SOLUTION: Find the exact value of the expression sin^-1(tan 5pi/4)

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Question 727492: Find the exact value of the expression sin^-1(tan 5pi/4)
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
5pi%2F4=pi%2Bpi%2F4 and since tangent has a period of pi
tan%285pi%2F4%29=tan%28pi%2F4%29
pi%2F4 (in radians) is 45%5Eo , half of a right angle.

You should know (or quickly figure out)
sine, cosine, and tangent of 30%5Eo , 45%5Eo and 60%5Eo ,
because a right triangle with 30%5Eo and 60%5Eo angles is half of an equilateral triangle,
and a right triangle with two 45%5Eo angles (an isosceles right triangle) is half of a square.

tan%2845%5Eo%29=a%2Fa=1 and sin%2845%5Eo%29=cos%2845%5Eo%29=a%2F%28sqrt%282%29a%29=1%2Fsqrt%282%29=sqrt%282%29%2F2
So tan%285pi%2F4%29=tan%28pi%2F4%29=tan%2845%5Eo%29=1
and you should know that sin%28pi%2F2%29=sin%2890%5Eo%29=1
The angle between 0 and pi %28180%5Eo%29 with a sine of 1 is highlight%28pi%2F2%29 and that is sin^-1(tan(5pi/4))