Chapter 4
4-1 Radian and degree measurement
Objectives
O Describe Angles
O Use radian measure
O Use degree measure and convert between
and radian measure
Angles
O As derived form the Greek language
O Trigonometry means “measurement of
triangles “
O Initially, trigonometry dealt with
relationships among the sides and angles of
triangles and was used in the development
of astronomy , navigation and surveying
O Today, the use has expanded to involve
rotations, orbits, waves, vibrations, etc.
Angles
O An angle is determined by rotating a ray
(half-line) about its endpoint.
Definitions
O The initial side of an angle is the starting
position of the rotated ray in the formation
of an angle.
O The terminal side of an angle is the position
of the ray after the rotation when an angle is
formed.
O The vertex of an angle is the endpoint of the
ray used in the formation of an angle.
Standard Position
O An angle is in standard position when the
angle’s vertex is at the origin of a coordinate
system and its initial side coincides with the
positive x-axis.
Positive and negative angles
O A positive angle is generated by a
counterclockwise rotation; whereas a
negative angle is generated by a clockwise
rotation.
Coterminal
O If two angles are coterminal, then they have
the same initial side and the same terminal
side.
Radian Measure
O The measure of an angle is determined by
the amount of rotation from the initial side
to the terminal side.
O One way to measure angles is in radians.
O To define a radian, you can use a central
angle of a circle, one whose vertex is the
center of the circle.
Radian Measure
O Arc length = radius when 𝜃= 1 radian
Radian
How many radians are in a
circle?
O *In general, the radian measure of a central
angle θ with radius r and arc length s is
O θ = s/r.
O We know that the circumference of a circle
is 2πr. If we consider the arc s as being the
circumference, we get θ = 2π r/r =2𝜋
Radians
O This means that the circle itself contains an
angle of rotation of 2π radians. Since 2π is
approximately 6.28, this matches what we
found above. There are a little more than 6
radians in a circle. (2π to be exact.)
Therefore: A circle contains 2π radians. A
semi-circle contains π radians of rotation. A
quarter of a circle (which is a right angle)
contains 𝜋2 radians of rotation
Radians
Coterminal angles
O Two angles are coterminal when they have
the same initial and terminal sides.
O For instance, the angle 0 and 2𝜋 are
𝜋
coterminal, as are the angles 𝑎𝑛𝑑 13𝜋/6.
6
O You can find an angle that is coterminal to a
given angle by adding or subtracting 2𝜋.
Example#1
O For the positive angle 13𝜋 / 6, subtract 2𝜋
to obtain a coterminal angle.
Example#2
O For the positive angle 3𝜋 / 4, subtract 2𝜋 to
obtain a coterminal angle.
Example#3
O For the negative angle –2𝜋 / 3, add 2𝜋 to
obtain a coterminal angle.
Student guided practice
O Do problems 25 and 26 in your book page
261
Degree Measure
O Definition: A degree is a unit of angle
measure that is equivalent to the rotation in
1/360th of a circle.
O Because there are 360° in a circle, and we
now know that there are also 2π radians in a
circle, then 2π = 360°.
O 360° = 2π radians 2π radians= 360°
O 180° = π radians
π radians = 180°
O 1° =
𝜋
radians
180
1 radian= 180/𝜋
Degree Measure
O To convert radians to degrees, multiply by
𝜋
180
.
O To convert degrees to radians, multiply by
O 180/𝜋.
Example
O Example: Convert 120° to radians.
O Example: Convert -315° to radians
Example
O Example: Convert
5𝜋
6
to degrees.
O Convert 7 to degrees.
Student guided practice
O Do odd problems form 55-65 in your book
page 262
Acute and Obtuse
O An acute angle has a measure between
O 0
𝜋
and .
2
O (or between 0° and 90°.)
O An obtuse angle has a measure between
𝜋
2
and π (or between 90° and 180°.)
Example
O Example: Find the supplement and
complement of 𝜋/5.
Arc Length
O Because we already know that with radian
measure θ =r /s,
O where s is the arc length, then s = r θ.
Example
O Example: Find the length of the arc that
subtends a central angle with measure
120° in a circle with radius 5 inches.
Example
O A circle has a radius of 4 inches. Find the
length of the arc intercepted by a central
angle of 240 degrees, as shown in Figure
4.15.
Student Guided practice
O Do problem 93 and 94 in your book page
263
Linear and angular speed
O Consider a particle moving at a constant
speed along a circular arc of radius r. If s is
the length of the arc traveled in time t, then
the linear speed of the particle is
O Linear speed = arc length/ time= s/t
𝑟𝜃
=
𝑡
Linear and angular speed
O Moreover, if θ is the angle (in radian
measure) corresponding to the arc length s,
then the angular speed of the particle is
Angular speed = central angle/ time=𝜃/ t.
Example
O Example: The circular blade on a saw rotates
at 2400 revolutions per minute.
O (a) Find the angular speed in radians per
second.
O (b) The blade has a diameter of 16 inches.
Find the linear speed of a blade tip.
Area of a sector
Homework
O Do problems
27,28,45,46,51,52,56,58,79,85
O In your book page 261 and 262
Closure
O Today we learned about radian and degree
measure
O Next class we are going to learn about the
unit circle