Question

The exact value of tan5pi/12 is

Answer 1

Explanation:

This is the sum identity for tangent(x + y), where x and y are angles measured in radians. The formula for tan(x + y) is
`(tan(x)+tan(y))/(1-tan(x)tan(y))`

The first order of business is to find which 2 angles, when expressed in terms of the denominator 12, add up to equal
`(5\pi)/(12)`

We will use the first quadrant angles, and only the "important" ones:

`(\pi)/(4),(\pi)/(3),(\pi)/(6),(\pi)/(2),\pi`

When expressed in terms of the denominator of 12, these angles have the equivalent angles, in order from above:

`(3\pi)/(12),(4\pi)/(12),(2\pi)/(12),(6\pi)/(12),(12\pi)/(12)`

We need to find the 2 whose numerators add up to a 5. That would be:

`(3\pi)/(12) +(2\pi)/(12)=(5\pi)/(12)`

Remember that

`(3\pi)/(12)=(\pi)/(4)` and
`(2\pi)/(12)=(\pi)/(6)` so

angle x is
`(\pi)/(4)` and angle y is
`(\pi)/(6)`, making our tangent sum:

`tan((\pi)/(4)+(\pi)/(6))`

Filling that into our formula for the sum of tan(x + y):

`tan((\pi)/(4)+(\pi)/(6))=(tan((\pi)/(4))+tan((\pi)/(6)) )/(1-tan((\pi)/(4))tan((\pi)/(6)) )`

It just so happens that

`tan((\pi)/(4))=1` and

`tan((\pi)/(6))=(√(3) )/(3)` so our formula then becomes

`tan((\pi)/(4)+ (\pi)/(6))=(1+(√(3) )/(3) )/(1-(1)((√(3) )/(3)) )` which simplifies to

`((3+√(3) )/(3) )/((3-√(3) )/(3) )` and then bring up the lower fraction and flip it to multiply giving you:

`tan((\pi)/(4) +(\pi)/(6) )=(3+√(3) )/(3-√(3) )`

I have the feeling that you need to rationalize that denominator, and if you do that, the final answer will be:

`2+√(3)`

I know I kind of left you hanging at the very end with rationalizing, but there was so much already that went into this problem in such depth, that I didn't want to risk possibly confusing you even more than I may have already done so. Try and follow the best that you can.