Question

HELP A GIRL IN NEED

Answer 1

Recall the half-angle identity:

cos²(x) = 1/2 (1 + cos(2x))

Let x = 75°, so that 2x = 150°. Then

cos²(75°) = 1/2 (1 + cos(150°))

You might already be aware that cos(150°) = -√3/2, so

cos²(75°) = 1/2 (1 - √3/2)

cos²(75°) = 1/2 - √3/4

cos²(75°) = (2 - √3)/4

But this is the square of the number we want, which we solve for by taking the square root of both sides. This introduces a second solution, however:

cos(75°) = ± √[(2 - √3)/4]

cos(75°) = ± √(2 - √3)/2

75° falls between 0° and 90°, and you should know that cos(x) is positive for x between these angles. This means cos(75°) must be positive, so we pick the positive root:

cos(75°) = √(2 - √3)/2