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

Example Questions

Example Question #1 : Inverse Sine And Cosine Functions

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 : Evaluate Expressions That Include The Inverse Sine Or Cosine Function

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 #1052 : Pre Calculus

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 : Evaluate Expressions That Include The Inverse Sine Or Cosine Function

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 #2 : Inverse Sine And Cosine Functions

Find the inverse of the function Make sure the final notation is only in the forms including  , and      Explanation:

The easiest way to solve this problem is to simplify the original expression.  To find its inverse, let's exchange and  Solving for   All Precalculus Resources 