Find the exact value for the expression under the given conditions. Write your answer in simplest form. Rationalize the denominator, if necessary. 11 63 tan (a+B); cos a = - for a in Quadrant III and sin ß=- for B in Quadrant IV. 65 tan (a + B)= 2 01/0 61
Find the exact value for the expression under the given conditions. Write your answer in simplest form. Rationalize the denominator, if necessary. 11 63 tan (a+B); cos a = - for a in Quadrant III and sin ß=- for B in Quadrant IV. 65 tan (a + B)= 2 01/0 61
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
![# Trigonometric Problem: Finding the Exact Value of \(\tan (\alpha + \beta)\)
**Problem Statement:**
Find the exact value for the expression under the given conditions. Write your answer in simplest form. Rationalize the denominator, if necessary.
Given:
\[ \tan (\alpha + \beta); \cos \alpha = -\frac{11}{61} \text{ for } \alpha \text{ in Quadrant III and } \sin \beta = -\frac{63}{65} \text{ for } \beta \text{ in Quadrant IV.} \]
\[ \tan (\alpha + \beta) = \boxed{\phantom{\frac{x}{y}}} \]
**Explanation:**
We are given the values of \(\cos \alpha\) and \(\sin \beta\) for angles \(\alpha\) and \(\beta\) located in specific quadrants. Here we need to find the value of \(\tan (\alpha + \beta)\).
### Steps to Solve:
1. **Define Trigonometric Identities:**
Utilize the identity of tangent addition formula:
\[
\tan (\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}
\]
2. **Determine \(\tan \alpha\) and \(\tan \beta\):**
- Given \(\cos \alpha = -\frac{11}{61}\) and \(\alpha\) is in Quadrant III, where both \(\sin \alpha\) and \(\cos \alpha\) are negative.
- Use the Pythagorean identity to find \(\sin \alpha\):
\[
\sin \alpha = -\sqrt{1 - \cos^2 \alpha} = -\sqrt{1 - \left(-\frac{11}{61}\right)^2} = -\sqrt{1 - \frac{121}{3721}} = -\sqrt{\frac{3600}{3721}} = -\frac{60}{61}
\]
- Then:
\[
\tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{-\frac{60}{61}}{-\frac{11}{61}} = \frac{60}{11}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe92e694a-8dce-4f5c-a0b3-7db66cedb1e5%2F8d1e29b6-fb0b-45ac-9139-0e049037d98d%2F8bpetff_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Trigonometric Problem: Finding the Exact Value of \(\tan (\alpha + \beta)\)
**Problem Statement:**
Find the exact value for the expression under the given conditions. Write your answer in simplest form. Rationalize the denominator, if necessary.
Given:
\[ \tan (\alpha + \beta); \cos \alpha = -\frac{11}{61} \text{ for } \alpha \text{ in Quadrant III and } \sin \beta = -\frac{63}{65} \text{ for } \beta \text{ in Quadrant IV.} \]
\[ \tan (\alpha + \beta) = \boxed{\phantom{\frac{x}{y}}} \]
**Explanation:**
We are given the values of \(\cos \alpha\) and \(\sin \beta\) for angles \(\alpha\) and \(\beta\) located in specific quadrants. Here we need to find the value of \(\tan (\alpha + \beta)\).
### Steps to Solve:
1. **Define Trigonometric Identities:**
Utilize the identity of tangent addition formula:
\[
\tan (\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}
\]
2. **Determine \(\tan \alpha\) and \(\tan \beta\):**
- Given \(\cos \alpha = -\frac{11}{61}\) and \(\alpha\) is in Quadrant III, where both \(\sin \alpha\) and \(\cos \alpha\) are negative.
- Use the Pythagorean identity to find \(\sin \alpha\):
\[
\sin \alpha = -\sqrt{1 - \cos^2 \alpha} = -\sqrt{1 - \left(-\frac{11}{61}\right)^2} = -\sqrt{1 - \frac{121}{3721}} = -\sqrt{\frac{3600}{3721}} = -\frac{60}{61}
\]
- Then:
\[
\tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{-\frac{60}{61}}{-\frac{11}{61}} = \frac{60}{11}
\]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Trigonometry (11th Edition)
Trigonometry
ISBN:
9780134217437
Author:
Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:
PEARSON
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Trigonometry (11th Edition)
Trigonometry
ISBN:
9780134217437
Author:
Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:
PEARSON
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning