# Precalculus : Evaluate Expressions That Include the Inverse Sine or Cosine Function

## Example Questions

### Example Question #1 : Evaluate Expressions That Include The Inverse Sine Or Cosine Function

Find angle A of the following triangle:

Explanation:

We are given the hypotenuse and the side opposite of the angle in question. The trig function that relates these two sides is SIN. Therefore, we can write:

In order to solve for A, we need to take the inverse sin of both sides:

which becomes

### Example Question #2 : Inverse Sine And Cosine Functions

Consider  , where theta is valid from .  What is a possible value of theta?

Explanation:

Solve for theta by taking the inverse sine of both sides.

Since this angle is not valid for the given interval of theta, add  radians to this angle to get a valid answer in the interval.

### Example Question #1 : Evaluate Expressions That Include The Inverse Sine Or Cosine Function

Evaluate:

Explanation:

First evaluate .

To evaluate inverse cosine, it is necessary to know the domain and range of inverse cosine.

For:

The domain  is only valid from .

is only valid from .

The part is asking for the angle where the x-value of the coordinate is .  The only possibility on the unit circle is the second quadrant.

Next, evaluate .

Using the same domain and range restrictions, the only valid angle for the given x-value is in the first quadrant on the unit circle.

Therefore:

### Example Question #1 : Evaluate Expressions That Include The Inverse Sine Or Cosine Function

Evaluate:

Explanation:

To find the correct value of , it is necessary to know the domain and range of inverse cosine.

Domain:

Range:

The question is asking for the specific angle when the x-coordinate is half.

The only possibility is located in the first quadrant, and the point of the special angle is

The special angle for this coordinate is .

### Example Question #1 : Inverse Sine And Cosine Functions

Find the value of .

Explanation:

In order to determine the value or values of , it is necessary to know the domain and range of the inverse sine function.

Domain:

Range:

The question is asking for the angle value of theta where the x-value is  under the range restriction.  Since  is located in the first and fourth quadrants, the range restriction makes theta only allowable from .  Therefore, the theta value must only be in the first quadrant.

The value of the angle when the x-value is  is  degrees.

### Example Question #3 : Evaluate Expressions That Include The Inverse Sine Or Cosine Function

Find the inverse of the function

Make sure the final notation is only in the forms including , and