Name: ________________
Date: ________________
Section 4.1
Angle and Angle Measure
By convention, angles measured in a counterclockwise direction are said to be
_________. Those measured in a clockwise direction are ____________.
The diagram angle AOB (see right) shows 1 radian.
Radians are one way that we can measure angles, although we are probably
much more familiar with degrees.
When thinking of radians it can be helpful to remember the formula for circumference (perimeter) of a
circle: ________
Some of the more common angles that we work with:
One full rotation is 360° or 2π radians.
One half rotation is 180° or ___ radians.
One quarter rotation is 90° or
𝜋
2
radians.
One eighth rotation is 45° or _____radians.
https://www.math.lsu.edu/~verrill/teaching/linearalgebra/math006/angles7.gif
When working with radians, we should be aware that many mathematicians omit units for radian measures.
For example,
2𝜋
3
radians may be written as
2𝜋
3
. Angle measures without units are considered to be in radians.
Definition of radian - one radian is the measure of the central angle subtended in a circle by an arc equal in
length to the __________________.
Example: 1
Draw each angle in standard position. Change each degree measure to radian measure and each radian
measure to degree measure. Give answers as both exact and approximate measures (if necessary) to the
nearest hundredth of a unit.
a. 2.57
b. 30°
a. 2.57
b. 30°
Coterminal Angles:
When you sketch an angle of 60° and an angle of 420° in
standard position, the terminal arms coincide. These are
________________________.
Example 2: Identify Coterminal Angles
Determine one positive and one negative angle measure that is coterminal with each angle. In which
quadrant does the terminal arm lie?
a. −430°
reminder quadrants
b.
8𝜋
3
a. the terminal arm of −430° is in quadrant IV
b.
are as follows →
Coterminal Angles in General Form
By adding or subtracting multiples of ________________, you can write an infinite number of angles that are
1
coterminal with any given angle. (e.g. 45° ±360° ±720° ±1080° … or 3 𝜋 ±2𝜋 ±4𝜋 ±6𝜋…)
Definition:
Any given angle has an infinite number of angles coterminal with it, since each time you make one full
rotation from the terminal arm, you arrive back at the same terminal arm. Angles coterminal with any
angle θ can be described using the expression:
θ ± (360°)n or θ ± 2πn,
where n is a natural number. This way of expressing an answer is called the general form.
Example 3: Express Coterminal Angles in General Form
a. Express the angles coterminal with 110° in general form. Identify the angles coterminal with 110°
that satisfy the domain −720° ≤ θ < 720°.
Solution: Angles coterminal with 110° occur at 110° ± (360°)𝑛, 𝑛 ∈ 𝑁.
n
110° − (360°)𝑛, 𝑛 ∈ 𝑁
1
2
3
110° + (360°)𝑛, 𝑛 ∈ 𝑁
b. Express the angles coterminal with
8𝜋
3
in general form. Identify the angles coterminal with
8𝜋
domain −4π ≤ θ < 4π.
Solution:
8𝜋
3
± 2𝜋𝑛, 𝑛 ∈ 𝑁 represents all angles coterminal with
n
1
2
3
8𝜋
3
4
8𝜋
− 2𝜋𝑛, 𝑛 ∈ 𝑁
3
8𝜋
+ 2𝜋𝑛, 𝑛 ∈ 𝑁
3
The angles in the domain −4π ≤ θ < 4π that are coterminal are −
10𝜋
3
,−
4𝜋
3
, and
2𝜋
3
3
in the
Arc Length of a Circle
𝜋
All arcs that subtend a right angle ( ) have the same central angle, but they have different arc lengths
2
depending on the radius of the circle. The arc length is proportional to the radius. This is true for any central
angle and related arc length.
Consider two concentric circles with centre O. The radius of the smaller
circle is 1, and the radius of the larger circle is r. A central angle of θ
radians is subtended by arc AB on the smaller circle and arc CD on the
larger one. You can write the following proportion, where x represents
the arc length of the smaller circle and a is the arc length of the larger
circle.
Consider the circle with radius 1 and the sector with central angle θ. The ratio of the arc length to the
circumference is equal to the ratio of the central angle to one full rotation.
This formula, a = θr, works for any circle, provided that θ is measured in radians and both a and r are
measured in the same units.
Example 4: Determine Arc Length in a Circle
Rosemarie is taking a course in industrial engineering. For an assignment, she is designing the interface of a
DVD player. In her plan, she includes a decorative arc below the on/off button. The arc has central angle
130° in a circle with radius 6.7 mm. Determine the length of the arc, to the nearest tenth of a millimetre.
Remember to convert to radians
Homework: 114-119: 1, 2, 3 a-b, 4, 5 a-b, 6, 8
4.1 Investigating Angle Measure
_____________________
Until now you have measured angles in degrees and never beyond 3600. We need to prepare ourselves for a new
angle measure and the idea that angles can be _________________ or _________________ and even bigger than
________
REFLECT AND RESPOND:
4.
Use your knowledge of circumference to show that your answer in step 3 is reasonable.
5.
Is angle AOB in step 3 greater than, equal to, or less than 600? ________________ Discuss this with the group.
6.
Determine the degree measure of angle AOB to the nearest tenth of a degree.
7.
Compare your results with those of other groups. Does the central angle AOB maintain its size if you use a
larger circle or a smaller circle?