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Which of the following fundamental identities is a reciprocal identity? O csc(8) = sine sin(8) O cos(-) = cos(8) O cos (8) = sin(-8) 7°F Mostly sunny cos 2(0)+sin 2 (0) = 1 Search

Which of the following fundamental identities is a reciprocal identity? O csc(8) = sine sin(8) O cos(-) = cos(8) O cos (8) = sin(-8) 7°F Mostly sunny cos 2(0)+sin 2 (0) = 1 Search

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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### Understanding Reciprocal Identities in Trigonometry **Question:** Which of the following fundamental identities is a reciprocal identity? **Options:** 1. \( \text{csc}(\theta) = \frac{1}{\sin(\theta)} \) 2. \( \cos(-\theta) = \cos(\theta) \) 3. \( \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \) 4. \( \cos^2(\theta) + \sin^2(\theta) = 1 \) **Explanation:** - The first option, \( \text{csc}(\theta) = \frac{1}{\sin(\theta)} \), is a reciprocal identity. It states that the cosecant function is the reciprocal of the sine function. - The second option, \( \cos(-\theta) = \cos(\theta) \), reflects the even property of the cosine function. - The third option, \( \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \), expresses a co-function identity. - The fourth option, \( \cos^2(\theta) + \sin^2(\theta) = 1 \), demonstrates the Pythagorean identity. Reciprocal identities are relationships where two trigonometric functions are reciprocals of each other. In this list, the correct reciprocal identity is the first option: \( \text{csc}(\theta) = \frac{1}{\sin(\theta)} \). For further understanding, note that reciprocal identities include: - \( \text{csc}(\theta) = \frac{1}{\sin(\theta)} \) - \( \sec(\theta) = \frac{1}{\cos(\theta)} \) - \( \cot(\theta) = \frac{1}{\tan(\theta)} \) These fundamental trigonometric identities are pivotal for solving various problems in trigonometry.
Transcribed Image Text:### Understanding Reciprocal Identities in Trigonometry **Question:** Which of the following fundamental identities is a reciprocal identity? **Options:** 1. \( \text{csc}(\theta) = \frac{1}{\sin(\theta)} \) 2. \( \cos(-\theta) = \cos(\theta) \) 3. \( \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \) 4. \( \cos^2(\theta) + \sin^2(\theta) = 1 \) **Explanation:** - The first option, \( \text{csc}(\theta) = \frac{1}{\sin(\theta)} \), is a reciprocal identity. It states that the cosecant function is the reciprocal of the sine function. - The second option, \( \cos(-\theta) = \cos(\theta) \), reflects the even property of the cosine function. - The third option, \( \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \), expresses a co-function identity. - The fourth option, \( \cos^2(\theta) + \sin^2(\theta) = 1 \), demonstrates the Pythagorean identity. Reciprocal identities are relationships where two trigonometric functions are reciprocals of each other. In this list, the correct reciprocal identity is the first option: \( \text{csc}(\theta) = \frac{1}{\sin(\theta)} \). For further understanding, note that reciprocal identities include: - \( \text{csc}(\theta) = \frac{1}{\sin(\theta)} \) - \( \sec(\theta) = \frac{1}{\cos(\theta)} \) - \( \cot(\theta) = \frac{1}{\tan(\theta)} \) These fundamental trigonometric identities are pivotal for solving various problems in trigonometry.
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