 When dealing with right triangles, if we want to compare the ratio of the opposite side to an angle and the hypotenuse of the triangle, we use the sine function.

opposite adjacent hypotenuse θ

 When we know the angle θ, using the sine function is just fine, however what happens if we do not know the angle θ?

 This is where the inverse sine function comes in.  The inverse sine function helps determine the angle θ, if we happen to know the appropriate side lengths of the right triangle.

 The inverse sine function is not the multiplicative inverse of the sine function.  The inverse sine function is the inverse function of the sine function.  So what is the difference?

Sine multiplied by its multiplicative inverse. Sine of inverse sine. Inverse sine of sine.

 When you multiply a function by its multiplicative inverse, you end up with 1.  When you plug the inverse function of some function into the function, you end up with the variable the function was analyzing.

 Because of this property, we can determine an angle of a right triangle if we happen to know the lengths of the side opposite the angle and the hypotenuse of the of the right triangle, using inverse sine.

θ Determine the measure of θ.

θ Plug-in values.

Sine and inverse sine cancel each other out. Use a calculator to make this calculation. Measure of θ

3 4 5 θ Determine the measure of θ.

θ 5√ Determine the measure of θ.

 For practice problem 1, θ is equal to approximately 37 degrees.  For practice problem 2, θ is equal to 30 degrees.