P.o.D. – Solve each triangle
using the Law of Cosines/Sines.
1.) a=7, b=4, C=102 degrees. Find
c.
2.) C=59 degrees, a=13, b=12.
Find c.
3.) a=13, b=6, c=15. Find A.
4.) a=7, b=7, C=85 degrees. Find
c.
5.) a=18, b=24.5, C=20 degrees.
Find c.
1.) 8.8
2.) 12.3
3.) 59.3 degrees
4.) 9.5
5.) 9.8
10-9: Radian Measure
Learning Target(s): I can
approximate values of
trigonometric functions using a
calculator; convert angle
measures from radians to
degrees or vice versa.
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 co-terminal.
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
𝑟
and r is 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
𝜋
radians.
2
𝜋
Obtuse Angles are between and
2
𝜋 radians.
EX: For the positive angle
9𝜋
subtract 2𝜋 to obtain a
coterminal angle.
9𝜋
9𝜋 8𝜋 𝜋
− 2𝜋 =
−
=
4
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
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
𝜋
2
<𝜃<𝜋
4
,
In Q3  𝜋 < 𝜃 <
In Q4 
3𝜋
2
3𝜋
2
< 𝜃 < 2𝜋
Complementary – two angles
𝜋
whose sum is radians or 90
2
degrees.
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 2𝜋 𝜋
− =
− =
=
2 6
6
6
6
3
Supplement:
𝜋 6𝜋 𝜋 5𝜋
𝜋− =
− =
6
6
6
6
EX: Find the complement and
supplement of
5𝜋
6
.
Complement:
𝜋 5𝜋 3𝜋 5𝜋 −2𝜋 −𝜋
−
=
−
=
=
2
6
6
6
6
3
Since the angle is negative, no
complement exists.
Supplement:
5𝜋 6𝜋 5𝜋 𝜋
𝜋−
=
−
=
6
6
6
6
*There are 360 degrees or 2𝜋
radians 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 60𝜋 𝜋
×
=
= 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
1
180°
180 3
EX: Convert 320 degrees to
radians in terms of pi.
320
𝜋
320𝜋 16𝜋
×
=
=
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
1
180
180
9
EX: Convert -30 degrees to
radians in terms of pi.
−30
𝜋
−30𝜋 −𝜋
×
=
=
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
1
180
180
6
*We could write this as a
positive angle.
−𝜋
−𝜋 12𝜋 11𝜋
+ 2𝜋 =
+
=
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
6
6
6
6
𝜋
EX: Express as a degree
6
measure.
𝜋 180 180
×
=
= 30°
6
𝜋
6
EX: Express
5𝜋
3
as a degree
measure.
5𝜋 180 900
×
=
= 300°
3
𝜋
3
EX: Express 3 radians as a degree
measure.
3 𝑟𝑎𝑑 180°
540
×
=
𝑑𝑒𝑔𝑟𝑒𝑒𝑠
1
𝜋 𝑟𝑎𝑑
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
160𝜋 8𝜋
×
=
=
1
180°
180
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
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
4𝜋 𝑐𝑚
⁄𝑠
𝑣= =
=
𝑡 60 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 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
=
= 80𝜋 𝑟𝑎𝑑/𝑠𝑒𝑐
60 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 3𝜋
𝜃=
×
=
𝑟𝑎𝑑.
1
180°
4
1
3𝜋
2
𝐴 = (75) ( ) =
2
4
16875𝜋
≈ 6626.797 𝑠𝑞𝑢𝑎𝑟𝑒 𝑓𝑒𝑒𝑡
8
*Let’s write a calculator
program to find the area of a
sector.
Do the following on your own:
a.) Convert 1 degree to
radians.
b.) Convert 60 degrees to
radians.
c.) Convert
5𝜋
6
radians to
degrees.
a.) 0.017 radians
𝜋
b.)
3
c.) 150 degrees
Use a calculator to evaluate each
of the following in radian mode.
a.) Cos(6)
b.) tan
7𝜋
6
a.) 0.9602
b.) 0.5774
Clever Uses of Trigonometry:
Radians vs. Degrees
Gang Sines
More appropriate on Pi Day
Upon completion of this lesson,
you should be able to:
1. Find co-terminal angles.
2. Find the complement and
supplement of angles.
3. Convert degrees to radians
and vice versa.
For more information, visit
http://www.mathwarehouse.com/trigonome
try/radians/convert-degee-to-radians.php
Now get against the wall. We
have a unit circle to recite!!!!!!
HW Pg. 715 1-28
Quiz 10.5-10.9 tomorrow