Download What is a RADIAN?

October 24, 2014
Chapter 4
Trigonometric Functions
Section 4.1
Radians and Degree Measure
Objective: Know how to describe an angle and convert between degree and radian measures.
Angles - 2 rays with same initial point.
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Measure of an angle - amount of
Initial
Vertex
rotation required to rotate initial
side to terminal side.
Measured in degrees or radians.
Standard position - vertex at origin, initial side lies along positive
Counterclockwise rotation positive
x-axis.
Clockwise rotation negative
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Vertex
Initial
Coterminal Angles - angles in standard position
that have the same terminal side.
To find coterminal angles, add or subtract 360
or 2π or integer multiples of 2 π.
Ex: Determine 2 coterminal angles for 30.
30
How
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What is a RADIAN?
Paper plate activity
s=r
θ
r
Radian - measure of a central angle
θ that intercepts an arc s equal in
length to the radius r of the circle.
Considered "natural" and "dimensionless" since
units of measure cancel out; radians is a ratio.
θ = 1 Radian
6.3 radians ≈2π
October 24, 2014
Note: If angles are not marked as degrees, then radians are
implied.
To convert...
Degrees
Radians
radians multiply by
π
180
Common Angles
180
degrees multiply by
π
Ex:
1 Convert to radians.
a) 120
b) -315
2 Convert to degrees.
a) 5π
6
b) 7
Geometry Review
acute - 0 < θ < π
2
obtuse - π < θ < π
2
Complementary - sum of angles = 90 or π .
2
Supplementary - sum of angles = 180 or π.
Ex: Find complement and supplement of π .
5
Complement
Supplement
Note: Angle measures sometimes given in degrees, minutes, seconds.
Ex:
46
a) 64 32 46 = 64 + 32 +
= 64.541...
60
b) 43.145 = 43 8 42
3600
October 24, 2014
Applications of Angles
Consider a particle moving at 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
rθ
particle is...
linear speed = (arc length)/(time) = s/t
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 or linear speed
radius
Ex:
A 6-inch-diameter gear makes 2.5 revolutions per second. Find the angular
speed of the gear in radians per second and the linear speed in inches per second.
Angular speed = 5π radians per second
Linear speed = 15π inches per second
Ex: A car is traveling at 65 mph. If each tire has a radius of 15 inches, at
what rate are the tires spinning in revolutions per minute (rpm)?
Hint: Need in/min
Linear speed = 68640 rev./30π min.≈728.29 rev./min.
Angular speed = 4,576 radians per minute.