Use a sketch to find the exact value of z. an (cos-120) 29 z =
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°.
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![**Problem Statement:**
Use a sketch to find the exact value of \( z \).
\[ z = \tan \left( \cos^{-1} \left(\frac{20}{29}\right) \right) \]
**Explanation:**
To find the exact value of \( z \), we can use trigonometric identities and a right triangle sketch.
1. **Understanding the Functions:**
- \(\cos^{-1}\left(\frac{20}{29}\right)\) finds an angle \(\theta\) such that \(\cos(\theta) = \frac{20}{29}\).
- \(\tan(\theta)\) then gives the tangent of that angle.
2. **Right Triangle Sketch:**
- Draw a right triangle where the adjacent side is 20 and the hypotenuse is 29 (from \(\cos(\theta) = \frac{20}{29}\)).
- Use the Pythagorean theorem to find the opposite side:
\[
\text{opposite} = \sqrt{29^2 - 20^2} = \sqrt{841 - 400} = \sqrt{441} = 21
\]
3. **Calculate \(\tan(\theta)\):**
- \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{21}{20}\).
Thus, the exact value of \( z \) is \(\frac{21}{20}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbcea6d79-c0c2-49cd-aa31-c3eca199518f%2F9a8976fc-1bd7-4325-98ab-c22df2f63761%2F860wp8t_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use a sketch to find the exact value of \( z \).
\[ z = \tan \left( \cos^{-1} \left(\frac{20}{29}\right) \right) \]
**Explanation:**
To find the exact value of \( z \), we can use trigonometric identities and a right triangle sketch.
1. **Understanding the Functions:**
- \(\cos^{-1}\left(\frac{20}{29}\right)\) finds an angle \(\theta\) such that \(\cos(\theta) = \frac{20}{29}\).
- \(\tan(\theta)\) then gives the tangent of that angle.
2. **Right Triangle Sketch:**
- Draw a right triangle where the adjacent side is 20 and the hypotenuse is 29 (from \(\cos(\theta) = \frac{20}{29}\)).
- Use the Pythagorean theorem to find the opposite side:
\[
\text{opposite} = \sqrt{29^2 - 20^2} = \sqrt{841 - 400} = \sqrt{441} = 21
\]
3. **Calculate \(\tan(\theta)\):**
- \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{21}{20}\).
Thus, the exact value of \( z \) is \(\frac{21}{20}\).
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