<< Chapter < Page Chapter >> Page >
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

Why is b in the answer
Dahsolar Reply
how do you work it out?
Brad Reply
answer
Ernest
heheheehe
Nitin
(Pcos∅+qsin∅)/(pcos∅-psin∅)
John Reply
how to do that?
Rosemary Reply
what is it about?
Amoah
how to answer the activity
Chabelita Reply
how to solve the activity
Chabelita
solve for X,,4^X-6(2^)-16=0
Alieu Reply
x4xminus 2
Lominate
sobhan Singh jina uniwarcity tignomatry ka long answers tile questions
harish Reply
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
ZAHRO Reply
If  , , are the roots of the equation 3 2 0, x px qx r     Find the value of 1  .
Swetha Reply
Parts of a pole were painted red, blue and yellow. 3/5 of the pole was red and 7/8 was painted blue. What part was painted yellow?
Patrick Reply
Parts of the pole was painted red, blue and yellow. 3 /5 of the pole was red and 7 /8 was painted blue. What part was painted yellow?
Patrick
how I can simplify algebraic expressions
Katleho Reply
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?
Mary Reply
23yrs
Yeboah
lairenea's age is 23yrs
ACKA
hy
Katleho
Ello everyone
Katleho
Laurene is 46 yrs and Mae is 23 is
Solomon
hey people
christopher
age does not matter
christopher
solve for X, 4^x-6(2*)-16=0
Alieu
prove`x^3-3x-2cosA=0 (-π<A<=π
Mayank Reply
create a lesson plan about this lesson
Rose Reply
Excusme but what are you wrot?

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask