Question

Use the sum and difference identities to find the exact values of the sine, cosine, and tangent of the angle.
165 = 135 + 30

Answer 1

2+2=4

Answer 2

Okay, so for sine, sin(x+y) = sin(x)cos(y) + cos(x)sin(y). With x=135 and y=30, we have
\[\sin(165) = {\sqrt{2} \over 2}{\sqrt{3} \over 2} +{\sqrt{2} \over 2}{1 \over 2} = {\sqrt{6} + \sqrt{2} \over 4}\]
Along the same lines, cos(x+y) = cos(x)cos(y) - sin(x)sin(y), so we have
\[\cos(165) = {\sqrt{2} \over 2}{\sqrt{3} \over 2} - {\sqrt{2} \over 2}{1 \over 2} = {\sqrt{6} - \sqrt{2} \over 4}\]
Now I guess you could use the identity for tangent, but it's easier at this point to say
\[\tan(165) = {\sin(165) \over \cos(165)} = {\sqrt{6} + \sqrt{2} \over 4} {4 \over \sqrt{6} - \sqrt{2}} = {\sqrt{6} + \sqrt{2} \over \sqrt{6} - \sqrt{2}} \]
Which, by multiplying both the top and bottom by sqrt(6)+sqrt(2), simplifies to\[{8 + 4\sqrt{3} \over 4} = 2 + \sqrt{3}\]I skipped a little arithmetic because it was kinda messy, but I hope this is good enough. If you actually have to use the tangent identity directly just say so and I'll go through it.