Parts of an Angle

(the fixed side)

(the rotating side)

alpha – common angle name

Each angle above is said to be in the “standard position” – the vertex is at the origin and the initial side is on the positive x-axis.

Example 1

(FYI: The ‘ is read as minutes; the “ is read as seconds)

Quadrantal AngleAn angle in the standard position in which the terminal side coincides with one of the axes.

Examples:

Example 2 (Past 360°)

Coterminal Angles

k360225

Two angles in standard position that have the same terminal side.

All angles have an infinite number of coterminal angles.

k360Coterminal angles are in the form of:

where k is some integer.

Example 3

Begin with the generic form to identify all coterminal angles:

Choose a positive integer for k to find one positive angle:

Choose a negative integer for k to find one negative angle:45 + 360(1) = 405°

45 + 360(-2) = -675°

b. 225°

a. 45° k36045

225 + 360(2) = 945°

45 + 360(-1) = -135°

45º

405º (1 loop)765º (2

loops)765)2(36045

405)1(36045

45)0(36045

Example 4

152777778.2360

775

55

360152777778.0

a. 775° k360

In other words, we need to find the value of alpha in

Find the number of rotations (k) by dividing the degree by 360:

Determine the leftover degrees:

Method 1 Method 2:

55

775720

7752360

775360

k(Partial rotation)

k = 2

b. -1297°

602777778.3360

1297

k = -3

217-

3606027777778.0

Implies that the coterminal angle should be positive.

217

12971080

12973360

775360

k

To convert to a positive angle:

143217360

Which quadrant does the terminal side of each lie in?

Quadrant 2

Example 5 Reference angle: an acute angle formed by the terminal side of a given angle and the x-axis.

a. 120°

Visualize it:

120°Since it’s in Quadrant II:

60120180180

b. -135°

-135°Since it’s in Quadrant III:

45180225180

Convert to a positive angle: 360 – 135 = 225

225°

HW: Page 280