is suitable for students in Grades 11-12, covering key concepts such as inverse functions and trigonometric identities.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation