SOLUTION: Use half-angle formulas to find the exact values. a. cos 15(degrees) b. sin 20(degress)30' I got {{{sqrt(3)}}}/2 for a and -{{{sqrt(2)}}}/2 for b which are wrong. What did I d

Algebra ->  Trigonometry-basics -> SOLUTION: Use half-angle formulas to find the exact values. a. cos 15(degrees) b. sin 20(degress)30' I got {{{sqrt(3)}}}/2 for a and -{{{sqrt(2)}}}/2 for b which are wrong. What did I d      Log On


   

Question 1159222: Use half-angle formulas to find the exact values.
a. cos 15(degrees)
b. sin 20(degress)30'
I got sqrt%283%29/2 for a and -sqrt%282%29/2 for b which are wrong. What did I do wrong?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

I'm not sure what part b is trying to say because the notation is a bit strange.

I'll show you how to do part a

cos%28x%2F2%29+=+sqrt%28+%281%2Bcos%28x%29%29%2F2%29 Half angle identity for cosine, when cosine is positive. Keep in mind that cos(15) is a positive value as the angle 15 degrees is in Q1 where cosine is positive.

cos%2830%2F2%29+=+sqrt%28+%281%2Bcos%2830%29%29%2F2%29 Plug in x = 30

cos%2815%29+=+sqrt%28+%281%2Bsqrt%283%29%2F2%29%2F2%29 Replace cos(30) with sqrt(3)/2

cos%2815%29+=+sqrt%28+%282%2F2%2Bsqrt%283%29%2F2%29%2F2%29 Rewrite 1 as 2/2

cos%2815%29+=+sqrt%28+%28%282%2Bsqrt%283%29%29%2F2%29%2F2%29 Combine the upper fractions

cos%2815%29+=+sqrt%28+%28%282%2Bsqrt%283%29%29%2F2%29%2A%281%2F2%29%29

cos%2815%29+=+sqrt%28+%282%2Bsqrt%283%29%29%2F4+%29

cos%2815%29+=+sqrt%28+2%2Bsqrt%283%29+%29%2Fsqrt%284%29 Break up the square root

cos%2815%29+=+sqrt%28+2%2Bsqrt%283%29+%29%2F2 Simplify sqrt(4) into 2. This is as simplified as it gets.


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Side note: if you use a calculator, you should find
cos%2815%29+=+0.96592582628907 (make sure your calculator is in degree mode)

sqrt%28+2%2Bsqrt%283%29+%29%2F2+=+0.96592582628907

both are approximations. Since we get the same approximation, this helps confirm the answer.

Another way to confirm the answer is to compute %28+sqrt%282%2Bsqrt%283%29%29+%29%2F%282%29-cos%2815%29 and you should get zero as a result (or some very small number due to rounding error). I'm using the idea that if x+=+y, then x-y+=+0