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In this section you will:
  • Draw angles in standard position.
  • Convert between degrees and radians.
  • Find coterminal angles.
  • Find the length of a circular arc.
  • Use linear and angular speed to describe motion on a circular path.

A golfer swings to hit a ball over a sand trap and onto the green. An airline pilot maneuvers a plane toward a narrow runway. A dress designer creates the latest fashion. What do they all have in common? They all work with angles, and so do all of us at one time or another. Sometimes we need to measure angles exactly with instruments. Other times we estimate them or judge them by eye. Either way, the proper angle can make the difference between success and failure in many undertakings. In this section, we will examine properties of angles.

Drawing angles in standard position

Properly defining an angle first requires that we define a ray. A ray    is a directed line segment. It consists of one point on a line and all points extending in one direction from that point. The first point is called the endpoint of the ray. We can refer to a specific ray by stating its endpoint and any other point on it. The ray in [link] can be named as ray EF, or in symbol form E F .

Illustration of Ray EF, with point F and endpoint E.

An angle    is the union of two rays having a common endpoint. The endpoint is called the vertex    of the angle, and the two rays are the sides of the angle. The angle in [link] is formed from E D and E F . Angles can be named using a point on each ray and the vertex, such as angle DEF , or in symbol form D E F .

Illustration of Angle DEF, with vertex E and points D and F.

Greek letters are often used as variables for the measure of an angle. [link] is a list of Greek letters commonly used to represent angles, and a sample angle is shown in [link] .

θ φ or ϕ α β γ
theta phi alpha beta gamma
Illustration of angle theta.
Angle theta, shown as θ

Angle creation is a dynamic process. We start with two rays lying on top of one another. We leave one fixed in place, and rotate the other. The fixed ray is the initial side     , and the rotated ray is the terminal side    . In order to identify the different sides, we indicate the rotation with a small arrow close to the vertex as in [link] .

Illustration of an angle with labels for initial side, terminal side, and vertex.

As we discussed at the beginning of the section, there are many applications for angles, but in order to use them correctly, we must be able to measure them. The measure of an angle    is the amount of rotation from the initial side to the terminal side. Probably the most familiar unit of angle measurement is the degree. One degree    is 1 360 of a circular rotation, so a complete circular rotation contains 360 degrees. An angle measured in degrees should always include the unit “degrees” after the number, or include the degree symbol ° . For example, 90  degrees = 90° .

To formalize our work, we will begin by drawing angles on an x - y coordinate plane. Angles can occur in any position on the coordinate plane, but for the purpose of comparison, the convention is to illustrate them in the same position whenever possible. An angle is in standard position    if its vertex is located at the origin, and its initial side extends along the positive x -axis. See [link] .

Graph of an angle in standard position with labels for the initial side and terminal side.  The initial side starts on the x-axis and the terminal side is in Quadrant II with a counterclockwise arrow connecting the two.

If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a positive angle    . If the angle is measured in a clockwise direction, the angle is said to be a negative angle    .

Questions & Answers

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
2x + 7 =19
2x +7=19. 2x=19 - 7 2x=12 x=6
because x is 6
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
4x=3-2 4x=1 x=1+4 x=5 5x
hi can you give another equation I'd like to solve it
what is the value of x in 4x-2+3
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
then x=-1/4
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
can you help me on this topic of Geometry if l help you
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
how do I attempted a trig number as a starter
Tumwe Reply
cos 18 ____ sin 72 evaluate
Het Reply

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