4 If tan(a) and cos(b) 3 answer in a simplified fraction. 8 with a in quadrant III and b in quadrant II. Find sin(a + b) Leave you 17' =
4 If tan(a) and cos(b) 3 answer in a simplified fraction. 8 with a in quadrant III and b in quadrant II. Find sin(a + b) Leave you 17' =
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°.
Related questions
Question
![**Problem:**
If \( \tan(a) = \frac{4}{3} \) and \( \cos(b) = -\frac{8}{17} \), with \( a \) in quadrant III and \( b \) in quadrant II, find \( \sin(a + b) \). Leave your answer in a simplified fraction.
**Explanation:**
To solve this problem, follow these steps:
1. **Find \(\sin(a)\) and \(\cos(a)\):**
Since \(\tan(a) = \frac{4}{3}\), and \(a\) is in quadrant III:
\[
\sin(a) = -\frac{4}{5}, \quad \cos(a) = -\frac{3}{5}
\]
In quadrant III, both sine and cosine are negative.
2. **Find \(\sin(b)\):**
Given \(\cos(b) = -\frac{8}{17}\) and \(b\) in quadrant II, where sine is positive and cosine is negative:
\[
\sin(b) = \frac{15}{17}
\]
3. **Use the sine addition formula:**
\[
\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)
\]
4. **Substitute the values:**
\[
\sin(a + b) = \left(-\frac{4}{5}\right)\left(-\frac{8}{17}\right) + \left(-\frac{3}{5}\right)\left(\frac{15}{17}\right)
\]
5. **Calculate:**
\[
\sin(a + b) = \frac{32}{85} - \frac{45}{85} = -\frac{13}{85}
\]
**Answer:**
\(\sin(a + b) = -\frac{13}{85}\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb5100fd-96bf-46d3-bc62-7120bd797ba5%2F5412df53-2815-41d4-8745-88c2587619dd%2Fzl69l8q.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
If \( \tan(a) = \frac{4}{3} \) and \( \cos(b) = -\frac{8}{17} \), with \( a \) in quadrant III and \( b \) in quadrant II, find \( \sin(a + b) \). Leave your answer in a simplified fraction.
**Explanation:**
To solve this problem, follow these steps:
1. **Find \(\sin(a)\) and \(\cos(a)\):**
Since \(\tan(a) = \frac{4}{3}\), and \(a\) is in quadrant III:
\[
\sin(a) = -\frac{4}{5}, \quad \cos(a) = -\frac{3}{5}
\]
In quadrant III, both sine and cosine are negative.
2. **Find \(\sin(b)\):**
Given \(\cos(b) = -\frac{8}{17}\) and \(b\) in quadrant II, where sine is positive and cosine is negative:
\[
\sin(b) = \frac{15}{17}
\]
3. **Use the sine addition formula:**
\[
\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)
\]
4. **Substitute the values:**
\[
\sin(a + b) = \left(-\frac{4}{5}\right)\left(-\frac{8}{17}\right) + \left(-\frac{3}{5}\right)\left(\frac{15}{17}\right)
\]
5. **Calculate:**
\[
\sin(a + b) = \frac{32}{85} - \frac{45}{85} = -\frac{13}{85}
\]
**Answer:**
\(\sin(a + b) = -\frac{13}{85}\)
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