# Precalculus : Angles in the Coordinate Plane

## Example Questions

### Example Question #1 : Find The Degree Measure Of An Angle

Explanation:

Write the conversion factor between radians and degrees.

Cancel the radians unit by using dimensional analysis.

### Example Question #1 : Find The Degree Measure Of An Angle

Convert  to degrees.

Explanation:

Write the conversion factor of radians and degrees.

Substitute the degree measure into .

### Example Question #3 : Find The Degree Measure Of An Angle

Determine the angle  in degres made in the plane by connecting a line segment from the origin to .

Assume

Explanation:

Firstly, since the point  is in the 3rd quadrant, it'll be between  and . If we take  to be the horizontal, we can form a triangle with base and leg of values  and . Solving for the angle in the 3rd quadrant given by

Since this angle is made by assuming  to be the horizontal, the total angle measure  is going to be:

### Example Question #4 : Find The Degree Measure Of An Angle

Find the degree measure of  radians.  Round to the nearest integer.

Explanation:

In order to solve for the degree measure from radians, replace the  radians with 180 degrees.

The nearest degree is .

### Example Question #5 : Find The Degree Measure Of An Angle

Given a triangle, the first angle is three times the value of the second angle.  The third angle is .  What is the value of the second largest angle in degrees?

Explanation:

A triangle has three angles that will add up to  degrees.

Convert the radians angle to degrees by substituting  for every .

The third angle is 60 degrees.

Let the second angle be .  The first angle three times the value of the second angle is .  Set up an equation that sums the three angles to .

Solve for .

Substitute  for the first angle and second angle.

The second angle is:

The first angle is:

The three angles are:

The second highest angle is:

### Example Question #1 : Find The Measure Of A Coterminal Angle

Find the coterminal angle of 15 degrees.

Explanation:

The coterminal angles can be positive or negative.  To find the coterminal angles, simply add or subtract 360 degrees as many times as needed from the reference angle.

All of these angles are coterminal angles.

### Example Question #2 : Find The Measure Of A Coterminal Angle

Of the given answers, what of the following is a coterminal angle of  radians?

Explanation:

To find the coterminal angle of an angle, simply add or subtract  radians, or 360 degrees as many times as needed.

These are all coterminal angles to  radians.

### Example Question #3 : Find The Measure Of A Coterminal Angle

Of the following choices, find a coterminal angle of .

Explanation:

In order to find a coterminal angle, simply add or subtract  radians to the given angle as many times as possible.

The possible angles after subtracting decrements of  radians are:

Out of the given possibilities, only  is a valid answer.

### Example Question #4 : Find The Measure Of A Coterminal Angle

Find the coterminal angle of 15 degrees in standard position from the following answers.

Explanation:

To determine the coterminal angle, simply add or subtract increments or decrements of 360 degrees to the given angle.

For :

These angles can all be coterminal to 15 degrees.  The only answer is .

### Example Question #5 : Find The Measure Of A Coterminal Angle

Find the coterminal angle of , if possible.