### All Precalculus Resources

## Example Questions

### Example Question #1 : Circular Functions

What is the exact value of ?

**Possible Answers:**

**Correct answer:**

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:

**Possible Answers:**

**Correct answer:**

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

Simplify by combining the three terms.

### Example Question #3 : Circular Functions

What is the value of ?

**Possible Answers:**

**Correct answer:**

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:**

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:

**Possible Answers:**

**Correct answer:**

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 #6 : Circular Functions

Find the value of

.

**Possible Answers:**

**Correct answer:**

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 .