Precalculus : Find the value of the sine or cosine functions of an angle

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Circular Functions

What is the exact value of ?

Possible Answers:

Correct answer:

Explanation:

The sine of an angle corresponds to the y-component of the triangle in the unit circle.  The angle  is a special angle.  In the unit circle, the hypotenuse is the radius of the unit circle, which is 1.  Since the angle is , the triangle is an isosceles right triangle, or a 45-45-90.  

Use the Pythagorean Theorem to solve for the leg. Both legs will be equal to each other.

Rationalize the denominator.

Therefore, .

Example Question #1 : Find The Value Of The Sine Or Cosine Functions Of An Angle

Evaluate: 

Possible Answers:

Correct answer:

Explanation:

To evaluate , break up each term into 3 parts and evaluate each term individually.

Simplify by combining the three terms.

 

Example Question #2 : Find The Value Of The Sine Or Cosine Functions Of An Angle

What is the value of  ?

Possible Answers:

Correct answer:

Explanation:

Convert  in terms of sine and cosine.

Since theta is  radians, the value of  is the y-value of the point on the unit circle at  radians, and the value of  corresponds to the x-value at that angle.

The point on the unit circle at  radians is .  

Therefore,  and .  Substitute these values and solve.

Example Question #4 : Circular Functions

Solve:  

Possible Answers:

Correct answer:

Explanation:

First, solve the value of .  

On the unit circle, the coordinate at  radians is .  The sine value is the y-value, which is .  Substitute this value back into the original problem.

 

Rationalize the denominator.

Example Question #3 : Find The Value Of The Sine Or Cosine Functions Of An Angle

Find the exact answer for:  

Possible Answers:

Correct answer:

Explanation:

To evaluate , solve each term individually.

 refers to the x-value of the coordinate at 60 degrees from the origin.  The x-value of this special angle is .

 refers to the y-value of the coordinate at 30 degrees.  The y-value of this special angle is .

 refers to the x-value of the coordinate at 30 degrees.  The x-value is .

Combine the terms to solve .

Example Question #4 : Find The Value Of The Sine Or Cosine Functions Of An Angle

Find the value of 

.

Possible Answers:

Correct answer:

Explanation:

The value of  refers to the y-value of the coordinate that is located in the fourth quadrant.

This angle  is also  from the origin.  

Therefore, we are evaluating .

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