/*! 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 46 Solve the following equations. ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 46

Solve the following equations. $$\tan ^{2} 2 \theta=1,0 \leq \theta<\pi$$

Short Answer

Expert verified
Based on the step by step solution, the possible values of 胃 that satisfy the given equation in the interval 0 鈮� 胃 < 蟺 are: 胃 = 蟺/8, 5蟺/8, 蟺/2, 3蟺/4

Step by step solution

01

Understand the equation and the interval

We have the equation \(\tan^2 2\theta = 1\). Since the square of tangent is equal to 1, this means the tangent of \(2\theta\) must be equal to either 1 or -1. So we need to find the values of \(\theta\) that satisfy the equation \(\tan(2\theta) = \pm 1\) in the interval \(0 \leq \theta < \pi\).
02

Solve the equation for positive tangent

First, let's focus on the case where \(\tan(2\theta) = 1\). From the equation, we get that \(2\theta = \arctan(1)\). We know that the arctangent of 1 occurs at angles that are odd multiples of \(\frac{\pi}{4}\), so we have: $$2\theta = \frac{\pi}{4} + k\pi$$ where k is any integer. Now, we divide both sides by 2 to get the possible values of \(\theta\): $$\theta = \frac{\pi}{8} + \frac{k\pi}{2}$$ We know that the interval for \(\theta\) is \(0 \leq \theta < \pi\). So, the values of k that satisfy this inequality are: $$k = 0 \Rightarrow \theta = \frac{\pi}{8}$$ $$k = 1 \Rightarrow \theta = \frac{5\pi}{8}$$
03

Solve the equation for negative tangent

Now, let's focus on the case where \(\tan(2\theta) = -1\). From the equation, we get that \(2\theta = \arctan(-1)\). We know that the arctangent of -1 occurs at angles that are even multiples of \(\frac{\pi}{4}\) plus \(\pi\), so we have: $$2\theta = \pi + \frac{k\pi}{2}$$ where k is any integer. Now, we divide both sides by 2 to get the possible values of \(\theta\): $$\theta = \frac{\pi}{2} + \frac{k\pi}{4}$$ We know that the interval for \(\theta\) is \(0 \leq \theta < \pi\). So, the values of k that satisfy this inequality are: $$k = 0 \Rightarrow \theta = \frac{\pi}{2}$$ $$k = 1 \Rightarrow \theta = \frac{3\pi}{4}$$
04

State the final solution

Combining the solutions from steps 2 and 3, we have the following values for \(\theta\) that satisfy the given equation: $$\theta = \frac{\pi}{8}, \frac{5\pi}{8}, \frac{\pi}{2}, \frac{3\pi}{4}$$

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