# Find the exact value of sin π/12 and tan π/12 by taking π/12 = π/3 - π/4Show complete solution and explain the answer.

sciencesolve | Certified Educator

You need to use the following formulas:

`sin (alpha - beta) = sin alpha*cos beta - sin beta*cos alpha`

`tan (alpha - beta) = (tan alpha - tan beta)/(1 + tan alpha*tan beta)`

Substituting `alpha = pi/3, beta = pi/4`  yields:

`sin (pi/3- pi/4) = sin pi/3*cospi/4 - sin pi/4*cos pi/3`

Substituting `sin pi/3,cos pi/3, sin pi/4, cos pi/4`  by its values yields:

`sin (pi/3- pi/4) = (sqrt3/2)*(sqrt2/2) - (sqrt2/2)*(1/2)`

`sin (pi/3- pi/4) = (sqrt6 - sqrt2)/4`

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

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

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

Expanding the binomial yields:

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

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

Hence, evaluating the values of`sin pi/12`  and`tan pi/12 ` yields: `sin (pi/12) = (sqrt6 - sqrt2)/4 ; tan (pi/12) = 1 - sqrt3.`