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If sin2x secx cotx = 3, find the value of csc x.

If sin2x secx cotx = 3, find the value of csc x.

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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Question
**Problem 5:** If \( \sin^2 x \cdot \sec x \cdot \cot x = 3 \), find the value of \( \csc x \). **Explanation:** This problem involves trigonometric identities and requires manipulation of the given equation to find the value of \( \csc x \), which is the cosecant of \( x \). Cosecant is the reciprocal of sine, given by \( \csc x = \frac{1}{\sin x} \). **Steps to Solve:** 1. Start by recalling the identities: - \( \sec x = \frac{1}{\cos x} \) - \( \cot x = \frac{\cos x}{\sin x} \) 2. Substitute these identities into the equation: - \( \sin^2 x \cdot \frac{1}{\cos x} \cdot \frac{\cos x}{\sin x} = 3 \) 3. Simplify the expression: - \( \sin x = 3 \) 4. Use the reciprocal identity to find \( \csc x \): - \( \csc x = \frac{1}{\sin x} \) This will allow you to find the specific value of \( \csc x \) based on the simplified equation.
Transcribed Image Text:**Problem 5:** If \( \sin^2 x \cdot \sec x \cdot \cot x = 3 \), find the value of \( \csc x \). **Explanation:** This problem involves trigonometric identities and requires manipulation of the given equation to find the value of \( \csc x \), which is the cosecant of \( x \). Cosecant is the reciprocal of sine, given by \( \csc x = \frac{1}{\sin x} \). **Steps to Solve:** 1. Start by recalling the identities: - \( \sec x = \frac{1}{\cos x} \) - \( \cot x = \frac{\cos x}{\sin x} \) 2. Substitute these identities into the equation: - \( \sin^2 x \cdot \frac{1}{\cos x} \cdot \frac{\cos x}{\sin x} = 3 \) 3. Simplify the expression: - \( \sin x = 3 \) 4. Use the reciprocal identity to find \( \csc x \): - \( \csc x = \frac{1}{\sin x} \) This will allow you to find the specific value of \( \csc x \) based on the simplified equation.
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