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.