To find the value of sin 75°, we can use the angle addition formula for sine. The angle 75° can be expressed as the sum of two angles that are easier to work with: 45° and 30°. Thus, we can write:
sin 75° = sin(45° + 30°)
Using the angle addition formula, which states that:
sin(a + b) = sin a * cos b + cos a * sin b
we can replace a with 45° and b with 30°:
sin 75° = sin 45° * cos 30° + cos 45° * sin 30°
Now, we need the sine and cosine values for 45°, 30°, and 60°:
- sin 45° = cos 45° = √2 / 2
- sin 30° = 1 / 2
- cos 30° = √3 / 2
Substituting these values into our equation:
sin 75° = (√2 / 2) * (√3 / 2) + (√2 / 2) * (1 / 2)
Now, simplify the expression:
sin 75° = (√6 / 4) + (√2 / 4)
Combining the fractions gives:
sin 75° = (√6 + √2) / 4
Therefore, the final value of sin 75° is:
sin 75° = (√6 + √2) / 4 ≈ 0.9659
This simplified form gives both an exact value as well as an approximate decimal representation, which is useful in different contexts.