SOLUTION: prove that 2arccos(4/5)=arcsin 24/25

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Question 476243: prove that 2arccos(4/5)=arcsin 24/25
Found 2 solutions by lwsshak3, ccs2011:
Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
prove that 2arccos(4/5)=arcsin 24/25
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Using a calculator:
2arccos(4/5)=2*36.8699º=73.7398º
arcsin 24/25=73.7398º
Therefore, 2arccos(4/5)=arcsin 24/25

Answer by ccs2011(207) About Me  (Show Source):
You can put this solution on YOUR website!
arccos(4/5) equals the angle which satisfies the equation: cos%28phi%29+=+4%2F5
arcsin(24/25) equals the angle which satisfies the equation: sin%28theta%29+=+24%2F25
So we are proving that 2phi+=+theta
Lets use a trig property for double angles.
cos%282phi%29+=+2%2Acos%5E2%28phi%29+-+1
Substitute cos%28phi%29+=+4%2F5
cos%282phi%29+=+2%2A%284%2F5%29%5E2+-+1
cos%282phi%29+=+2%2A%2816%2F25%29+-+1
cos%282phi%29+=+32%2F25+-+1
cos%282phi%29+=+32%2F25+-+25%2F25+=+7%2F25
Now assume 2phi+=+theta is true and check if it works.
cos%282phi%29+=+cos%28theta%29+=+7%2F25
Now use a trig identity:
sin%5E2%28theta%29+%2B+cos%5E2%28theta%29+=+1
%2824%2F25%29%5E2+%2B+%287%2F25%29%5E2+=+1
%2824%5E2+%2B+7%5E2%29%2F25%5E2+=+1
625%2F625+=+1
It works
Therefore 2phi is equivalent to theta.