Verified answer

sin(105) = sin(75) = sin(45+30) = sin(45)cos(30) + cos(45)sin(30) = [sqrt(2)/2][sqrt(3)/2 + 1/2] = (1/4)[sqrt(6) + sqrt(2)]

cos(105) = -cos(75) = -cos(45+30) = sin(45)sin(30) - cos(45)cos(30) = [sqrt(2)/2][1/2 - sqrt(3)/2] = (1/4)[sqrt(2) - sqrt(6)]

tan(105) = sin(105)/cos(105) = [sqrt(6) + sqrt(2)]/[sqrt(2) - sqrt(6)] = -2 - sqrt(3)

Any idiot can enter these into a calculator and get some crappy rounded-off decimal, but obviously that's not what you're being asked to do here. You CAN calculate the exact values of these.

105 is 60+45. So take sin(60+45) and apply the addition formula for sine. You get

sin(60)cos(45) + sin(45)cos(60) =

(√3 /2)(√2 /2) + (√2 /2)(1/2) =

√6 /4 + √2 / 4 =

(√6 + √2) / 4

Similarly, take cos(60+45) and apply the addition formula for cosine:

cos(60)cos(45) - sin(60)sin(45) =

(1/2)(√2 /2) - (√3 /2)(√2 /2) = etc.

Or you could just use the identity sin^2(x) + cos^2(x) = 1. Since angles in the second quadrant are positive for sine and negative for cosine, you have cos(x) = -√(1 - sin^2(x)), not cos(x) = +√(1 - sin^2(x)).

To find tan(105), just divide sin(105) by cos(105).

type "sin(105)" into google and google's calculator will answer it for you.