Final answer:
The exact value of sin(75°) is found using the angle sum identity for sine with the sum of 45° and 30°, resulting in (√2 + √3)/2√2.
Step-by-step explanation:
To find the exact value of sin(75°), we can use the angle sum identity for sine, which is sin(a + b) = sin(a)cos(b) + cos(a)sin(b). Since 75° can be expressed as the sum of 45° and 30°, which are angles whose sine and cosine values are known, we can calculate sin(75°) using these special angles.
Let's break it down:
- sin(75°) = sin(45° + 30°)
- sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°)
- sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2)
- sin(75°) = √2√3/4 + √2/4
- sin(75°) = (√6 + √2)/4
- sin(75°) = (√2 + √3)/2√2
The exact value of sin(75°) is therefore (√2 + √3)/2√2.