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Solve the following equation.

2sin^2 2x = 1

User Eek
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Final answer:

To solve the equation 2sin² 2x = 1, we first simplify it to sin² 2x = 1/2, then take the square root of both sides to get two sets of solutions: 2x = 45° + n⋅1° or 2x = 135° + m⋅1°. We then solve for x by dividing by 2, resulting in the final solutions x = 22.5° + n⋅180° or x = 67.5° + m⋅180°, where n and m are integers.

Step-by-step explanation:

We need to solve the following equation: 2sin² 2x = 1. Our first step is to divide both sides by 2 to simplify the equation.

sin² 2x = ½

Now, we take the square root of both sides, remembering that we will get both positive and negative roots:

sin 2x = ±√½

sin 2x = ±√(1/2)

Since √(1/2) is the sine of both 45° (or π/4 radians) and 135° (or 3π/4 radians), we consider both when seeking solutions for 2x.

2x = 45° + n⋅1°, where n is an integer (since the sine function is periodical with period 360°).

or

2x = 135° + m⋅1°, where m is an integer.

Finally, we divide by 2 to solve for x:

x = 22.5° + n⋅180°

or

x = 67.5° + m⋅180°

The values of x are all the angles that satisfy the original equation 2sin² 2x = 1.

User Foosion
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