Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Problem 24
**Question:**
If the secant of an angle is 4.2, then the cotangent of that angle is \(*\)
This problem involves the relationship between the secant and cotangent of an angle in trigonometry.
**Concepts Involved:**
1. **Secant (sec):** The secant of an angle in a right triangle is the reciprocal of the cosine of that angle. If \( \sec(\theta) = 4.2 \), then \( \cos(\theta) = \frac{1}{4.2} \).
2. **Cotangent (cot):** The cotangent of an angle is the reciprocal of the tangent of that angle. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), making \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \).
3. To find \( \cot(\theta) \), we need the values of \( \cos(\theta) \) and \( \sin(\theta) \).
**Steps to Solve:**
1. Given \( \sec(\theta) = 4.2 \), calculate \( \cos(\theta) \):
\[
\cos(\theta) = \frac{1}{4.2}
\]
Evaluate \( \cos(\theta) \approx 0.2381 \).
2. Using the Pythagorean identity:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
Determine \( \sin(\theta) \):
\[
\sin^2(\theta) = 1 - \cos^2(\theta)
\]
3. Substitute \( \cos(\theta) \approx 0.2381 \):
\[
\sin^2(\theta) = 1 - (0.2381)^2
\]
4. Calculate:
\[
\sin(\theta) = \sqrt{1 - 0.2381^2} \approx \sqrt{1 - 0.0567} \approx \sqrt{0.9433} \approx 0.9712
\]
5. Now, compute](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba874e82-f331-4b1e-9400-f6613fffe075%2F7ed33f8c-f8b2-41b7-be6e-8248748fd23f%2F1oohqk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 24
**Question:**
If the secant of an angle is 4.2, then the cotangent of that angle is \(*\)
This problem involves the relationship between the secant and cotangent of an angle in trigonometry.
**Concepts Involved:**
1. **Secant (sec):** The secant of an angle in a right triangle is the reciprocal of the cosine of that angle. If \( \sec(\theta) = 4.2 \), then \( \cos(\theta) = \frac{1}{4.2} \).
2. **Cotangent (cot):** The cotangent of an angle is the reciprocal of the tangent of that angle. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), making \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \).
3. To find \( \cot(\theta) \), we need the values of \( \cos(\theta) \) and \( \sin(\theta) \).
**Steps to Solve:**
1. Given \( \sec(\theta) = 4.2 \), calculate \( \cos(\theta) \):
\[
\cos(\theta) = \frac{1}{4.2}
\]
Evaluate \( \cos(\theta) \approx 0.2381 \).
2. Using the Pythagorean identity:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
Determine \( \sin(\theta) \):
\[
\sin^2(\theta) = 1 - \cos^2(\theta)
\]
3. Substitute \( \cos(\theta) \approx 0.2381 \):
\[
\sin^2(\theta) = 1 - (0.2381)^2
\]
4. Calculate:
\[
\sin(\theta) = \sqrt{1 - 0.2381^2} \approx \sqrt{1 - 0.0567} \approx \sqrt{0.9433} \approx 0.9712
\]
5. Now, compute
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