SOLUTION: Excet velu of sin22.5?

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Question 907355: Excet velu of sin22.5?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
if you are looking for the exact value of sin(22.5), you would do that as follows.

2 * 22.5 = 45
since you know that the sin of 45 is sqrt(2)/2, you can use the pythagorean identity for a half angle formula to find the exact value of sin(22.5).

the half angle formula is:

sin%28a%2F2%29 = plus or minus sqrt%28%281-cos%28a%29%29%29%2F2

a = 45 degrees.
cos(a) = sqrt(2)/2

replace cos(a) in the equation with sqrt%282%29%2F2 and you get:

sin%28a%2F2%29 = plus or minus sqrt%28%281-sqrt%282%29%2F2%29%2F2%29

this can be simplified to sin%28a%2F2%29 = plus or minus sqrt%28%28%282-sqrt%282%29%29%2F2%29%2F2%29 which can be further simplified to sin%28a%2F2%29 = plus or minus sqrt%28%282-sqrt%282%29%29%2F4%29 which can be even further simplified to sin%28a%2F2%29 = plus or minus sqrt%28%282-sqrt%282%29%29%29%2F2

since your angle is in the first quadrant, then your solution will be positive.

your solution is sin(22.5) = sqrt%28%282-sqrt%282%29%29%29%2F2
you can confirm with your calculator.

using the calculator, sin(22.5) = .3826834324

also using the calculator, sqrt%28%282-sqrt%282%29%29%29%2F2 = .3826834324

most of the double and half angle formulas you need in high school trigonometry can be found in the following reference.

http://regentsprep.org/Regents/math/algtrig/ATT14/formulalesson.htm