Final answer:
The exact value of sin(165°) can be found using the sum formula for sine, which, after substituting the known values for sin(120°), cos(45°), cos(120°), and sin(45°), yields the answer √6 - √2 / 4.
Step-by-step explanation:
The question asks for the exact value of sin(165°). To find this, we can use the sum formula for sine, which is sin(a + b) = sin(a)cos(b) + cos(a)sin(b). In this case, we can think of 165° as 120° + 45° or alternatively 60° + 105°. This approach will involve using known values for sine and cosine at these angles.
Let's proceed with the first option (120° + 45°):
- sin(120°) = √3/2
- cos(45°) = √2/2
- cos(120°) = -1/2
- sin(45°) = √2/2
Now, substitute these values into the sine sum formula:
sin(165°) = sin(120°)cos(45°) + cos(120°)sin(45°)
= (√3/2)(√2/2) + (-1/2)(√2/2)
= √6/4 - √2/4
= (√6 - √2)/4
Therefore, the correct answer is (b) √6 - √2 / 4.