Angles in Standard Position

1. Drawing Angles in Standard Position

1a. The Basics of Angles

Angles are virtually everywhere in the world around us. When an airplane takes off, it does so at some angle with the horizontal. When a golfer wishes to hit a golf ball out of a sand trap, they need to determine the best angle to do so.

Given your exposure to angles before taking this course, we need to look at a more specific way to define angles, which is standard position. Before doing so, we need to define some terms.

A ray consists of the endpoint of a line and all points extending in one direction from the endpoint. An angle is the figure formed by two rays with a common endpoint, called the vertex.

When constructing an angle, start with two rays, both pointing in the same direction. Leave one ray fixed, and rotate the other. The fixed ray is called the initial side, and the rotated side is called the terminal side. To indicate the direction of the rotation, an arrow is used as follows.

In order to properly use angles, we have to know how to measure them. An angle measure is the amount of rotation from the initial side to the terminal side.

Recall that one full circular rotation measures Then, one degree is of a full rotation. It is very important to write the degree symbol when referencing the degree measure of an angle. For example, a 30-degree angle is written as

terms to know

Standard Position

An angle that has its vertex at the origin and its initial side extended along the positive x-axis.

Ray

Consists of the endpoint of a line and all points extending in one direction from the endpoint.

Angle

The figure formed by two rays with a common endpoint, called the vertex.

Angle Measure

The amount of rotation from the initial side to the terminal side.

1b. Angles in Standard Position

To better visualize angles, we use the xy-plane, where angles are drawn in standard position.

Greek letters are often used to represent angles. The most popular of these is which is pronounced “thay-ta” and spelled theta. An angle in standard position, labeled with the letter is shown in the figure.

An angle can be positive, negative, or zero:

• when the rotation is counterclockwise from the initial side to the terminal side.
• when the rotation is clockwise from the initial side to the terminal side.
• when there is no rotation between the initial side and the terminal side.

Shown below are the sketches of and

The angles below are the most recognizable in standard position since their terminal sides are one of the axes. These angles are called quadrantal angles. Note their measures:

hint

Notice that the angles and all have the same initial and terminal sides. This is a big idea coming up later!

A quadrantal angle’s terminal side coincides with the x- or y-axis.

The quadrantal angles can be referenced in order to sketch other angles. When an angle is a multiple of or we can more easily visualize the angle as a fraction of a full rotation.

For example, we know that is a quarter of a full rotation since This means that using the initial side along the positive x-axis, rotate counterclockwise until reaching the positive y-axis. This angle is as shown in the picture above.

EXAMPLE

Sketch the angle in standard position.

Since is of this means that the rotation is the size of the rotation for

Consider this picture, which shows the right angle split into three equal angles:



This means that the sketch of the angle in standard position is:

watch

In this video, we’ll sketch the angle in standard position.

Here are a few more examples of angles drawn in standard position.

Angles Drawn in Standard Position

try it

Consider the angle sketched in standard position.

What is the measure of the angle?

try it

Consider the angle

Sketch the angle in standard position.

term to know

Quadrantal Angle

An angle whose terminal side coincides with the x- or y-axis.

2. Finding Coterminal Angles

Consider the angles and both in standard position, shown in the figure.

Notice how they both have the same terminal side. When two angles in standard position share the same terminal side, the angels are called coterminal.

So far, we have only considered angles formed by rotating at most in either direction. When the measure of an angle has absolute value more than it means that the ray is rotated more than one full rotation from its initial side.

Consider the angles given below, which all share the same terminal side.

Note: is the Greek letter alpha, is the Greek letter beta, and is the Greek letter phi, pronounced “fee.”

All four of these angles are coterminal since they share the same terminal side.

Considering the angle notice the following:

• Starting at the terminal side of applying one full counterclockwise rotation to get angle which has measure
• Starting at the terminal side of applying one full clockwise rotations to get angle which has measure
• Starting at the terminal side of applying two full counterclockwise rotations to get angle which has measure

try it

Consider the angle

What is the measure of the angle obtained after applying three full counterclockwise rotations to θ?

Each counterclockwise rotation is 360°. Therefore, the measure of the angle is

What is the measure of the angle obtained after applying four full clockwise rotations to θ?

Each clockwise rotation is -360°. Therefore, the measure of the angle is

Thus, when the terminal ray is rotated by any multiple of counterclockwise or clockwise, the resulting angle has the same terminal side. This means that the measures of two coterminal angles differ by a multiple of

This gives a general rule for finding coterminal angles.

formula to know

Finding Coterminal Angles

Given an angle adding or subtracting any integer multiple of results in an angle coterminal to That is, if k is an integer, then is coterminal to

EXAMPLE

Find two angles, one positive and one negative, that are coterminal to

While you could add or subtract any multiples of we’ll add and subtract

Add

Subtract

Two coterminal angles are and

In many applications, angles that are either negative or more than can be awkward to work with. As a result of the relationship we established between coterminal angles, every angle is coterminal with an angle between and which we will find is the most convenient range of values to use.

Thus, given a negative angle or an angle larger than we can find a coterminal angle between and as follows:

• If the angle is negative, add to the angle. If the angle is positive, you are done. If it is negative, continue to add until the result is positive.
• If the angle is positive, subtract until the result is an angle between and

EXAMPLE

Find an angle coterminal to that is between and

Since is positive, subtract until we get a number between and

This is larger than so keep subtracting.

This is between and ; stop here.

Thus, the requested coterminal angle is

try it

Consider the angle

Find an angle coterminal to the given angle that is between 0° and 360°.

To find the angle, add 360° until obtaining an angle between 0° and 360°.

Therefore, the coterminal angle is 180°.

hint

When given an angle and you wish to find another angle coterminal to add or subtract to until getting an angle in the desired range.

EXAMPLE

Find an angle coterminal to that is between and

Since the desired range is between two negative angles, we subtract until our angle falls into that range.

	Subtract Note the result is not in our desired range, so we will subtract again in the next step.
	Subtract again. This time, is between and so we have found our coterminal angle.

The coterminal angle is

term to know

Coterminal Angles

Angles that share the same terminal side.

summary

In this lesson, you learned about drawing an angle in standard position, which means that its initial side (the fixed ray) is along the positive x-axis, vertex is at the origin, and the terminal side is rotated in a counterclockwise or clockwise direction. You covered some additional basics of angles, recalling that an angle measure is the amount of rotation from the initial side to the terminal side, and that one full circular rotation measures You also learned that we draw angles in standard position on the xy-plane, and angles are often represented by the Greek letter . Depending on the direction of the rotation from the initial side to the terminal side, an angle can be positive (counterclockwise rotation), negative (clockwise rotation), or zero (no rotation). Finally, you learned that two angles in standard position which share the same terminal side are called coterminal; to find a coterminal angle, add or subtract an integer multiple of to the given angle. This is especially useful to find an angle between and that is coterminal to an angle that is either negative or larger than

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.