Final answer:
The exact value of sin(75°) is found using the sum of angles formula for sine, resulting in (sqrt(6) + sqrt(2))/4.
Step-by-step explanation:
To find the exact value of sin(75°), we can use the sum of angles formula for sine, which states that sin(A + B) = sin(A)cos(B) + cos(A)sin(B). The angle 75° can be expressed as the sum of two angles whose sine and cosine values we know exactly, such as 45° and 30°. Therefore, using the formula,
sin(75°) = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°).
Knowing that sin(45°) = √2/2, cos(30°) = √3/2, cos(45°) = √2/2, and that sin(30°) = 1/2, we can substitute these values into the equation:
sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4.
Thus, the exact value of sin(75°) is (√6 + √2)/4.