Pre- Calculus
Lesson 5-1: Angles and Degree Measure
Vertex:
The endpoint of two rays that form an angle.
Terminal Side
Initial Side:
Terminal Side
The side of the angle that is fixed.
The side of the angle that will rotate or move.
Standard Position:
Vertex
Initial Side
An angle with its vertex at the origin and its initial side along the positive x-axis.
Common measure used to measure an angle. Degree is
Degree: based on 60 rather than 10 as we use in decimals. Therefore
1
1o is equal to of the measure of equilateral triangle.
60
Minutes:
The degree is subdivided into 60 equal parts called minutes.
Seconds:
The minute is divided into 60 equal parts known as seconds.
Example 1:
a.
Geographic locations are typically expressed in terms of latitude and longitude.
Las Vegas, Nevada, is located at about 36.175° north latitude. Change to
36.175° to degrees, minutes, and seconds.
36.175 = 36 o + (0.175 . 60)’
= 36o + 10.5’
36o 10’ 30”
= 36o + 10’ + (0.5 . 60)”
= 36o + 10’ + 30”
b.
Las Vegas is also located at 115° 8 11 west longitude. Write 115 ° 8 11 as a
decimal rounded to the nearest thousandth.
1
1
115o + (8 . 60)’ + (11 . 3600 )”
About 115.1363889…
115o + 0. 133333…. + 0.003055555…
One Rotation of a Circle =
Clockwise =
About 115.136
360o
If rotation is clockwise it is a negative angle.
Counterclockwise =
If rotation is counterclockwise it is a positive angle.
Example 2:
a.
Give the angle measure represented by each rotation.
3.75 rotations clockwise
b.
4.2 rotations counterclockwise
3.75 . (-360)
4.2 . 360
-1350o
Coterminal angles:
1512o
Two angles in standard position, if they have the same terminal side.
If ∝ is the degree measure of an angle, then all angles measuring ∝ + 360k, where k is an integer, are
coterminal with ∝.
Example 3:
a.
Identify all angles that are coterminal with each angle. Then find one
positive angle and on negative angle that are coterminal with the angle.
42°
b.
•
All angles having a measure of
where k is an integer.
•
Positive Angle is 42o + 360(2) = 42 + 720
= 762o
42o
+
360ko,
128°
• All angles having a measure of 128o + 360k,
where k is an integer.
• Positive Angle is 128o + 360(3) = 128 + 1080
= 1208o
• Negative Angle is 42o + 360(-2) = 42 + (-720)• Negative Angle is 128 + 360(-3) = 128 + -1080
= -952o
= -678o
Example 4:
a.
If each angle is in standard position, determine a coterminal angle that is
between 0° and 360°. State the quadrant in which the terminal side lies.
445°
b.
445
Take
to determine how many complete
360
rotations there are.
1.236111…
Now determine the remaining degrees of .236111
by .236111 ∙ 360 = About 85o, Quadrant I
Reference Angle:
-2408°
Take
−2408
360
=
-6.68888888…
-.68888888 ∙ 360 =
About -248
Coterminal angle needs to be positive:
360 – 248 = 112o , Quadrant II
The acute angle formed by the terminal side of the given angle and the x-axis.
Reference Angle Rule
For any angle ∝, 0o < ∝ < 3600, its reference angle ∝′ is defined by:
a. ∝ , when the terminal side is in Quadrant I
b. 180 - ∝ , when the terminal side is in Quadrant II
c.
∝ - 180 , when the terminal side is in Quadrant III
d. 360 - ∝ , when the terminal side is in Quadrant IV
Example 5:
Find the measure of the reference angle for each angle.
a. 240°
Since 240o is between 180 and 270
the terminal side of the angle lies in
Quadrant III.
240 – 180= 60o
b. -305°
A coterminal angle of -305 is 360 – 305 = 55
Since 55o is between 90 and 0, the terminal
side of the angle lies in Quadrant I.
The reference angle is 55o.