Angles

I. Definitions

Ray : A part of a line with a single endpoint

Angle : Two rays with a common endpoint

Initial Side

Terminal Side

Vertex

QUADRANTS IIIIVIII

An angle with its vertex at the origin and its initial side on the positive x-axis is in standard form.

Positive Angles: Angles measured from the initial side counterclockwise to its terminal side.

Negative Angles: Angles measured from the initial side clockwise to its terminal side.

II. Measuring Angles

Acute Angle : Angles between 0 and 90o

Right Angle : Angles measuring 90o

Obtuse Angle : Angles between 90o and 180o

Straight Angle : Angles that measure 180o

The size of an angle is determined by the space between the two sides (rays).These are measured in two different units:

Degrees Radians

Degrees Radians360o 2p

Degrees Radians180o p

So we can concluded180o = p radians

III. Converting between Radians and Degrees

30o30o ∙𝜋180o

radians6 =

30

120o ∙𝜋180o

120o radians3

= 2

120120o

225o ∙𝜋180o

225o radians4

= 5

225

225o

330o ∙𝜋180o

330o radians6

= 11

330330o

∙180o

𝜋

60𝜋3 ¿60o

60𝝅𝟑

∙180o

𝜋

305𝜋6 ¿150o

150𝟓𝝅𝟔

∙180o

𝜋

604𝜋3 ¿240o240

𝟒𝝅𝟑

∙180o

𝜋

457𝜋4

𝟕𝝅𝟒 315

¿315o

IV. Reference & Co-terminal Angles

Reference angle: the acute angle measured from the terminal side to the nearest x-axis.

80∘

Find the reference angle.

65∘

−295∘

80∘

260∘

30∘

−69 0∘

60∘2𝜋3

∙180o

𝜋

602𝜋3 ¿120o

2

11

120o

IV. Arc Lengths

𝑠=𝑟 ∙𝜃𝑖𝑛𝑟𝑎𝑑𝑖𝑎𝑛𝑠

q

𝑠=12 𝑓𝑡 ∙4𝜋3

12 𝑓𝑡

4

1𝑠=16𝜋 𝑓𝑡

𝑠=𝑟 ∙𝜃

4𝜋3𝑟𝑎𝑑

𝑠=4𝑚∙ 45°

45°

4𝑚

∙𝜋𝑟𝑎𝑑180∘

4

1

𝑠=4𝑚∙𝜋4

𝑠=𝜋𝑚𝑒𝑡𝑒𝑟𝑠

𝑠=𝑟 ∙𝜃