Question

Find the exact value, without a

calculator.
710
6
sin
2
tan
12
6
2
7Tt/6
COS

Answer 1

`-2 -√(3)`

Explanation:

First consider numerator

`sin ((7\pi)/(6))/(2) = sin (7\pi)/(12)= sin ((\pi)/(4) + (\pi)/(3))`

Using the formula : sin (A + B) = sin A cos B + cos A sin B

`sin (\pi)/(4) = (√(2) )/(2), \ cos (\pi)/(4) = (√(2) )/(2)\\\\sin (\pi)/(3) = (√(2) )/(2), \ cos (\pi)/(3) = (1)/(2)`

`sin((\pi)/(4) + (\pi)/(3)) = sin (\pi)/(4) \cdot cos (\pi)/(3) + cos (\pi)/(4) \cdot sin (\pi)/(3)`

`=(√(2) )/(2) \cdot (1)/(2) + (√(2) )/(2) \cdot (√(3) )/(2) \\\\= (√(2) )/(4) + (√(6) )/(4)\\\\=(√(2) +√(6) )/(4)`

Second consider denominator

`cos ((7\pi)/(6))/(2) = cos (7\pi)/(12)= cos ((\pi)/(4) + (\pi)/(3))`

Using the formula : cos (A + B) = cos A cos B - sin A sin B

`sin (\pi)/(4) = (√(2) )/(2), \ cos (\pi)/(4) = (√(2) )/(2)\\\\sin (\pi)/(3) = (√(2) )/(2), \ cos (\pi)/(3) = (1)/(2)`

`cos((\pi)/(4) + (\pi)/(3)) = cos (\pi)/(4) \cdot cos (\pi)/(3) -sin (\pi)/(4) \cdot sin (\pi)/(3)`

`=(√(2) )/(2) \cdot (1)/(2) - (√(2) )/(2) \cdot (√(3) )/(2)\\\\=(√(2))/(4) - (√(6))/(4)\\\\= (√(2) -√(6) )/(4)`

Therefore,

`tan (7\pi)/(12) = (sin (7\pi)/(12))/(cos(7\pi)/(12))`

`= \frac{(√(2) +√(6) )/(4)} {(√(2) -√(6) )/(4) }\\\\=(√(2) +√(6) )/(4) * (4 )/(√(2) -√(6))\\\\=(√(2) +√(6) )/(√(2) -√(6))`

Either we can stop here or Rationalize the denominator:

`(√(2) +√(6) )/(√(2) -√(6)) * (√(2) +√(6) )/(√(2) +√(6)) = ((√(2) +√(6))^(2) )/((√(2))^2 -(√(6))^2) = (2 + 6 +2√(12) )/(2-6) = (8+2√(12) )/(-4) = (8+ 4√(3) )/(-4) = -2-√(3)`