To find the value of sin 75°, we can use the sine addition formula. The sine addition formula states that:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
In this case, we can express 75° as the sum of two angles whose sine and cosine values we know. A common choice is:
75° = 45° + 30°
Now, applying the sine addition formula:
sin(75°) = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°)
Next, we need the sine and cosine values for 45° and 30°:
- sin(45°) = √2/2
- cos(45°) = √2/2
- sin(30°) = 1/2
- cos(30°) = √3/2
Substituting these values into our formula:
sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2)
This simplifies to:
sin(75°) = (√6/4) + (√2/4) = (√6 + √2)/4
Thus, the value of sin 75° is:
sin 75° = (√6 + √2)/4