P.o.D.
1.) Divide 6𝑥 3 − 4𝑥 2 by 2𝑥 2 + 1.
2.) Solve the inequality
1
𝑥+1
≥
1
.
𝑥+5
3.) Find ALL the zeros of
𝑓(𝑥 ) = 2𝑥 4 − 11𝑥 3 + 30𝑥 2 − 62𝑥 − 40
1.) 3𝑥 − 2 −
3𝑥−2
2𝑥 2 +1
2.) 𝑥 < −5 or x> -1
3.) 4, −1/2, 1+3i, 1-3i
Don’t Forget, today is Veteran’s Day.
4.1 – Radian and Degree Measure
Learning Target: Be able to describe
angles in both radians and degrees.
*Chapter 4 begins the study of
Trigonometry – measures of Triangles.
Angles have two sides:
1. An initial side
2. A terminal side
Terminal Side
Initial Side
- The endpoint of the two rays is
known as the vertex.
- An angle centered at the origin is said
to be in Standard Position.
- Positive angles are measured
counterclockwise.
- Negative angles are measure
clockwise.
- If two angles have the same position,
then they are said to be coterminal.
Radian vs. Degree:
- Just as distance may be measured in
feet and centimeters, angles can be
measured in both radians and
degrees.
Definition of a Radian:
𝑠
𝜃 = , where s is the arc length and r is
𝑟
the radius.
Conversion Factors:
2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 360°
𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 180°
1 𝑟𝑎𝑑𝑖𝑎𝑛 ≈ 57.3°
Some Other Common Radian Measures:
𝜋
45° = 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
4
𝜋
60° = 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
3
𝜋
30° = 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
6
𝜋
90° = 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
2
Acute Angles are between 0 and
radians.
𝜋
2
𝜋
Obtuse Angles are between and 𝜋
2
radians.
EX: For the positive angle
9𝜋
4
subtract 2𝜋
to obtain a coterminal angle.
9𝜋
9𝜋 8𝜋 𝜋
− 2𝜋 =
−
=
4
4
4
4
EX: For the positive angle
5𝜋
6
subtract 2𝜋
to obtain a coterminal angle.
5𝜋
5𝜋 12𝜋
7𝜋
− 2𝜋 =
−
=−
6
6
6
6
EX: For the negative angle −
3𝜋
4
, add 2𝜋
to find a coterminal angle.
3𝜋
−3𝜋 8𝜋 5𝜋
−
+ 2𝜋 =
+
=
4
4
4
4
Recall your Quadrants for Geometry:
In Q1  0 < 𝜃 <
In Q2 
𝜋
2
3𝜋
2
2
<𝜃<𝜋
In Q3  𝜋 < 𝜃 <
In Q4 
𝜋
3𝜋
2
< 𝜃 < 2𝜋
Complementary – two angles whose
𝜋
sum is radians or 90 degrees.
2
Supplementary – two angles whose sum
is 𝜋 radians or 180 degrees.
EX: Find the complement and
𝜋
supplement of .
6
Complement:
𝜋 𝜋 3𝜋 𝜋
− =
−
2 6
6
6
2𝜋 𝜋
=
=
6
3
Supplement:
𝜋 6𝜋 𝜋
𝜋− =
−
6
6
6
5𝜋
=
6
EX: Find the complement and
supplement of
5𝜋
6
.
Complement:
𝜋 5𝜋
−
=
2
6
3𝜋 5𝜋
−
=
6
6
−2𝜋 −𝜋
=
6
3
Since the angle is negative, no
complement exists.
Supplement:
5𝜋
𝜋−
6
6𝜋 5𝜋
=
−
6
6
𝜋
=
6
*There are 360 degrees in a circle.
Conversions Between Degrees and
Radians:
1. To convert degrees to radians,
multiply degrees by
𝜋
.
180
2. To convert radians to degrees,
multiply radians by
180
𝜋
.
EX: Convert 60 degrees to radians in
terms of pi.
60° 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
×
=
1
180°
60𝜋
180
𝜋
= 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
3
EX: Convert 320 degrees to radians in
terms of pi.
320
𝜋
×
1
180
320𝜋
=
180
16𝜋
=
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
9
EX: Convert -30 degrees to radians in
terms of pi.
−30
𝜋
×
1
180
−30𝜋
=
180
−𝜋
=
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
6
*We could write this as a positive angle.
−𝜋
−𝜋 12𝜋
+ 2𝜋 =
+
6
6
6
11𝜋
=
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
6
𝜋
EX: Express as a degree measure.
6
𝜋 180
×
6
𝜋
180
=
6
= 30°
EX: Express
5𝜋
3
as a degree measure.
5𝜋 180
×
3
𝜋
900
=
3
= 300°
EX: Express 3 radians as a degree
measure.
3 𝑟𝑎𝑑 180°
×
1
𝜋 𝑟𝑎𝑑
540
=
𝑑𝑒𝑔𝑟𝑒𝑒𝑠
𝜋
≈ 171.887°
*We could also have done 3(57.3) =
171.9°
*Let’s write a calculator program to
convert radians to degrees and vice
versa.
Recall: 𝜃 =
𝑠
𝑟
Arc Length:
𝑠 = 𝑟𝜃, where r is the radius and theta is
the measure of the central angle.
- It is important to note that theta
must always be in radians when used
in a formula.
EX: A circle has a radius of 27 inches.
Find the length of the arc intercepted by
a central angle of 160 degrees.
First, convert 160 degrees to radian
measure.
160°
𝜋
×
1
180°
160𝜋
=
180
8𝜋
=
9
Next, apply the formula for arc length,
𝑠 = 𝑟𝜃.
8𝜋
𝑠 = 27 ( )
9
= 24𝜋
≈ 75.398 𝑖𝑛𝑐ℎ𝑒𝑠
Linear Speed (v):
Linear Speed v =
𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ
𝑡𝑖𝑚𝑒
=
𝑠
𝑡
Angular Speed 𝜔 (omega):
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑎𝑛𝑔𝑙𝑒 𝜃
𝜔=
=
𝑡𝑖𝑚𝑒
𝑡
*Study and memorize the tan box in the
lower left hand corner of page 287.
EX: The second hand of a clock is 8
centimeters long. Find the linear speed
of the tip of this second hand as it passes
around the clock face.
We first need to find the distance (arc
length) traveled by the second hand as it
makes one complete revolution.
𝑠 = 𝑟𝜃
= 8(2𝜋)
= 16𝜋
Now find its linear speed.
𝑠
16𝜋
𝑣= =
𝑡 60 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
4𝜋 𝑐𝑚
⁄𝑠
=
15
≈ 0.8378 𝑐𝑚/𝑠
EX: The circular blade on a saw rotates at
2400 revolutions per minute. Find the
angular speed in radians per second.
We can use a process known as
dimensional analysis.
2400 𝑟𝑒𝑣 2𝜋 𝑟𝑎𝑑 1 𝑚𝑖𝑛
×
×
1 𝑚𝑖𝑛
1 𝑟𝑒𝑣
60 𝑠𝑒𝑐
4800𝜋 𝑟𝑎𝑑
=
60 𝑠𝑒𝑐
= 80𝜋 𝑟𝑎𝑑/𝑠𝑒𝑐
EX: Referring to the previous problem,
the blade has a radius of 4 inches. Find
the linear speed of a blade tip in inches
per second.
Linear Velocity is Angular Velocity
multiplied by Radius, 𝑣 = 𝜔𝑟
𝑣 = 80𝜋(4) = 320𝜋
≈ 1005.3096 𝑖𝑛/𝑠𝑒𝑐
Area of a Sector:
1 2
𝐴= 𝑟 𝜃
2
EX: A sprinkler on a golf course is set to
spray water over a distance of 75 feet
and rotates through an angle of 135
degrees. Find the area of the fairway
watered by the sprinkler.
Let’s first draw a picture of the situation.
135° 𝜋 𝑟𝑎𝑑
𝜃=
×
1
180°
3𝜋
=
𝑟𝑎𝑑.
4
1
3𝜋
2
𝐴 = (75) ( )
2
4
16875𝜋
=
8
≈ 6626.797 𝑠𝑞𝑢𝑎𝑟𝑒 𝑓𝑒𝑒𝑡
*Let’s write a calculator program to find
the area of a sector.
Upon completion of this lesson, you
should be able to:
1. Convert from radians to degrees
and vice versa.
2. Find the supplement and/or
complement of an angle.
3. Find the arc length of a sector.
4. Compute linear and angular speed.
5. Find the area of a sector.
For more information, visit
http://academics.utep.edu/Portals/1788/CALCULUS%20
MATERIAL/4_1%20RADIAN%20N%20DEGREES%20MEAS
URES.pdf
HW Pg.290 6-90 6ths, 102, 106, 117120.