/*! 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

Expert verified
Question: Solve the equation $$\sin^2{\theta}-1=0$$ and find the angles in degrees or radians. Answer: The solutions are $$\theta = 90^{\circ} + k \cdot 360^{\circ}$$ and $$\theta = 270^{\circ} + k \cdot 360^{\circ}$$ in degrees, or $$\theta = \dfrac{\pi}{2} + k \cdot 2\pi$$ and $$\theta = \dfrac{3\pi}{2} + k \cdot 2\pi$$ in radians, where k is any integer.

Step by step solution

01

Add 1 to both sides of the equation

Add 1 to both sides of the equation to isolate \(\sin^2{\theta}\): $$\sin^2{\theta} - 1 + 1 = 0 + 1$$ This simplifies to: $$\sin^2{\theta} = 1$$
02

Take the square root of both sides of the equation

To further isolate \(\theta\), take the square root on both sides of the equation. Keep in mind that there are two possible solutions, as this involves the use of trigonometric functions of the angle: $$\sqrt{\sin^2{\theta}} = \sqrt{1}$$ That leads to: $$\sin{\theta} = \pm 1$$
03

Find the angle whose sine values are 1 and -1

Now, we need to find out which angles give a sine value equal to 1 and -1. The angles are: $$\theta = \arcsin(1)$$ and $$\theta = \arcsin(-1)$$ When the angle's sine value is 1, the angle is 90 degrees (or pi/2 radians) along with its coterminal angles, i.e., angles that differ by 360 degrees or (2pi radians). For the equation where the sine value is -1, the angle is 270 degrees (or 3pi/2 radians) and the coterminal angles.

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