How do you express sin(pi/ 6 ) * cos( ( pi) / 8 ) sin(π6)cos(π8) without using products of trigonometric functions?

1 Answer
Feb 17, 2017

sqrt(2 + sqrt2)/42+24

Explanation:

P = sin (pi/6).cos (pi/8)P=sin(π6).cos(π8)
Trig table gives --> sin (pi/6) = 1/2
P = (1/2)cos (pi/8)P=(12)cos(π8).
We can evaluate cos (pi/8)cos(π8) by using trig identity:
2cos^2 a = 1 + cos 2a2cos2a=1+cos2a
In this case:
2cos^2 (pi/8) = 1 + cos (pi/4) = 1 + sqrt2/2 = (2 + sqrt2)/22cos2(π8)=1+cos(π4)=1+22=2+22
cos^2 (pi/8) = (2 + sqrt2)/4cos2(π8)=2+24
cos (pi/8) = +- sqrt(2 + sqrt2)/2cos(π8)=±2+22.
Since cos (pi/8)cos(π8) is positive, take the positive answer.
Finally,
P = (1/2)cos (pi/8) = sqrt(2 + sqrt2)/4P=(12)cos(π8)=2+24