/*! 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 44 Solve the following equations. ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 44

Solve the following equations. $$\sin ^{2} \theta-1=0$$

Short Answer

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Question: Solve the trigonometric equation \(\sin^2 \theta - 1 = 0\) and find the general solution for \(\theta\). Answer: The general solution for \(\theta\) in the equation \(\sin^2 \theta - 1 = 0\) is \(\theta = 2n \pi \pm \frac{\pi}{2}\), where \(n\) is an integer.

Step by step solution

01

Rewrite the equation in terms of sine

Given, $$\sin^2 \theta - 1 = 0$$ Rewrite this equation like this: $$\sin^2 \theta = 1$$
02

Use properties of sine function

We know that the sine function can take values between -1 and 1, and is equal to 1 at the following angles (in the general case): $$\theta = 2n \pi + \pi k$$ where \(n\) and \(k\) are integers. Now, we can solve the equation \(\sin^2 \theta = 1\) for \(\theta\): First, take the square root of both sides: $$\sin \theta = \pm 1$$
03

Identify the angles that correspond to the given sine values

When \(\sin \theta = 1\), the general solution can be expressed as: $$\theta = 2n\pi + \frac{\pi}{2}$$ When \(\sin \theta = -1\), the general solution can be expressed as: $$\theta = 2n\pi - \frac{\pi}{2}$$ Both of these solutions can be combined into one general solution: $$\theta = 2n \pi \pm \frac{\pi}{2}$$
04

Conclusion

The general solution to the given trigonometric equation \(\sin^2 \theta - 1 = 0\) is: $$\theta = 2n \pi \pm \frac{\pi}{2}$$ where \(n\) is an integer.

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