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Which of the following fundamental identities is a reciprocal identity? 7 O sin(-8)= -sin(e) O csc(8) = sec (5-8) - O tan² (8) + 1 = sec²(0) cot( 8 ) = tan(0)

Which of the following fundamental identities is a reciprocal identity? 7 O sin(-8)= -sin(e) O csc(8) = sec (5-8) - O tan² (8) + 1 = sec²(0) cot( 8 ) = tan(0)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Identifying Reciprocal Trigonometric Identities** **Question:** Which of the following fundamental identities is a reciprocal identity? **Options:** 1. \( \sin(-\theta) = -\sin(\theta) \) 2. \( \csc(\theta) = \sec\left(\frac{\pi}{2} - \theta\right) \) 3. \( \tan^2(\theta) + 1 = \sec^2(\theta) \) 4. \( \cot(\theta) = \frac{1}{\tan(\theta)} \) **Explanation:** Reciprocal identities in trigonometry are relationships where a trigonometric function is defined as the reciprocal of another trigonometric function. From the given options, let's analyze each one: 1. **Option 1:** \( \sin(-\theta) = -\sin(\theta) \) - This identity is an odd function property of sine, not a reciprocal identity. 2. **Option 2:** \( \csc(\theta) = \sec\left(\frac{\pi}{2} - \theta\right) \) - This represents a co-function identity, where the secant of the complementary angle is equal to the cosecant. 3. **Option 3:** \( \tan^2(\theta) + 1 = \sec^2(\theta) \) - This is a Pythagorean identity, not a reciprocal one. 4. **Option 4:** \( \cot(\theta) = \frac{1}{\tan(\theta)} \) - This is a reciprocal identity because cotangent is defined as the reciprocal of the tangent function. **Correct Answer:** The reciprocal identity is \( \cot(\theta) = \frac{1}{\tan(\theta)} \).
Transcribed Image Text:**Identifying Reciprocal Trigonometric Identities** **Question:** Which of the following fundamental identities is a reciprocal identity? **Options:** 1. \( \sin(-\theta) = -\sin(\theta) \) 2. \( \csc(\theta) = \sec\left(\frac{\pi}{2} - \theta\right) \) 3. \( \tan^2(\theta) + 1 = \sec^2(\theta) \) 4. \( \cot(\theta) = \frac{1}{\tan(\theta)} \) **Explanation:** Reciprocal identities in trigonometry are relationships where a trigonometric function is defined as the reciprocal of another trigonometric function. From the given options, let's analyze each one: 1. **Option 1:** \( \sin(-\theta) = -\sin(\theta) \) - This identity is an odd function property of sine, not a reciprocal identity. 2. **Option 2:** \( \csc(\theta) = \sec\left(\frac{\pi}{2} - \theta\right) \) - This represents a co-function identity, where the secant of the complementary angle is equal to the cosecant. 3. **Option 3:** \( \tan^2(\theta) + 1 = \sec^2(\theta) \) - This is a Pythagorean identity, not a reciprocal one. 4. **Option 4:** \( \cot(\theta) = \frac{1}{\tan(\theta)} \) - This is a reciprocal identity because cotangent is defined as the reciprocal of the tangent function. **Correct Answer:** The reciprocal identity is \( \cot(\theta) = \frac{1}{\tan(\theta)} \).
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