Answered: Rewrite the expression cos (0) without…

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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$**Problem Statement:** Rewrite the expression $ \cos^4(\theta) $ without using any powers of sine or cosine. **Explanation:** The goal is to express $ \cos^4(\theta) $ using trigonometric identities that do not include powers of the sine or cosine functions directly. This typically involves using identities such as the double angle formulas or the Pythagorean identity. Here's a step-by-step approach: 1. **Start with the Pythagorean identity:** \[ \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \] 2. **Square both sides to find $ \cos^4(\theta) $:** \[ \cos^4(\theta) = \left(\frac{1 + \cos(2\theta)}{2}\right)^2 \] 3. **Expand the square:** \[ \cos^4(\theta) = \frac{(1 + \cos(2\theta))^2}{4} \] 4. **Expand the numerator using the formula for squaring a binomial:** \[ (1 + \cos(2\theta))^2 = 1 + 2\cos(2\theta) + \cos^2(2\theta) \] 5. **Recall the identity for $ \cos^2(2\theta) $:** \[ \cos^2(2\theta) = \frac{1 + \cos(4\theta)}{2} \] 6. **Substitute back:** \[ \cos^4(\theta) = \frac{1 + 2\cos(2\theta) + \frac{1 + \cos(4\theta)}{2}}{4} \] 7. **Simplify the expression:** \[ \cos^4(\theta) = \frac{2 + 2\cos(2\theta) + 1 + \cos(4\theta)}{8} \] 8. **Simplify further:** \[ \cos^4(\theta) = \frac{3 + 2\cos(2\theta) + \cos(4\theta)}{8} \] Thus, the expression$

Transcribed Image Text:**Problem Statement:** Rewrite the expression $ \cos^4(\theta) $ without using any powers of sine or cosine. **Explanation:** The goal is to express $ \cos^4(\theta) $ using trigonometric identities that do not include powers of the sine or cosine functions directly. This typically involves using identities such as the double angle formulas or the Pythagorean identity. Here's a step-by-step approach: 1. **Start with the Pythagorean identity:** \[ \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \] 2. **Square both sides to find $ \cos^4(\theta) $:** \[ \cos^4(\theta) = \left(\frac{1 + \cos(2\theta)}{2}\right)^2 \] 3. **Expand the square:** \[ \cos^4(\theta) = \frac{(1 + \cos(2\theta))^2}{4} \] 4. **Expand the numerator using the formula for squaring a binomial:** \[ (1 + \cos(2\theta))^2 = 1 + 2\cos(2\theta) + \cos^2(2\theta) \] 5. **Recall the identity for $ \cos^2(2\theta) $:** \[ \cos^2(2\theta) = \frac{1 + \cos(4\theta)}{2} \] 6. **Substitute back:** \[ \cos^4(\theta) = \frac{1 + 2\cos(2\theta) + \frac{1 + \cos(4\theta)}{2}}{4} \] 7. **Simplify the expression:** \[ \cos^4(\theta) = \frac{2 + 2\cos(2\theta) + 1 + \cos(4\theta)}{8} \] 8. **Simplify further:** \[ \cos^4(\theta) = \frac{3 + 2\cos(2\theta) + \cos(4\theta)}{8} \] Thus, the expression

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