Pre-Cal Question
 

Arg, I've been working at this for like half an hour and I can't figure it out. I'm getting a couple different answers, but I'm not sure which is right. I've done most of the work, it's just the simplifying the answer I'm stuck on. The question is this:

Use the sum or difference identity to find the exact value of the trigonometric function tan(165)

I've gotten it reduced to this
http://www.boomspeed.com/spskelly/math.GIF
Sorry for the sloppiness, I just scribbled that in paint, but I think it is understandable. I asked a few friends and got varying answers, so I thought I might come here and see if anyone here has any good input. What I can't figure out is how exactly to simplify that. I know it can get more simplified, I'm just not sure how. Thanks a lot! :)

You can simplify it to (-3+sqrt(3))/(3+sqrt(3)), but that's about it.

They might also want you to rationalize the denominator. This involves multiplying the top and bottom by (3-sqrt(3)). Even it has zero effect on the value of your answer and ever though all you're really doing is fucking around with algebra, some teachers prefer the denominator to be rationalized.

Yeah, do what filtherton said, then you get a much prettier answer without any fractions.

Simplification may make things look nice, but it sure is a bitch.

I'd check over my work again, according to pg. 39 of this - http://homepage.smc.edu/geltner_peter/new4to6.pdf

The '1' constant in the numerator shouldn't be negative.

BTW, with that in mind, you end up with 2 + Sqrt(3) as the answer.

[If you don't change the negative sign, you get -2 + sqrt(3)

I'm not sure what you read but in that example the '1' in the numerator is negative. The answer has to yield a negative answer because the tangent of an angle is negative in quadrants II and IV, and 165 degrees is in quadrant II.

so we have tan(165^o)
=tan(135^o + 30^o)

tan(a + b) = (tan(a) + tan(b))/(1 - tan(a)tan(b))

=(tan(135^o) + tan(30^o)) / (1 - tan(135^o)tan(30^o))

tan(135^o) = -1
tan(30^o) = 3^1/2 / 3

=(-1 + 3^1/2/3)/(1 - (-1)(3^1/2/3))
=(-1 + 3^1/2/3)/(1 + 3^1/2/3)
=(-3 + 3^1/2)/(3 + 3^1/2)
=(-3 + 3^1/2)(3 - 3^1/2) / (3 + 3^1/2)(3 - 3^1/2)

= 3^1/2 - 2