/*! 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
Answer: The general solutions for the given equation can be expressed as $$\theta = 0, \ \pm \dfrac{2\pi}{3} + 2\pi k_1, \ \pm \dfrac{4\pi}{3} + 2\pi k_2$$, where $$k_1$$ and $$k_2$$ are integer values.

Step by step solution

01

Factor the equation

To begin, let's try to factor the given equation. Factoring allows us to simplify the equation and find its roots more easily. Notice that both terms in the equation have a common factor of $$\theta$$, so we can factor it out: $$\theta(2 \cos \theta + 1) = 0$$,
02

Find the values of $$\theta$$

Now we have the factored equation, we can see that the equation equals zero either when $$\theta = 0$$ or when $$2 \cos \theta + 1 = 0$$. First, let's find when $$2 \cos \theta + 1 = 0$$ : $$2 \cos \theta = -1$$ Divide by 2: $$\cos \theta = -\dfrac{1}{2}$$ Now we should find the angles which cosine is $$-\dfrac{1}{2}$$
03

Solve for $$\theta$$ in $$\cos \theta = -\dfrac{1}{2}$$

Recall that the cosine function repeats every $$2\pi$$, so we are looking for the general solution of the equation $$\cos \theta = -\dfrac{1}{2}$$, which can be written as: $$\theta= \pm \dfrac{2\pi}{3} + 2\pi k_1 \ \ \ or \ \ \ \theta=\pm \dfrac{4\pi}{3} + 2\pi k_2$$ Here, $$k_1$$ and $$k_2$$ are integer values.
04

Combine the solutions

We found two types of solutions for the given equation: 1. $$\theta = 0$$ 2. $$\theta= \pm \dfrac{2\pi}{3} + 2\pi k_1 \ \ \ or \ \ \ \theta=\pm \dfrac{4\pi}{3} + 2\pi k_2$$ So, the general solution of the equation $$2 \theta \cos \theta+\theta=0$$ can be written as: $$\theta = 0, \ \pm \dfrac{2\pi}{3} + 2\pi k_1, \ \pm \dfrac{4\pi}{3} + 2\pi k_2$$ where $$k_1$$ and $$k_2$$ are integer values.

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