SOLUTION: how do i find the exact value of sin(tan^-1 3/8) without using a calculator?

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Question 958899: how do i find the exact value of sin(tan^-1 3/8) without using a calculator?
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
sin[tan-1(3/8)]

This says "Find the sin of the angle whose tangent is 3/8"

So we sketch a right triangle that has an angle whose
tangent is 3/8.

Since the tangent is the opposite over the adjacent, and
3/8 is 3 over 8, we the opposite side as 3, and the adjacent
side as 8.


 
The angle marked with the red arc is tan-1(3/8),
the angle whose tangent is 3/8.

Now we just need the sine of that angle marked.  (Its tangent is 3/8.)

Since the sine is the opposite over the hypotenuse, we will need
to find the hypotenuse:

We use the Pythagorean theorem:

hypotenuse%5E2=adjacent%5E2%2Bopposite%5E2

hypotenuse%5E2=8%5E2%2B3%5E2

hypotenuse%5E2=64%2B9

hypotenuse%5E2=73

hypotenuse=sqrt%2873%29

So now we can label the hypotenuse:



Now since the original problem is

sin[tan-1(3/8)]

We find the sine by putting the opposite side 3 over the hypotenuse √73

So the answer is 

sin[tan-1(3/8)] = 3/√73

If you like you can rationalize the denominator and get 3√73/73

Edwin