/*! 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 40 Solve the following equations. ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 40

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

Short Answer

Expert verified
Answer: The angles 胃 that satisfy the equation are \(\frac{\pi}{4}\) and \(\frac{3\pi}{4}\).

Step by step solution

01

Given equation and interval

We are given the equation, \(\cos^2\theta=\frac{1}{2}\) and we need to solve for \(\theta\) in the interval \(0 \leq \theta < 2\pi\).
02

Take square root on both sides

To find the values of \(\cos\theta\), we need to take the square root of both sides of the given equation. We get: $$\cos\theta = \pm\sqrt{\frac{1}{2}}$$
03

Simplify the square root

We simplify the square root as follows: $$\cos\theta = \pm\frac{\sqrt{2}}{2}$$
04

Use inverse trigonometric function to find the possible values of 胃

Now, we use the inverse cosine function (arccos) to find the possible values of 胃 for both positive and negative values of \(\frac{\sqrt{2}}{2}\): $$\theta_1 = \arccos\left(\frac{\sqrt{2}}{2}\right)$$ $$\theta_2 = \arccos\left(-\frac{\sqrt{2}}{2}\right)$$
05

Solve for the two initial angles 胃

We get the following two values for the angles 胃: $$\theta_1 = \frac{\pi}{4}$$ $$\theta_2 = \frac{3\pi}{4}$$
06

Find all angles within the given interval

Since the cosine function is periodic with a period of \(2\pi\), there might be more angles that satisfy the equation within the given interval. We can find these angles by adding or subtracting \(2\pi\) to the two initial angles 胃. However, since the interval is \(0 \leq \theta < 2\pi\), we only need to add \(2\pi\) to each initial angle: $$\theta_3 = \theta_1 + 2\pi = \frac{\pi}{4}+2\pi$$ $$\theta_4 = \theta_2 + 2\pi = \frac{3\pi}{4}+2\pi$$
07

Check if the new angles are within the given interval

We check if the new angles are within the interval \(0 \leq \theta < 2\pi\): $$\theta_3 = \frac{\pi}{4}+2\pi = \frac{9\pi}{4}$$ $$\theta_4 = \frac{3\pi}{4}+2\pi = \frac{11\pi}{4}$$ Both the new angles are outside the given interval.
08

State the solution

The two angles that satisfy the equation \(\cos^2\theta=\frac{1}{2}\) in the interval \(0 \leq \theta < 2\pi\) are: $$\theta_1 = \frac{\pi}{4}$$ $$\theta_2 = \frac{3\pi}{4}$$

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