SOLUTION: Use the sum or difference identities to find the exact value of the trigonometric function. tan([23pi]/[12])

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Question 1032966: Use the sum or difference identities to find the exact value of the trigonometric function.
tan([23pi]/[12])
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Look up half angle formulas on Wikipedia.

Answer by ikleyn(52795) About Me  (Show Source):
You can put this solution on YOUR website!
.
Use the sum or difference identities to find the exact value of the trigonometric function.
tan([23pi]/[12])
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

tan%2823pi%2F12%29 = tan%28-pi%2F12%29 = -tan%28pi%2F12%29 = -tan(15°) = -sin%2815%5Eo%29%2Fcos%2815%5Eo%29.

sin(15°) = sqrt%28%281-cos%2830%5Eo%29%29%2F2%29 =   (formula of half argument for sine, see the lesson  Trigonometric functions of half argument  in this site)

= sqrt%28%281+-+sqrt%283%29%2F2%29%2F2%29 = sqrt%28%282-sqrt%283%29%29%2F4%29 = sqrt%282-sqrt%283%29%29%2F2.


cos(15°) = sqrt%28%281%2Bcos%2830%5Eo%29%29%2F2%29 =   (formula of half argument for sine, see the lesson  Trigonometric functions of half argument  in this site)

= sqrt%28%281+%2B+sqrt%283%29%2F2%29%2F2%29 = sqrt%28%282%2Bsqrt%283%29%29%2F4%29 = sqrt%282%2Bsqrt%283%29%29%2F2.


Therefore,  tan(15°) = sqrt%282-sqrt%283%29%29%2Fsqrt%282%2Bsqrt%283%29%29.

You can rationalize the denominator in the last expression by multiplying the numerator and the denominator by sqrt%282-sqrt%283%29%29. You will get 

tan(15°) = sqrt%282-sqrt%283%29%29%2Fsqrt%282%2Bsqrt%283%29%29 =  = %282-sqrt%283%29%29%2Fsqrt%284-sqrt%283%29%5E2%29 = 2-sqrt%283%29.

Finally,  tan%2823pi%2F12%29 = -tan%2815%5Eo%29 = -%282-sqrt%283%29%29 = sqrt%283%29-2  (negative value).

It is your answer. The problem is solved.


See also the lesson  Trigonometric functions of half argument - Examples,  Example 2.