Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem Statement**
5. Evaluate \( \sin \left( \cos^{-1} \frac{\sqrt{3}}{2} + \tan^{-1} \frac{5}{4} \right) \) exactly (no calculators).
**Explanation of Terms**
- \( \cos^{-1} \) represents the inverse cosine function, also known as arccosine. It gives the angle whose cosine is the specified value.
- \( \tan^{-1} \) represents the inverse tangent function, also known as arctangent. It gives the angle whose tangent is the specified value.
- \( \sin \) represents the sine function, which is a trigonometric function of an angle.
**Approach**
1. Find the angle for \( \cos^{-1} \frac{\sqrt{3}}{2} \). Since \( \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} \), then \( \cos^{-1} \frac{\sqrt{3}}{2} = \frac{\pi}{6} \).
2. Find the angle for \( \tan^{-1} \frac{5}{4} \). This requires knowledge of trigonometric values or constructing a right triangle with opposite side 5 and adjacent side 4 to find the angle.
3. Use the angle sum identity for sine:
\[
\sin(A + B) = \sin A \cos B + \cos A \sin B
\]
where \( A = \cos^{-1} \frac{\sqrt{3}}{2} \) and \( B = \tan^{-1} \frac{5}{4} \).
4. Calculate each component using known values or from derived angles above without a calculator.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61fd18cc-da03-4701-b352-9d1cb0ad2c40%2Fd487e3d2-006e-4513-8c29-b8f019b1c24a%2Fq869r4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
5. Evaluate \( \sin \left( \cos^{-1} \frac{\sqrt{3}}{2} + \tan^{-1} \frac{5}{4} \right) \) exactly (no calculators).
**Explanation of Terms**
- \( \cos^{-1} \) represents the inverse cosine function, also known as arccosine. It gives the angle whose cosine is the specified value.
- \( \tan^{-1} \) represents the inverse tangent function, also known as arctangent. It gives the angle whose tangent is the specified value.
- \( \sin \) represents the sine function, which is a trigonometric function of an angle.
**Approach**
1. Find the angle for \( \cos^{-1} \frac{\sqrt{3}}{2} \). Since \( \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} \), then \( \cos^{-1} \frac{\sqrt{3}}{2} = \frac{\pi}{6} \).
2. Find the angle for \( \tan^{-1} \frac{5}{4} \). This requires knowledge of trigonometric values or constructing a right triangle with opposite side 5 and adjacent side 4 to find the angle.
3. Use the angle sum identity for sine:
\[
\sin(A + B) = \sin A \cos B + \cos A \sin B
\]
where \( A = \cos^{-1} \frac{\sqrt{3}}{2} \) and \( B = \tan^{-1} \frac{5}{4} \).
4. Calculate each component using known values or from derived angles above without a calculator.
Expert Solution
Step 1
We use
sin(A+B)=sin(x)cos(y)+cos(x)sin(y)
Step by step
Solved in 5 steps with 4 images
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