Answer:
Explanation:
To find the exact value of sin(-165 degrees), we can use the trigonometric identity:
sin(-θ) = -sin(θ)
Since sin(165 degrees) is positive, sin(-165 degrees) will be negative. Therefore:
sin(-165 degrees) = -sin(165 degrees)
Now, let's calculate the exact value of sin(165 degrees). We can use the fact that sin(180 degrees - θ) = sin(θ) and sin(θ) = sin(180 degrees + θ). Therefore:
sin(165 degrees) = sin(180 degrees - 15 degrees) = sin(15 degrees)
To find the exact value of sin(15 degrees), we can use the half-angle formula:
sin(θ/2) = sqrt((1 - cosθ)/2)
Let's calculate it:
θ = 15 degrees
cosθ = cos(15 degrees)
Using the half-angle formula:
sin(15 degrees) = sqrt((1 - cos(30 degrees))/2)
Now, we need to calculate cos(30 degrees). We know that cos(30 degrees) = sqrt(3)/2. Substituting this value:
sin(15 degrees) = sqrt((1 - sqrt(3)/2)/2)
To simplify this expression further, we can rationalize the denominator:
sin(15 degrees) = sqrt(2 - sqrt(3))/2
Finally, substituting this value into the equation for sin(-165 degrees):
sin(-165 degrees) = -sin(165 degrees) = -sqrt(2 - sqrt(3))/2
So, the exact value of sin(-165 degrees) is -sqrt(2 - sqrt(3))/2.