Use sum and difference identities to evaluate trig functions

Question

Use sum and difference identities to evaluate trig functions

1. find the exact value of sin 165 degree
2. find the exact value of cos -7pi/12

Answer 1

sin165 = sin15

You can see that by sketching them on a unit circle

sin15 = sin30/2 Use the half angle formula, at least to check your answer even if you want to do it a harder way.

sinA/2 = square root of (1-cosA)/2

sin15 = square root of (1-cos30)/2

Then you might notice cos(7pi/12) = cos255 = -cos75 = -sin15

so part 2 of the question has a solution equal to the negative of part 1

But you want to use the difference formulas

sin165 = sin(135+30) = sin135cos30 + cos135sin30 = sqr2/2 x sqr3/2 - sqr2/2 x 1/2

=sqr6-sqr2 all over 4

sin135=sin45=sqr2/2

cos135=-cos45 = -sqr2/2

cos 30 =sqr3/2

sin30 = 1/2

same with cos(-7pi/12) where you can break -7pi/12 = 255 degrees into 2 parts

cos(180+75) or cos(pi + 5pi/12) and use the formula

cos(A+B) = cos^2(A)cos^2(B)- sin^2(A)sin^2(B) where A=pi or 180 and B = 75 or 5pi/12

-7pi/12 has the same cosine as the angle -7pi/12 +2pi = pi+5pi/12. Add 360 degrees and all the trigonometric values are the same.