Evaluate the expression cos sin 6 16 + tan 3 . Give an exact answer.
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![**Title: Evaluating Trigonometric Expressions Involving Inverse Functions**
**Instruction: Evaluate the expression \( \cos \left( \sin^{-1} \left( \frac{6}{16} \right) + \tan^{-1} \left( \frac{3}{7} \right) \right) \). Give an exact answer.**
To solve this problem, follow these steps:
1. Start by defining the angles for the inverse sine and inverse tangent functions.
- Let \( \theta = \sin^{-1} \left( \frac{6}{16} \right) \).
- Let \( \phi = \tan^{-1} \left( \frac{3}{7} \right) \).
2. Evaluate the trigonometric expressions:
- \( \sin \theta = \frac{6}{16} = \frac{3}{8} \).
- Since \( \theta \) is an angle in a right triangle with opposite side 3 and hypotenuse 8, the adjacent side \( a \) can be found using the Pythagorean theorem:
\[
a = \sqrt{8^2 - 3^2} = \sqrt{64 - 9} = \sqrt{55}.
\]
Thus,
\[
\cos \theta = \frac{\sqrt{55}}{8}.
\]
3. For \( \phi \):
- \( \tan \phi = \frac{3}{7} \).
- This represents a right triangle where the opposite side is 3 and the adjacent side is 7. The hypotenuse \( h \) is:
\[
h = \sqrt{7^2 + 3^2} = \sqrt{49 + 9} = \sqrt{58}.
\]
Thus,
\[
\cos \phi = \frac{7}{\sqrt{58}}.
\]
4. Combine the two angles:
- Use the cosine angle addition formula:
\[
\cos (\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi.
\]
- Plug in the known values:
\[
\cos (\theta + \phi) = \left( \frac{\sqrt{55}}{8} \right)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b559ab3-5119-482d-94d5-a98a530a62e8%2Fd2f77a96-68b1-43c4-8ea4-73fb836fb2b0%2Fzmcm0xp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Evaluating Trigonometric Expressions Involving Inverse Functions**
**Instruction: Evaluate the expression \( \cos \left( \sin^{-1} \left( \frac{6}{16} \right) + \tan^{-1} \left( \frac{3}{7} \right) \right) \). Give an exact answer.**
To solve this problem, follow these steps:
1. Start by defining the angles for the inverse sine and inverse tangent functions.
- Let \( \theta = \sin^{-1} \left( \frac{6}{16} \right) \).
- Let \( \phi = \tan^{-1} \left( \frac{3}{7} \right) \).
2. Evaluate the trigonometric expressions:
- \( \sin \theta = \frac{6}{16} = \frac{3}{8} \).
- Since \( \theta \) is an angle in a right triangle with opposite side 3 and hypotenuse 8, the adjacent side \( a \) can be found using the Pythagorean theorem:
\[
a = \sqrt{8^2 - 3^2} = \sqrt{64 - 9} = \sqrt{55}.
\]
Thus,
\[
\cos \theta = \frac{\sqrt{55}}{8}.
\]
3. For \( \phi \):
- \( \tan \phi = \frac{3}{7} \).
- This represents a right triangle where the opposite side is 3 and the adjacent side is 7. The hypotenuse \( h \) is:
\[
h = \sqrt{7^2 + 3^2} = \sqrt{49 + 9} = \sqrt{58}.
\]
Thus,
\[
\cos \phi = \frac{7}{\sqrt{58}}.
\]
4. Combine the two angles:
- Use the cosine angle addition formula:
\[
\cos (\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi.
\]
- Plug in the known values:
\[
\cos (\theta + \phi) = \left( \frac{\sqrt{55}}{8} \right)
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