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Math 140 Lecture 15
Angles, arcs, and radians
p is approximately 3.14.
In answers, use the exact p not the decimal 3.14.
A circle of radius r has circumference 2pr.
Unit circle circumference = 2pr = 2p1 = 2p.
Greek letters: q is “theta”, w is “omega”.
DEFINITION. An angle intercepts an arc of length s on a
circle of radius r. The angle’s radian measure is
q = s /r
r
q s
For unit circles, r = 1 and radian measure = arc length:
q = s.
Radians are dimensionless and “radians” can be omitted,
10 = 10 radians.
`Radians and degrees on the unit circle .
p/2 = 90 o
p/4= 45
o
1
o
o
o
0 = 0, 360 = 2p
180 = p
-p/4= -45
o
Clockwise angles ?are negative.
CONVERSION FORMULAS.
180o = p radians Hence the following conversion ratios are 1.
 radians
180 o
180o  1 and  radians  1
 radians
`18  18  1 18
180 o
o
o
o

 radians
10


10 radians
To convert to radians, conversion factor must have radians on top.
To cancel degrees, conversion factor needs degrees on the bottom.
To convert between radians and degrees, use the ratio
 radians
180 o
,
180 o
 radians
which cancels the current units.
` Convert 50 o to radians. Which formula cancels
degrees and gives radians?
(A) 50o
 radians
180 o
o
180
(B) 50o  radians
` Convert 5 radians to degrees. Which formula cancels
radians and gives degrees?
 radians
(A) 5 radians 180 o
180 o
(B) 5 radians  radians
`Convert 100o to radians.
... 5/9 rad.
`Convert 40o to radians.
.
.
.
`Convert p/6 radians
to degrees.
`Convert p/5 radians
to degrees.
.
.
If q is in radians, then q = s/r
`Find the length
of a 36o arc on circle
of radius 10 inches.
Convert 36o to radians.
qr = s
s = qr
`Find the length
of a 30o arc on a circle
of radius 12 inches. ... 2
.
.
Find the length of the arc.
.
.
`Find the degree measure
of an angle intercepting
a 5 inch arc on a
circle of radius 12 inches.
`Find the degree measure
of an angle which
intercepts a 7 inch arc on a
circle of radius 18 inches.
Find the radian measure
q = s/r .
... 75 0 /
.
.
Then convert to degrees.
.
.
Speed
DEFINITION.
If an object travels a distance d in time t,
its linear speed is d/t. E.g., miles/hour.
If an object rotates through an angle q in time t,
its rotational speed is w = q/t. E.g., radians/second.
THEOREM. If a point rotates around a circle of radius r
with rotational speed w, then its linear speed is wr.
PROOF.
Suppose the point rotates through angle q in time t.
t w = q/t
The distance d it travels
= the length of the arc it traces
= qr .
tits linear speed = d/t = qr/t = (q/t)r = wr .
`A point revolves around
a circle of radius 10 feet at
4 revolutions per minute.
`A point revolves around
circle of radius 3 feet at
10 revolutions per minute.
... 60
.
Find its rotational speed
in radians/min.
.
Find its linear speed.
.
What are the units
for the linear speed?
.
`The tip of a clock’s
minute hand is 6 inches
from the center. How
fast is the tip moving?
`A point on the equator
is 4000 miles from
the center of the earth.
Distance moved per day?
How many inches does
it move in one hour?
Speed in miles/day?
How fast is the tip
moving in inches/hour?
Speed in miles/hour?
... 1000mi/hr.
How fast is the tip
moving in inches/minute?
Trigonometric functions
The unit circle has radius one and center (0, 0).
There are four quadrants I, II, III, IV as pictured.
An angle is in standard position if
its vertex is the origin (0,0) and
its initial side is the positive x-axis.
The other side of the angle is the terminal side.
The terminal side intersects the unit circle at a point Pq.
Pq
II
q
I
(0,0)
III
IV
DEFINITION. Given a standard-position angle of radian
measure q, let (x, y) be the coordinates of Pq.
tanq=y/x=slope
Pq= (x, y)
1
y = sin q
q
x=cos q
The six trigonometric functions of q are:
sin q = y
sine csc q = 1/ sin q
cos q = x
tan q = sin q / cos q
cosine
tangent
sec q = 1/cos q
cosecant
secant
cot q = cos q / sin q cotangent
`Draw an angle of p/2 radians in standard position.
`sin /2 
`cos /2 
`tan /2 
`A point (x, y) on the unit circle is in the second
quadrant and y = 34 .
 7
3
Find sin , cos , tan  .
... 34
4
7
1
q
3/4
x
sin 
cos  
tan 
b
Know the sin, cos and tan of: 0, p/6, p/4, p/3, p/2.
(0,1)
p/2=90o
1
(1/ 2, l3/ 2)
p/3=60o
o (1/l2, 1/l2)
p/4=45
o
p/6=30 (l3/2,1/2)
0o (1,0)
sin(0) =
0
cos(0) =
1
tan(0) =
0
sin(p/6) = 1/2
cos(p/6) =
sin(p/4) = 1/ 2
cos(p/4) = 1/ 2
tan(p/4) = 1
sin(p/3) = l3̄/2
cos(p/3) = 1/2
tan(p/3) = l3̄
sin(p/2) = 1
cos(p/2) = 0
tan(p/2) = undef.
bb
3 /2
tan(p/6) = 1/l3̄
p/2
y sin
q
___
_
slope of line = =
x cos q
1
y = sin q
q
x =cos q
= tan q