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Chapter 1: Problem 42

Solve the following equations. $$\sin 3 x=\frac{\sqrt{2}}{2}, 0 \leq x<2 \pi$$

Short Answer

Expert verified
Answer: The values of x that satisfy the given equation in the interval [0, 2蟺) are: $$x = \left\{\frac{\pi}{12}, \frac{\pi}{4}, \frac{3\pi}{4}, \frac{11\pi}{12}\right\}$$

Step by step solution

01

Rewrite the equation

For clarity, we change the equation from $$\sin 3 x=\frac{\sqrt{2}}{2}$$ to $$\sin(3x) = \sin{\left(\frac{\pi}{4}\right)}$$ since sin(蟺/4) = 鈭�2/2.
02

Apply the general solution formula for the sine function

To solve sin(3x) = sin(蟺/4), we can use the general solution for the sine function: $$3x = n\pi + (-1)^n \left(\frac{\pi}{4}\right)$$ where n is an integer.
03

Determine the values of 3x in the given interval

We need to find all the values of 3x in the interval [0, 6蟺), so we will try different n values to get our results: 1. When n = 0, we get 3x = 蟺/4 2. When n = 1, we get 3x = 蟺 - 蟺/4 = 3蟺/4 3. When n = 2, we get 3x = 2蟺 + 蟺/4 = 9蟺/4 4. When n = 3, we get 3x = 3蟺 - 蟺/4 = 11蟺/4 5. When n = 4, we get 3x = 4蟺 + 蟺/4 = 17蟺/4 (this value is outside the given interval for 3x) So the possible values for 3x are 蟺/4, 3蟺/4, 9蟺/4, and 11蟺/4.
04

Determine the values of x

We now divide each value of 3x by 3 to find the corresponding values of x: 1. For 3x = 蟺/4, we get x = (蟺/4) / 3 = 蟺/12 2. For 3x = 3蟺/4, we get x = (3蟺/4) / 3 = 蟺/4 3. For 3x = 9蟺/4, we get x = (9蟺/4) / 3 = 3蟺/4 4. For 3x = 11蟺/4, we get x = (11蟺/4) / 3 = 11蟺/12
05

Write the solution as a set

Now that we have found all the values of x that satisfy the given equation, we write the solution as a set: $$x = \left\{\frac{\pi}{12}, \frac{\pi}{4}, \frac{3\pi}{4}, \frac{11\pi}{12}\right\}$$

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