SOLUTION: What is the exact value of tan345*

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Question 731435: What is the exact value of tan345*
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
We use the fact that 345° = 315°+30°, two special angles, and the identity

,

From our knowledge of special angles:

tan(315°) = -1 and tan(30°) = 1%2Fsqrt%283%29 = sqrt%283%29%2F3

tan(345°) = tan(315°+30°) =   = %28%28-1%29%2B%28sqrt%283%29%2F3%29%29%2F%281-%28-1%29%28sqrt%283%29%2F3%29%29 = %28-1%2Bsqrt%283%29%2F3%29%2F%281%2Bsqrt%283%29%2F3%29

To simplify that, first we multiply top and bottom by 3 and distribute
to clear of fractions in the numerator and denominator:

%283%28-1%2Bsqrt%283%29%2F3%29%29%2F%283%281%2Bsqrt%283%29%2F3%29%29 = %28-3%2Bsqrt%283%29%29%2F%283%2Bsqrt%283%29%29

Now we rationalize the denominator by multiplying top and bottom by 3-√3,
multiplying out and simplifying:

 =  =  =  = %28-9%2B6sqrt%283%29-3%29%2F%289-3%29 =

%28-12%2B6sqrt%283%29%29%2F6 = %286%28-2%2Bsqrt%283%29%29%29%2F6 = %28cross%286%29%28-2%2Bsqrt%283%29%29%29%2Fcross%286%29 = -2+√3

It's always a good idea to check such answers with a calculator,
getting and comparing the decimal approximations:

By calculator tan(345°) = -.2679491924

and by calculator

-2+√3 = -.2679491924

Edwin