Evaluate the following expression in terms of x Evaluate the

Evaluate the following expression in terms of x.

Evaluate the following expression in terms of x. sin[tan^-1(x/2)]

Solution

We can use the principles of \"SOH-CAH-TOA\":

tan1(x/2)= is the angle when tan()=x / 2.

Since tan()=opposite / adjacent, we know that opposite=x and adjacent=2

Using the Pythagorean Theorem, we can see that the hypotenuse of a right triangle with legs x and 2 has hypotenuse=x2+4..

Now, to find sin(tan1(x/2)), find sin for the triangle where

opposite=x
adjacent=2
hypotenuse=x2+4

Since sin=opposite / hypotenuse, we see that

==> x / x2+4