Determine the exact value of tan 7pi/12 by using the angles pi/4 an pi/3. Please show steps on how to solve.

Expert Answers

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You should use the sum `pi/3 + pi/4`  instead of result `7pi/12`   such that:

`tan 7pi/12 = tan (pi/3 + pi/4)`

You need to use the following formula such that:

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

Reasoning by analogy yields:

`tan (pi/3 + pi/4) = (tan (pi/3)+ tan (pi/4))/(1 - tan (pi/3)*tan (pi/4))`

Using `tan (pi/3) = sqrt3`  and `tan (pi/4) = 1`  yields:

`tan (pi/3 + pi/4) = (sqrt3 + 1)/(1 - sqrt3)`

`tan (pi/3 + pi/4) = ((sqrt3 + 1)^2)/(1 - 3)`

`tan (pi/3 + pi/4) = (4 + 2sqrt3)/(-2)`

`tan (pi/3 + pi/4) = -2 - sqrt3`

Hence, evaluating the tangent of angle `7pi/12 ` using the sum `pi/3 + pi/4`  yields `tan 7pi/12 =-2 - sqrt3` .

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