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Chapter 1: Problem 41

Solve the following equations. $$\sqrt{2} \sin x-1=0$$

Short Answer

Expert verified
Answer: The general solution for the given trigonometric equation can be expressed as $$x = n\pi \pm \frac{\pi}{4}$$, where n is an integer.

Step by step solution

01

Add 1 to both sides of the equation

To isolate sin(x) and solve for x, we will first add 1 to both sides of the equation: $$\sqrt{2} \sin x - 1 + 1 = 0 + 1$$ This simplifies to: $$\sqrt{2} \sin x = 1$$
02

Isolate sin(x)

Next, divide both sides of the equation by $$\sqrt{2}$$ to isolate sin(x): $$\sin x = \frac{1}{\sqrt{2}}$$
03

Take the inverse sine of both sides

Now, we will take the inverse sine (arcsin) of both sides to find the value(s) of x: $$x = \arcsin \left(\frac{1}{\sqrt{2}}\right)$$
04

Find the general solutions

The general solutions to this equation can be expressed as: $$x = n\pi \pm \arcsin \left(\frac{1}{\sqrt{2}}\right)$$ where n is an integer.
05

Simplify and find specific solutions

We know that $$\arcsin \left(\frac{1}{\sqrt{2}}\right)$$ is equal to $$\frac{\pi}{4}$$. So the general solutions can be simplified to: $$x = n\pi \pm \frac{\pi}{4}$$ Here, x can be any angle that satisfies the general solution, with n being any integer value. For specific solutions, plug in the values of n (0, 1, -1, and so on) and find the respective values of x.

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