Question

Find the value of the product. 2 sin(37.5°) sin(82.5°)

Answer 1

The value of the product 2 sin(37.5°) sin(82.5°) is (√2 + 1)/4.

Step-by-step explanation:

To find the value of the product 2 sin(37.5°) sin(82.5°), we can use the product-to-sum identity for sine: sin(A)sin(B) = (cos(A-B) - cos(A+B))/2. Plugging in the given angles, we have:

2 sin(37.5°) sin(82.5°) = (cos(37.5° - 82.5°) - cos(37.5° + 82.5°))/2.
Next, we simplify the angles: 37.5° - 82.5° = -45° and 37.5° + 82.5° = 120°.

Substituting these values into the equation, we get:
2 sin(37.5°) sin(82.5°) = (cos(-45°) - cos(120°))/2.
Using the trigonometric values for cosine, we find:
cos(-45°) = -√2/2 and cos(120°) = -1/2.
Substituting these values back into the equation, we get:
2 sin(37.5°) sin(82.5°) = (-(-√2/2) - (-1/2))/2.
Simplifying further, we have:
2 sin(37.5°) sin(82.5°) = (√2/2 + 1/2)/2 = (√2 + 1)/4.