**Unformatted text preview: **4.1 Angles and Their Measures
Converting Between Units of Angle Measure
Engineers in the ancient world discovered ratios within right triangles, as they learned fundamental
Conversion Formulas
engineering skills. The secrets and patterns necessary to understanding circles and spheres helped
in the development of trigonometry. Our chapter begins the journey into trigonometry with an
introduction into angles and their measures.
Convert Degrees to Radians multiply degrees by:
180
Convert Radians to Degrees multiple radians by:
180
Angle Characteristics
An angle is determined by rotating a ray about its endpoint.
EX $4: Convert each angle from radians to degrees or from degrees to radians.
The _
Vertex is the endpoint of the ray.
( Rotates )
Terminal side
30
180
= /2109
D. 1350
The
Initial
side is the starting position.
Vertex
- 16 TT
The
Terminal
side is the position after
180
2. 20 #1 4
Initial side
B.
27
-3200
rotation.
(Doesn't)
move
- LIT
Standard Position Angles
15
Angles that are in standard position are placed on a coordinate
plane, with the vertex at the origin and the initial side on the
Terminal side
Practice With Radians
positive x-axis.
How many radians is each angle of an equilateral triangle?
Positive
angles are determined by a
counterclockwise rotation.
2. The minute hand of a clock travels how many radians in 15 minutes?
Negative
Vertex
Initial side
angles are determined by a
clockwise rotation.
3 . The minute hand of a clock travels how many radians in 10 minutes?
Angle in Standard Position
1 60 . 7805
60
Coterminal Angles
4. The minute hand of a clock travels how many radians in 5 minutes?
30#
Two angles in standard position that have the same terminal side are called coterminal angles. We
30
can find an angle that is coterminal to a given angle by adding or subtracting one revolution. So, any
5. What is the measure of O in radians, if
* = 45?
5. What is the measure of 0 in radians?
given angle has many coterminal angles. For example B = 36 is coterminal to all of the following:
396, 756, -324,-6840
* + 5 X = 180
Beta
360 - 45 = 315 0
6 * = 180
5 x
EX #1: Draw 0 = 220 and label 0' as the angle that is coterminal.
O could also be
0 = 220'
315
Theta
220'
580 or any full
150
=
220
30
* = 30'
+ 3 10
revolution
580 '
140'
2 20" + 340'ny
n is an
insever
O'= - 140
2017 Jean Adams (Flamingo Math Tha)
2017 Jean Adams (Flamingo Math TA)
Either will add 340 or sulstreet
Positive and negative
are different never...

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- Fall '08
- ZIEGLER
- coterminal angles, minute hand of a clock travels