Find the exact value of sin(p/12). Use the half-angle formula.

Question

Please help

Answer 1

The half-angle formula for sin is:

 

sin (x/2) = ±√((1 - cos x) / 2)

 

Since π/12 is half of π/6, you can use π/6 as x in the formula.

 

sin π/12 = ±√((1 - cos π/6) / 2)

= ±√((1 - √3/2) / 2)          (since cos π/6 = √3/2)

= ±√(1 - √3/2) / √2

= ±√(2(1 - √3/2)) / √4      (doing this to get the radical out of the denominator)

= ±√(2 - √3) / 2

 

Then the question is, is it + or -?  Since π/12 is in quadrant I, sin π/12 must be positive.  If you work out the expression on a calculator, you'll find that it's around 0.26, so you have to use the positive value.  So the answer is:

 

√(2 - √3) / 2