Download Angles and their Measures

Section 5.1 – Day 1
Angles and their Measures
Date______________
Angle: a shape or measure made between two rays that share a fixed endpoint/vertex
 Initial Side – the fixed/stationary ray (This will usually be on the positive x-axis.)
 Terminal Side – the ray that rotates
 Positive Angle – occurs when the terminal side rotates counterclockwise
 Negative Angle – occurs when the terminal side rotates clockwise
 Standard Position – the vertex of the angle is at the origin and the initial side
is on the positive x-axis

Quadrantal Angle – an angle whose initial side is in standard position and
whose terminal side is on an axis

Co-terminal Angle: an angle whose terminal side coincides with another
angle’s terminal side from standard position
Angle Measures: Two types – degrees and radians
Converting between Degrees, Minutes & Seconds (D°M’S”) & Decimal Degrees:
Parts of degrees are measure in minutes and seconds.
______ of a degree.
One second = 1” = ______ of a degree.

One minute = 1’ =


Conversion Ideas: 1 counterclockwise revolution = _____. 1° = _____. 1’ = ____.
Radians: (Think of angles inside circles whose vertex is at the center of the circle.)
Central angle is an angle whose ___________ is at the center of a circle.
One radian is the measure of the angle you get from a central angle when the arc on the
circle is the same length as the radius of the circle.
The radian measure of an angle is the _________ of the arc length to the radius of a
circle that the central angle intersects.
Derivation of a radian:
Theorem: For a circle of radius r, a central angle of θ radians subtends an arc whose
length s is
_______.
Converting between Degrees and Radians:
Think of circumference from Geometry. ________. Now relate it to arc length, s = rθ.
One revolution = circumference = the distance around a circle = arc length of one circle
So, ... 1 revolution = ____ radians = ______ degrees
½ revolution = ___ radians = _____ degrees
Therefore: 360° = 2π radians; 180° = 1π radians; 1° =

180
radians;
1 radian =
180

°
Ex 1 Change the following degree measures to radians in terms of π.
a) 45°
b) -150°
c) 270°
d) 345°
Ex 2 Change the following radian measures to degrees.

4
3
a)
b)
c) 
3
6
4
d)
7
8
Common Conversions of degrees and radians: (See p.331 of your book for a table format.)