How do you express cos(pi/ 3 ) * sin( ( pi) / 8 ) cos(π3)sin(π8) without using products of trigonometric functions?

1 Answer
Jul 11, 2016

P = (1/4)sqrt(2 - sqrt2)P=(14)22

Explanation:

P = cos (pi/3).sin (pi/8)P=cos(π3).sin(π8)
Trig table --> cos (pi/3) = 1/2.
There for, P can be expressed as P = (1/2)sin (pi/8)P=(12)sin(π8)
We can evaluate sin (pi/8) by using trig identity:
cos 2a = 1 - 2sin^2 a
cos (2pi)/8 = cos (pi/4) = sqrt2/2 = 1 - 2sin^2 (pi/8)cos(2π)8=cos(π4)=22=12sin2(π8)
2sin^2 (pi/8) = 1 - sqrt2/2 = (2 - sqrt2)/22sin2(π8)=122=222
sin^2 (pi/8) = (2 - sqrt2)/4sin2(π8)=224
sin (pi/8) = +- sqrt(2 - sqrt2)/2sin(π8)=±222
Since sin (pi/8)sin(π8) is positive, then,
P = (1/4)sqrt(2 - sqrt2)P=(14)22