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the secant of an angle nt f that c ngle ie *

the secant of an angle nt f that c ngle ie *

Algebra and Trigonometry (6th Edition)
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
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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