# What is the sum and difference of sin 165 degrees?

Though I'm not sure I fully grasp your intentions, here's my attempt:

and so

By signing up, you agree to our Terms of Service and Privacy Policy

The sum and difference identities for sine are as follows:

Sum: sin(A + B) = sin(A)cos(B) + cos(A)sin(B) Difference: sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

For sin(165°), we can express it as sin(150° + 15°).

Using the sum identity: sin(150° + 15°) = sin(150°)cos(15°) + cos(150°)sin(15°)

sin(150°) = -√3/2 cos(15°) = √(1 - cos^2(15°)) = √(1 - (cos(30°))^2) = √(1 - (√3/2)^2) = √(1 - 3/4) = √(1/4) = 1/2 cos(150°) = -√3/2 sin(15°) = sin(30° - 15°) = sin(30°)cos(15°) - cos(30°)sin(15°) = (√3/2)(1/2) - (1/2)(1/2) = √3/4 - 1/4 = (√3 - 1)/4

Substituting the values: sin(165°) = (-√3/2)(1/2) + (-√3/2)((√3 - 1)/4) = -√3/4 - √(3 - 1)/4 = -√3/4 - √(2)/4 = -(√3 + √2)/4

Therefore, the sum of sin(165°) is -(√3 + √2)/4.

Using the difference identity: sin(150° - 15°) = sin(150°)cos(15°) - cos(150°)sin(15°)

Substitute the values: sin(150° - 15°) = (-√3/2)(1/2) - (-√3/2)((√3 - 1)/4) = -√3/4 + √(3 - 1)/4 = -√3/4 + √(2)/4 = (-√3 + √2)/4

Therefore, the difference of sin(165°) is (-√3 + √2)/4.

By signing up, you agree to our Terms of Service and Privacy Policy

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7