What is the sum and difference of sin 165 degrees?

Question

Oliver Atkinson · Accepted Answer

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

#sin165# could also be written as #sin(120+45)#

and so

#sin126= sin(120+45)= sin120cos45+sin45cos120#

#sin165# as a difference is

#sin(210-45) -=sin210cos45-sin45cos210#

Asher Friedman · Answer

undefined 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.