Evaluate the exact value of 4 /5 tan sec 4
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 Statement:**
Evaluate the exact value of
\[
\tan\left(\sec^{-1}\left(\frac{4 \cdot \sqrt{5}}{4}\right)\right)
\]
**Solution Steps:**
1. Recognize that \(\sec^{-1}(x)\) represents the angle \(\theta\) where \(\sec(\theta) = x\).
2. Simplify the expression \(\frac{4 \cdot \sqrt{5}}{4}\) to \(\sqrt{5}\).
3. Set \(\sec(\theta) = \sqrt{5}\). Recall the identity \(\sec(\theta) = \frac{1}{\cos(\theta)}\), so \(\cos(\theta) = \frac{1}{\sqrt{5}}\).
4. Use the Pythagorean identity: \(\tan^2(\theta) = \sec^2(\theta) - 1\).
5. Substitute \(\sec^2(\theta) = 5\) into the identity to get \(\tan^2(\theta) = 5 - 1 = 4\).
6. Find \(\tan(\theta) = \pm 2\). Since \(\sec^{-1}(x)\) returns an angle where the tangent is non-negative, we choose \(\tan(\theta) = 2\).
Thus, the exact value is \(2\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1680afc1-b9ed-4705-8271-5c5463bd69a2%2F1aa459fc-b931-41fe-bc31-8df479d795c2%2Fedajmlj_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Evaluate the exact value of
\[
\tan\left(\sec^{-1}\left(\frac{4 \cdot \sqrt{5}}{4}\right)\right)
\]
**Solution Steps:**
1. Recognize that \(\sec^{-1}(x)\) represents the angle \(\theta\) where \(\sec(\theta) = x\).
2. Simplify the expression \(\frac{4 \cdot \sqrt{5}}{4}\) to \(\sqrt{5}\).
3. Set \(\sec(\theta) = \sqrt{5}\). Recall the identity \(\sec(\theta) = \frac{1}{\cos(\theta)}\), so \(\cos(\theta) = \frac{1}{\sqrt{5}}\).
4. Use the Pythagorean identity: \(\tan^2(\theta) = \sec^2(\theta) - 1\).
5. Substitute \(\sec^2(\theta) = 5\) into the identity to get \(\tan^2(\theta) = 5 - 1 = 4\).
6. Find \(\tan(\theta) = \pm 2\). Since \(\sec^{-1}(x)\) returns an angle where the tangent is non-negative, we choose \(\tan(\theta) = 2\).
Thus, the exact value is \(2\).
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