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

Study concepts, example questions & explanations for Precalculus

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Example Questions

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

Find angle A of the following triangle:

Using_inverse_sin_to_find_angle_of_triangle

Possible Answers:

None of the other answers

Correct answer:

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?

Possible Answers:

Correct answer:

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

Evaluate:  

Possible Answers:

Correct answer:

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 #4 : Evaluate Expressions That Include The Inverse Sine Or Cosine Function

Evaluate:  

Possible Answers:

Correct answer:

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 #5 : Evaluate Expressions That Include The Inverse Sine Or Cosine Function

Find the value of .

Possible Answers:

Correct answer:

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 #6 : 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 

Possible Answers:

Correct answer:

Explanation:

The easiest way to solve this problem is to simplify the original expression. 

To find its inverse, let's exchange  and 

Solving for 

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