 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.

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 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.

 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.

5 12 13 θ Determine the measure of θ.

5 12 13 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√3 5 10 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.

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