SOLUTION: How do I solve this? Use the exact values of the sine and cosine of 2π/3 and π /4, and the angle sum identity for sine, to find the exact value of sin(11 π /12).

Algebra ->  Trigonometry-basics -> SOLUTION: How do I solve this? Use the exact values of the sine and cosine of 2π/3 and π /4, and the angle sum identity for sine, to find the exact value of sin(11 π /12).      Log On


   

Question 873813: How do I solve this? Use the exact values of the sine and cosine of 2π/3 and π /4, and the angle sum identity for sine, to find the exact value of sin(11 π /12).
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The angle sum identity for sine is
sin%28A%2BB%29=sin%28A%29%2Acos%28B%29%2Bcos%28A%29%2Asin%28B%29
Since 2pi%2F3%2Bpi%2F4=8pi%2F12%2B3pi%2F12=11pi%2F12 ,
you can use the angle sum identity (above),
and the values for sine and cosine of 2pi%2F3 and pi%2F4
to find sin%2811pi%2F12%29=sin%282pi%2F3%2Bpi%2F4%29 .
sin%28pi%2F4%29 and cos%28pi%2F4%29 are easy. pi%2F4 is 45%5Eo ,
and sin%28pi%2F4%29=cos%28pi%2F4%29=sqrt%282%29%2F2 .

2pi%2F3 is a little harder because it is in quadrant II.

sin%282pi%2F3%29=sin%28pi-2pi%2F3%29=sin%28pi%2F3%29=sqrt%283%29%2F2
cos%282pi%2F3%29=-cos%28pi-2pi%2F3%29=-cos%28pi%2F3%29=-1%2F2

So, applying sin%28A%2BB%29=sin%28A%29%2Acos%28B%29%2Bcos%28A%29%2Asin%28B%29 with A=2pi%2F3 and B=pi%2F4

=
That could be "simplified" different ways:

or

Those expressions are the exact value of sin%2811pi%2F12%29