Question

Sin²x/4-cos²x/4=1/2

Answer 1

Решение
sin²x/4-cos²x/4=1/2
- (cos²x/4 - sin²x/4) = 1/2
cos²x/4 - sin²x/4 = - 1/2
cos[2*(x/4)] = - 1/2
cosx/2 = - 1/2
x/2 = +-arccos(-1/2) + 2πk, k ∈ Z
x/2 = +- [π - arccos(1/2)] + 2πk, k ∈ Z
x/2 = +- [π - π/3)] + 2πk, k ∈ Z
x/2 = +- [2π/3)] + 2πk, k ∈ Z
x = +- [4π/3)] + 4πk, k ∈ Z

Answer 2

To solve the equation sin²x/4 - cos²x/4 = 1/2, we can follow these steps:

Step 1: Combine the terms on the left side of the equation. Step 2: Use trigonometric identities to simplify the equation. Step 3: Solve for x.

Step 1: Combine the terms on the left side of the equation: (sin²x - cos²x)/4 = 1/2

Step 2: Use the trigonometric identity sin²θ - cos²θ = -cos(2θ): -cos(2x)/4 = 1/2

Step 3: Now, we need to isolate the cosine term. Multiply both sides by -4 to get rid of the fraction:

cos(2x) = -2

Step 4: To find x, we'll need to take the inverse cosine (also known as arccos) of both sides:

2x = arccos(-2)

Step 5: Since the range of the inverse cosine function is limited to [-π, π], we'll need to check if -2 is within that range. However, the range of the cosine function is [-1, 1], so there's no real value of x that makes cos(2x) equal to -2.

Therefore, there are no real solutions to the equation sin²x/4 - cos²x/4 = 1/2.