/*! 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 values of 胃 that satisfy the given equation in the specified range are \(胃 = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4},\) and \(\frac{7\pi}{4}\).

Step by step solution

01

Analyze and rewrite the given equation

The given equation is: $$\cos^2 \theta = \frac{1}{2}$$ This can be rewritten as: $$2\cos^2 \theta = 1$$
02

Use trigonometric identity

We know the double angle formula for cosine: $$\cos(2\alpha) = 2\cos^2\alpha - 1$$ Re-arrange the formula to make \(\cos^2 \alpha\) the subject: $$\cos^2 \alpha = \frac{\cos(2\alpha)+1}{2}$$ Now replace \(\alpha\) with \(\theta\) in the equation: $$\cos^2 \theta = \frac{\cos(2\theta)+1}{2}$$
03

Substitution

Substitute the given equation: $$\frac{\cos(2\theta)+1}{2} = \frac{1}{2}$$
04

Solve for 胃

Multiply both sides by 2 to get rid of the fraction: $$\cos(2\theta) + 1 = 1$$ Subtract 1 from both sides: $$\cos(2\theta) = 0$$ Now we need to determine the angles 胃 that satisfy this equation in the range \(0 \leq \theta < 2 \pi\). Notice that the angle input for the cosine function is \(2\theta\). To determine the solutions, we can first find the values of \(2\theta\) and the divide those by 2 to get the values of 胃.
05

Find the values for \(2\theta\) within the given range

Since the cosine function is 0 at odd multiples of \(\frac{\pi}{2}\), we need to find such multiples within the range \(0 \leq 2\theta < 4\pi\). These multiples are \(\frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2},\) and \(\frac{7\pi}{2}\).
06

Obtain the values of 胃

To find the values of 胃 for each multiple, we divide by 2: $$\theta_1 = \frac{\frac{\pi}{2}}{2} = \frac{\pi}{4}$$ $$\theta_2 = \frac{\frac{3\pi}{2}}{2} = \frac{3\pi}{4}$$ $$\theta_3 = \frac{\frac{5\pi}{2}}{2} = \frac{5\pi}{4}$$ $$\theta_4 = \frac{\frac{7\pi}{2}}{2} = \frac{7\pi}{4}$$ So the angles that satisfy the given equation in the specified range are: $$\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$$

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