/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 38 Solve the following equations. ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 38

Solve the following equations. $$2 \theta \cos \theta+\theta=0$$

Short Answer

Expert verified
Question: Solve the trigonometric equation \(2 \theta \cos \theta+\theta=0\) for 胃. Answer: The solutions to the equation are \(\theta = 0\) and \(\theta = 2n\pi \pm \frac{2\pi}{3}\) for all integers n.

Step by step solution

01

Factor the equation

Factor out the common term 胃 from the given equation: $$\theta(2\cos \theta + 1)=0$$
02

Set each factor equal to zero

Set each factor equal to zero to solve for 胃: $$\theta=0 \quad \text{or} \quad 2\cos \theta + 1 = 0$$
03

Solve for 胃 in each case

For the first case, we already have the solution: $$\theta=0$$ For the second case, isolate the cos(胃) term: $$2\cos\theta = -1 \quad \Rightarrow \quad\cos\theta = -\frac{1}{2}$$ Now, we need to find the angles with cosine equal to \(-\frac{1}{2}\). The general formula to find these angles is: $$\theta = 2n\pi \pm \frac{2\pi}{3}$$ where n is an integer. Therefore the solution to the given equation is: $$\theta = 0, \quad 2n\pi \pm \frac{2\pi}{3}, \quad \text{for all integers n}$$

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