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

Example Questions

Example Question #1 : Circular Functions

What is the exact value of ?      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 #2 : Circular Functions

Evaluate:       Explanation:

To evaluate , break up each term into 3 parts and evaluate each term individually.   Simplify by combining the three terms. Example Question #1 : Circular Functions

What is the value of ?      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 #1 : Circular Functions

Solve:       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 #5 : Circular Functions

Find the exact answer for:       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 #1 : Circular Functions

Find the value of .      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 . All Precalculus Resources 