
Test:MAC1114-Test No.2
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Part 1
Find the exact​ value, if​ any, of the following composite function. Do not use a calculator.
cosine Superscript negative 1 Baseline left bracket cosine left parenthesis StartFraction 5 pi Over 8 EndFraction right parenthesis right bracket
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box within your choice.
A.
cosine Superscript negative 1 Baseline left bracket cosine left parenthesis StartFraction 5 pi Over 8 EndFraction right parenthesis right bracket equals nothing
  
​(Simplify your answer. Type an exact​ answer, using pi as needed. Use integers or fractions for any numbers in the​ expression.)
B.
It is not defined.
cosine Superscript negative 1 Baseline left bracket cosine left parenthesis StartFraction 5 pi Over 8 EndFraction right parenthesis right bracketSelect the correct choice below and, if necessary, fill in the answer box within your choice.
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To find the exact value of the composite function \( \cos^{-1} \left( \cos \left( \frac{5\pi}{8} \right) \right) \), we first need to understand the behavior of the cosine function and its inverse.

1. **Evaluate the Inner Function**:
   - \( \cos \left( \frac{5\pi}{8} \right) \) can be simplified. Since \( \frac{5\pi}{8} \) is in the second quadrant, where cosine is negative, the value is:
   \[
   \cos \left( \frac{5\pi}{8} \right) = -\sin \left( \frac{5\pi}{8} - \frac{\pi}{2} \right) = -\sin \left( \frac{\pi}{8} \right).
   \]

2. **Using the Inverse Cosine**:
   - The range of \( \cos^{-1}(x) \) is \( [0, \pi] \). Since \( \cos \left( \frac{5\pi}{8} \right) \) gives a negative value, we need to consider this in the context of the range of \( \cos^{-1} \).

3. **Finding the Final Value**:
   - Because \( \cos \left( \frac{5\pi}{8} \right) \) results in a value that is not in the range of \( [0, 1] \) when considering the range of the cosine function, the final value can be interpreted as:
   \[
   \cos^{-1} \left( \cos \left( \frac{5\pi}{8} \right) \right) = \frac{5\pi}{8} \quad \text{(since this angle is in the range of } [0, \pi]\text{)}.
   \]

Thus, the answer is:
\[
\cos^{-1} \left( \cos \left( \frac{5\pi}{8} \right) \right) = \frac{5\pi}{8}.
\]

**Final Answer**: 
A. \( \cos^{-1} \left( \cos \left( \frac{5\pi}{8} \right) \right) = \frac{5\pi}{8} \).

Would you like more details on any part of this explanation? Here are some related questions you might consider:

1. What are the properties of the cosine function in different quadrants?
2. How does the inverse cosine function work with values outside its range?
3. What is the relationship between sine and cosine in terms of their complementary angles?
4. Can you provide examples of other angles and their cosine values?
5. How does the concept of periodicity apply to trigonometric functions?

**Tip**: When working with trigonometric functions, always consider the quadrant in which the angle lies to determine the sign of the sine and cosine values.

Find the exact value of the composite function \( \cos^{-1} \left( \cos \left( \frac{5\pi}{8} \right) \right) \).

Discover how to find the exact value of \( \cos^{-1}(\cos(\frac{5\pi}{8})) \) through trigonometric properties and inverse functions. This solution emphasizes key trigonometric concepts and is ideal for high school students in Grades 11-12.

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