SOLUTION: 10. find the exact value of the following by using the specified method. a. sin 157.5 degrees by using a half-angle identity b. cos 7pi/12 by using a sum identity

Question 1171885: 10. find the exact value of the following by using the specified method.
a. sin 157.5 degrees by using a half-angle identity
b. cos 7pi/12 by using a sum identity
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
sin (315/2)
sin (x/2)=+/-sqrt((1-cos x)/2); cosine is positive in 4th quadrant and sine is positive in the second quadrant.
=sqrt((1-cos 315)/2)
= sqrt (1- sqrt(2)/2)/2
= sqrt [(2-sqrt(2)/4)
=(1/2)(sqrt (2- sqrt(2))
-
cos 7π/12=cos (4π/12+3π/12)
this is cos π/3*cosπ/4-sin π/3*sin π/4
=(1/2)*sqrt(2)/2)-sqrt(3)/2*sqrt(2)/2
=(1/2-sqrt(3/2))*sqrt(2)/2
=(1-sqrt(3))/2*sqrt(2)/2
=sqrt(2)/4-sqrt(6)/4 or (1/4)*(sqrt(2)-sqrt(6))

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