Chapter 5-1 Angles and Their Measure
Obj: to change from radians to degrees and vice versa, find angles that are co-terminal with a given angle, and find
the reference angle for a given angle.
An Angle is formed by the rotation of two rays and one fixed endpoint (the
vertex).
Fixed ray: Initial Side
Rotating ray: Terminal Side
Standard Position: Vertex at the origin, Initial side along the positive x-axis
Positive Angle:
Counterclockwise
Rotation
Negative Angle:
Clockwise Rotation
Quadrant Angle:
Terminal side
coincides with one of
the axes
Angle Measure - Most common units: Degree & Radians* (*we will use OFTEN)
Degree
1° =
1
360
of a complete revolution in the positive direction
1° = 60 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 = 3600 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
Ex. 1
1) Change 29° 45′ 26" to a decimal number of degrees (to the nearest thousandth)
2) Change 34° 29′ 19" to a decimal number of degrees (to the nearest thousandth)
Radians and the unit circle – (The definition of a radian is based on the unit circle)
Unit Circle – a circle of radius 1 whose center is at the origin of a rectangular coordinate
system. The circle is symmetric with respect to the x-axis, the y-axis and the origin
A point (𝑥, 𝑦) is on the unit circle if and only if its distance from the
origin is 1.
and…
For each point (𝑥, 𝑦) on the unit circle:
𝑥2 + 𝑦2 = 1
The measure of an angle in standard position is defined by the
corresponding arc on the unit circle.
𝑚∡𝛼 = 𝑠 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 ∗ (𝑠 𝑤𝑖𝑙𝑙 𝑖𝑛𝑐𝑙𝑢𝑑𝑒 𝜋) → ∗think circumference!
Degree ⟺ Radians Conversions
One complete revolution:
360° = 2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
Half way around:
180° = 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
(about 57.3°)
Ex.2
1) Change 30° to radian measure in terms of 𝜋
2) Change
3𝜋
4
radians to degree measure
3) Change 45° to radian measure in terms of 𝜋
4) Change
7𝜋
8
radians to degree measure
(about 0.017 radians)
Coterminal Angles - are angles in standard position with the same
terminal side.
For example, angles measuring 120° 𝑎𝑛𝑑 − 240° (𝑎𝑛𝑑 480° ) are ALL
coterminal. There are infinitely many coterminal angles
To find a measure of a coterminal angle is to add or subtract:
Multiples of 360° to angle 𝛼
𝜶 ± 𝟑𝟔𝟎°𝑲
Multiple of 2𝜋 when in radians to angle 𝛽 .
𝜷 ± 𝟐𝝅 ∙ 𝑲
Find the measures of a positive angle and a negative angle that are coterminal with each
given angle.
1) 𝜃 = 380°
2) 𝜃 = −120°
3)
11𝜋
4
Identify ALL the angles that are coterminal with a 60° angle.
Identify ALL the angles that are coterminal with a
NOTE: Commonly Used Angle in Trigonometry
𝜋
3
angle.
4)
13𝜋
4
Reference Angles
Reference Angle - for an angle 𝜃 in standard position, the reference
angle is the positive acute angle formed by the terminal side of 𝜃 and
the x-axis. (given angle is nonquadrantal)
Examples: Find the measures of the reference angle for each given angle (tip…sketch the angle)
𝑎) 150°
𝑑)
5𝜋°
4
5-1: Angles and Their Measure
𝑏) 280°
𝑒)
13𝜋
3
p.245/ 25-61 odds
𝑐) 425°
𝑓) −
17𝜋
6
5-1: Angles and Their Measure
p.245/ 25-61 odds
Name __________________________________________________ Date _________ Per ____
Ch 5-1 Angles: Degree and Radians
If each angle has the given measures and is in standard position, determine the quadrant in
which the terminal side lies.
1) −
8𝜋
3
2)
7𝜋
3)
8
13𝜋
3
4) −
3𝜋
5
Change each degree to radian measure in terms of 𝜋.
5) 200°
6) 75°
7) −570°
8) 405°
11) 1.75
12) 17.46
Change each radian to degree measure
9)
𝜋
3
10)
4𝜋
3
Find one positive angle and one negative angle that are coterminal with each angle.
13) −60°
14)
11𝜋
6
Find the reference angle for each angle with the given measures.
15) 30°
16) 130°
17)
9𝜋
4
18)
23𝜋
6
Change each degree measure to radian measure to the nearest thousandth (change to
degree decimal for first, then...convert to radian)
19) 55° 22′
20) 250° 49’ 15”