1 7.1 Angles and Their Measure In this section, we will study the following topics: Terminology used...
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Transcript of 1 7.1 Angles and Their Measure In this section, we will study the following topics: Terminology used...
1
7.1 Angles and Their Measure
In this section, we will study the following topics:
Terminology used to describe angles
Degree measure of an angle
Radian measure of an angle
Converting between radian and degree measure
3
Standard Position
Vertex at origin
The initial side of an angle in standard position is always located on the positive x-axis.
5
Measuring Angles
The measure of an angle is determined by the amount of
rotation from the initial side to the terminal side.
There are two common ways to measure angles, in
degrees and in radians.
******************************************************************
We’ll start with degrees, denoted by the symbol º.
One degree (1º) is equivalent to a rotation of
of one revolution. 1
360
7
Acute and Obtuse Angles
Acute angles have measure between _____º and _____º.
Obtuse angles have measure between ____º and _____º.
Straight angles measure _______º.
8
Angles are often classified according to the
QUADRANT in which their terminal sides lie.
Example:
50º is a ____ quadrant angle.
208º is a ____ quadrant angle. II I
-75º is a _____ quadrant angle. III IV
Classifying Angles
9
Classifying Angles
Standard position angles that have their terminal side on
one of the axes are called QUADRANTAL ANGLES.
For example, 0º, 90º, 180º, 270º, 360º, … are quadrantal
angles.
10
1. Sketch in standard position. In which quadrant is located?
2. Sketch in standard position. In which quadrant is located?
194
278.1
Sketching Angles (Degrees)
11
Complementary and Supplementary Angles
Complementary Angles
Two positive angles are complementary if their sum is ______º
Angles that measure 22º and ____º are complements.
Supplementary Angles
Two positive angles are supplementary if their sum is _______º
Angles that measure 137º and ____º are supplements.
12
In general, for in radians,
A second way to measure angles is in radians.
Radian Measure
s
r
Definition of Radian:
One radian is the measure of a central angle that intercepts arc s equal in length to the radius r of the circle.
13
Radian Measure
2 radians corresponds to 360
radians corresponds to 180
radians corresponds to 902
2 6.28
3.14
1.572
15
Conversions Between Degrees and Radians
1. To convert degrees to radians, multiply degrees by
2. To convert radians to degrees, multiply radians by
180
180
Example
Convert from Degrees to Radians: 210º
210
16
Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
b) Convert from radians to degrees: 3.8
3
4
3
4
3.8
17
Conversions Between Degrees and Radians
Try it!
c) Convert from degrees to radians (exact):
d) Convert from radians to degrees:13
6
13
6
675
675
18
Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places):
f) Convert from radians to degrees (to nearest tenth): 1 rad
5252
1
20
1. Sketch in standard position. In which quadrant is located?
2. Sketch in standard position. In which quadrant is located?
2
3
Sketching Angles (Radians)
3
4
21
1 minute (1’) = degree OR 1° = ______ minutes
1 second (1”) = _____ minute OR 1’ = _______ seconds
1 second (1”) = _____ degree OR 1° = ______ seconds
Example
Convert to decimal degrees: 52 15'42"
Degrees, minutes, and seconds
22
Conversions between decimal degrees and degrees, minutes,seconds can be easily accomplished using your TI graphingcalculator.
The ANGLE menu on your calculator has built-in features for converting between decimal degrees and DMS.
Degrees, minutes, and seconds
Note that the seconds (“) symbol is not in the ANGLE menu.
Use for “ symbol.
23
Using your TI graphing calculator,
Convert to decimal degrees to the nearest hundredth of a degree.
Convert 57.328° to an equivalent angle expressed to the nearest second.
14 32 '18"
Practice