1. Parts of angles and dms
Dr. Shildneck

2. Angles and Their Measures
Recall that an angle is defined as two non‐colinear rays that share a common endpoint, called the v ertex.
An angle in (circle) trigonometry has a fixed (starting) side, called the i nitial side ‐ which points in the direction of the positive x‐axis.
The v ertex is the origin.
The second side is rotated (like the hands of a clock) about the vertex. This side is called the t erminal side.
An angle with this placement is said to be in S tandard Position.

3. y
terminal side
initial side x

4. The measure of the angle describes the amount and direction of the rotation necessary to move from the initial side to the terminal side.
A positive angle indicates that the terminal side has been rotated counter‐clockwise.
A negative angle indicates that the terminal side has been rotated clockwise.

5. y x
Positive Angle
Negative Angle

6. The most common angular unit of measure is called the "degree," which is equivalent to 1/360 of a full rotation around a circle.
Key (basic) Angle Measures (counterclockwise)
0o ‐ No rotation
90o ‐ ¼ of a rotation
180o ‐ ½ of a rotation
270o ‐ ¾ of a rotation
360o ‐ 1 full rotation (back to the initial side)
These angles are called the "Quadrantal Angles" since they end on the boundaries between the quadrants of the coordinate plane.

7. Angles based on rotation can be any size and either sign (positive or negative).
If you were to continue to rotate the terminal side past 360o, you would simply continue to add (or subtract if negative) to the angle.
120o
-240o
**Note that Additional rotations are indicated by a rotation mark.

8. Degrees – Not exact
Obviously the terminal side can end up not on a "degree" mark. Thus, you can have measures that use parts of a degree.
There are two ways that this might be indicated.
1. Decimal form: This form simply indicates the "fractional" part of the degree.
Example: 35.75o ‐ means 35 degrees and 3/4 more of a degree

9. 2. Degrees-Minutes-Seconds (DMS)
A more formal "part form" of a degree uses minutes and seconds.
The conversion ratio between decimal form and DMS is like that of a clock.
1 degree is made up of 60 minutes ( 1o = 60') 1 minute is made up of 60 seconds (1' = 60")

10. Converting by Hand – Dimensional Analysis
(1) Convert o into DMS form
Answer: _____o ____’ _____”
How many “whole” degrees?
How many “whole” seconds?
How many “whole” minutes? 65
0.755 degrees x 60 minutes/1 degree = 45.3 minutes
0.3 minutes x 60 seconds/1 minute = 18 seconds 18 45
(2) Convert 20o 21' into Decimal form
Answer: ___. ___ o
How many “part” degrees?
How many “whole” degrees? 20
21 minutes x 1 degree/60 minutes = 0.35 degrees 35

11. Type 65.75382 into your calculator. Press [2nd][APPS] (angle)
Converting by Calculator
Convert o into DMS form
Type into your calculator.
Press [2nd][APPS] (angle)
Press [4] (>DMS)
Press [ENTER]
ANSWER: 65o 45’ ”
Convert 20o15' 54" into Decimal form
Type 20, Press [2nd][APPS][1]
Type 15, Press [2nd][APPS][2]
Type 54, Press [ALPHA][+]
Press [ENTER]
ANSWER: o 10

12. ASSIGNMENT 2
WS ‐ Measures of Angles 11