Answer: We can use the fact that the sine function is an odd function, which means that:
sin(-x) = -sin(x)
So, we have:
sin(-15 degrees) = -sin(15 degrees)
We can use the sum and difference identity for sine, which states that:
sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
Letting a = 45 degrees and b = 30 degrees, we get:
sin(45 degrees - 30 degrees) = sin(45 degrees)cos(30 degrees) - cos(45 degrees)sin(30 degrees)
We can use the fact that cos(45 degrees) = sin(45 degrees) = √2 / 2 and cos(30 degrees) = √3 / 2 and sin(30 degrees) = 1 / 2 to simplify:
sin(15 degrees) = (√2 / 2)(√3 / 2) - (√2 / 2)(1 / 2)
sin(15 degrees) = (√6 - √2) / 4
Therefore, we have:
sin(-15 degrees) = -sin(15 degrees) = -[(√6 - √2) / 4] = (-√6 + √2) / 4
So the exact value of sin(-15 degrees) is (-√6 + √2) / 4.
Explanation: