ind the exact value of tan 2 cos y² = 25 y = 5 12 [2 COB-¹ (-17/13)] - y Since x=-12 and y = 5, tan 0 = X tan (20). tan (20) = 2 tan 0 1- tan ²0 2 (-=-512 1-(-=-172) 120 119 5 -12 100 Substitute tan 8 = 5
ind the exact value of tan 2 cos y² = 25 y = 5 12 [2 COB-¹ (-17/13)] - y Since x=-12 and y = 5, tan 0 = X tan (20). tan (20) = 2 tan 0 1- tan ²0 2 (-=-512 1-(-=-172) 120 119 5 -12 100 Substitute tan 8 = 5
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
How do you get negative 120 over 119?
![**Example Problem: Finding the Exact Value of a Trigonometric Expression**
**Problem:**
Find the exact value of \( \tan \left[ 2 \cos^{-1} \left( -\frac{12}{13} \right) \right] \).
**Solution:**
1. **Determine \( y \):**
\[
y^2 = 25 \quad \Rightarrow \quad y = 5
\]
2. **Find \( \tan \theta \):**
Given \( x = -12 \) and \( y = 5 \):
\[
\tan \theta = \frac{y}{x} = \frac{5}{-12} = -\frac{5}{12}
\]
Substitute \( \tan \theta = -\frac{5}{12} \) into the formula for \( \tan (2\theta) \).
3. **Use the Double Angle Formula for Tangent:**
\[
\tan (2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta}
\]
Substitute \( \tan \theta = -\frac{5}{12} \):
\[
\tan (2\theta) = \frac{2 \left( -\frac{5}{12} \right)}{1 - \left(-\frac{5}{12}\right)^2}
\]
\[
= \frac{-\frac{10}{12}}{1 - \frac{25}{144}}
\]
\[
= \frac{-\frac{10}{12}}{\frac{119}{144}}
\]
Simplify to:
\[
= -\frac{120}{119}
\]
4. **Conclusion:**
\[
\tan \left[ 2 \cos^{-1} \left( -\frac{12}{13} \right) \right] = -\frac{120}{119}
\]
This solution applies trigonometric identities and properties to solve for the exact value of the given expression.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff462e736-7a38-4abe-8728-e08b1ac68dd9%2F1abeee6a-645e-48fb-a5b3-9f524e5b9579%2F9lapi3c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Example Problem: Finding the Exact Value of a Trigonometric Expression**
**Problem:**
Find the exact value of \( \tan \left[ 2 \cos^{-1} \left( -\frac{12}{13} \right) \right] \).
**Solution:**
1. **Determine \( y \):**
\[
y^2 = 25 \quad \Rightarrow \quad y = 5
\]
2. **Find \( \tan \theta \):**
Given \( x = -12 \) and \( y = 5 \):
\[
\tan \theta = \frac{y}{x} = \frac{5}{-12} = -\frac{5}{12}
\]
Substitute \( \tan \theta = -\frac{5}{12} \) into the formula for \( \tan (2\theta) \).
3. **Use the Double Angle Formula for Tangent:**
\[
\tan (2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta}
\]
Substitute \( \tan \theta = -\frac{5}{12} \):
\[
\tan (2\theta) = \frac{2 \left( -\frac{5}{12} \right)}{1 - \left(-\frac{5}{12}\right)^2}
\]
\[
= \frac{-\frac{10}{12}}{1 - \frac{25}{144}}
\]
\[
= \frac{-\frac{10}{12}}{\frac{119}{144}}
\]
Simplify to:
\[
= -\frac{120}{119}
\]
4. **Conclusion:**
\[
\tan \left[ 2 \cos^{-1} \left( -\frac{12}{13} \right) \right] = -\frac{120}{119}
\]
This solution applies trigonometric identities and properties to solve for the exact value of the given expression.
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