SOLUTION: use an identity to evaluate sin (2 invesre of sin 5/13)

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Question 595833: use an identity to evaluate sin (2 invesre of sin 5/13)
Answer by jsmallt9(3759) About Me  (Show Source):
You can put this solution on YOUR website!
sin%282%2Asin%5E-1%285%2F13%29%29
Broadly speaking this expression is a reference to the sin of twice some angle. This suggests that the identity to use is the one for sin(2x):
sin(2x) = 2*sin(x)*cos(x)

Applying this pattern to your expression we get:
2%2Asin%28sin%5E-1%285%2F13%29%29%2Acos%28sin%5E-1%285%2F13%29%29

sin%28sin%5E-1%285%2F13%29%29 is "the sine of the angle whose sin is 5/13". So it obviously has a value of 5/13.

cos%28sin%5E-1%285%2F13%29%29 is "the cosine of the angle whose sin is 5/13". Its value is not so obvious. To find it you can use the Pythagorean identity:
sin%5E2%28x%29+%2B+cos%5E2%28x%29+=+1
Substituting in our sin value this becomes:
%285%2F13%29%5E2+%2B+cos%5E2%28x%29+=+1
To solve for cos(x) we start by simplifying:
25%2F169+%2B+cos%5E2%28x%29+=+1
Subtract the fraction:
cos%5E2%28x%29+=+1+-+25%2F169
cos%5E2%28x%29+=+169%2F169+-+25%2F169
cos%5E2%28x%29+=+144%2F169
Find the square root of each side (discarding the negative square root) we get:
cos%28x%29+=+12%2F13

Now that we have values for sin%28sin%5E-1%285%2F13%29%29 and cos%28sin%5E-1%285%2F13%29%29 we can substitute them into 2%2Asin%28sin%5E-1%285%2F13%29%29%2Acos%28sin%5E-1%285%2F13%29%29:
2%2A%285%2F13%29%2A%2812%2F13%29
Simplifying we get:
%2810%2F13%29%2A%2812%2F13%29
120%2F169
This fraction does not reduce so
sin%282%2Asin%5E-1%285%2F13%29%29+=+120%2F169