Download radians to degrees. - Fort Thomas Independent Schools

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.
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.
2.
An _____________ side
A ______________ side
Terminal Side
Initial Side
- The _____________ of the two __________ is known as the
____________.
- An angle centered at the ____________ is said to be in
________________ Position.
- ___________ angles are measured ____________________________.
- ___________ angles are measured ____________________.
- If two angles have the same _____________, then they are said
to be ______________________.
Radian vs. Degree:
- Just as distance may be measured in feet and centimeters,
angles can be measured in both ____________ and ____________.
Definition of a Radian:
𝑠
𝜃 = , where s is the ________ length and r is the ______________.
𝑟
Conversion Factors:
2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 =
𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 =
1 𝑟𝑎𝑑𝑖𝑎𝑛 ≈
Some Other Common Radian Measures:
45° =
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
60° =
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
30° =
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
90° =
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
Acute Angles are between 0 and
Obtuse Angles are between
EX: For the positive angle
angle.
9𝜋
4
and
radians.
radians.
subtract 2𝜋 to obtain a coterminal
EX: For the positive angle
5𝜋
6
subtract 2𝜋 to obtain a coterminal
angle.
EX: For the negative angle −
3𝜋
4
, add 2𝜋 to find a coterminal angle.
Recall your Quadrants for Geometry:
In Q1 
In Q2 
In Q3 
In Q4 
Complementary – two angles whose sum is
degrees.
radians or
Supplementary – two angles whose sum is radians or
𝜋
EX: Find the complement and supplement of .
6
EX: Find the complement and supplement of
5𝜋
6
.
degrees.
*There are _____ degrees or
radians in a circle.
Conversions Between Degrees and Radians:
1.
2.
To convert degrees to radians, multiply degrees by
To convert radians to degrees, multiply radians by
EX: Convert 60 degrees to radians in terms of pi.
EX: Convert 320 degrees to radians in terms of pi.
EX: Convert -30 degrees to radians in terms of pi.
𝜋
EX: Express as a degree measure.
6
EX: Express
5𝜋
3
as a degree measure.
EX: Express 3 radians as a degree measure.
Recall: 𝜃 =
𝑠
𝑟
Arc Length:
.
.
𝑠 = ____, where r is the ___________ and theta is the measure of the
central ____________.
- It is important to note that ______ must always be in
____________ 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.
Linear Speed (v):
Linear Speed v =
Angular Speed 𝜔 (omega):
𝜔=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑎𝑛𝑔𝑙𝑒
=
𝑡𝑖𝑚𝑒
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.
EX: The circular blade on a saw rotates at 2400 revolutions per
minute. Find the angular speed in radians per second.
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.
Area of a Sector:
𝐴=
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.
Do the following on your own:
a.)
b.)
Convert 1 degree to radians.
Convert 60 degrees to radians.
c.)
Convert
5𝜋
6
radians to degrees.
Use a calculator to evaluate each of the following in radian mode.
a.)
Cos(6)
b.)
tan
7𝜋
6
HW Pg. 715
1-28
Quiz 10.5-10.9 tomorrow