Help with compound angle formulae (exact value) for angles over 120 degrees?

[misplaced1]

1. I don't understand coumpound angle formulae, for example sin(x+y), sin(x-y) etc. I'm supposed to solve using angles in the special triangles--so I use angles like 45, 30, 60, or those angles converted into radians--and add them to get the answer. For example: sin15 would be solved using sin(x-y) or sin (45-30). But when I have a question like "find the exact value of tan165", I'm lost as to what to do.

Question:
Solve the following using exact values:
a)tan165

b)tan 13π/12

2. Relevant Eqns:
tan(x+y)= tanx+tany / 1-tanxtany
tan(x-y)= tanx-tany / 1+tanxtany 3. I'm not at all sure how to go about solving it.

 

[Mark44]

1. I don't understand coumpound angle formulae, for example sin(x+y), sin(x-y) etc. I'm supposed to solve using angles in the special triangles--so I use angles like 45, 30, 60, or those angles converted into radians--and add them to get the answer. For example: sin15 would be solved using sin(x-y) or sin (45-30). But when I have a question like "find the exact value of tan165", I'm lost as to what to do.

Question:
Solve the following using exact values:
a)tan165

b)tan 13π/12




2. Relevant Eqns:
tan(x+y)= tanx+tany / 1-tanxtany
tan(x-y)= tanx-tany / 1+tanxtany





3. I'm not at all sure how to go about solving it.

There are only a few angles that have "nice" trig functions that we can represent exactly - 0, 30, 45, 60, and 90 degrees are the ones in the first quadrant. Using the sum and difference formulas and the double-angle and half-angle formulas, we can get a few more.

Notice that 165 degrees = 180 degrees - 15 degrees. Does that suggest an identity that you could use to get the exact value of tan(165 degrees)?

13π/12 = π + π/12, and π/12 = 15 degrees, so tan(13π/12) = tan(π + π/12) = ?

When you say "I don't understand coumpound angle formulae" there's not a whole lot we can do. Do you have any specific questions on these formulas?

 

[misplaced1]

Thanks for your reply.
In answer to "Notice that 165 degrees = 180 degrees - 15 degrees. Does that suggest an identity that you could use to get the exact value of tan(165 degrees)?", I asked someone about this and they explained to me that I could use the special triangle angles, but just use them in a different quadrant. So 165 degrees would be in the second quadrant with an acute angle of 15 degrees. If this is the case, would the answer to tan165 would be negative because it is in the second and therefore cosine quadrant? And is solving for tan15 in the second quadrant the answer or do I have to add anything to that?