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
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.
2.
An ________________ side
A _________________ side
Terminal Side
Initial Side
-
- The endpoint of the two rays is known as the _____________.
- An angle centered at the ______________ is said to be in _____________
Position.
- Positive angles are measured _______________________________.
- Negative angles are measured ___________________________.
- If two angles have the same position, 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 arc length and r is the radius.
𝑟
Conversion Factors:
2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = __________
𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = _____________
1 𝑟𝑎𝑑𝑖𝑎𝑛 ≈ _______________
Some Other Common Radian Measures:
_____° =
𝜋
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
4
_____° =
𝜋
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
3
𝜋
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
6
𝜋
_____° = 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
2
_____° =
𝜋
__________ Angles are between 0 and radians.
2
𝜋
___________ Angles are between and 𝜋 radians.
2
EX: For the positive angle
EX: For the positive angle
9𝜋
4
5𝜋
6
EX: For the negative angle −
subtract 2𝜋 to obtain a coterminal angle.
subtract 2𝜋 to obtain a coterminal angle.
3𝜋
4
, add 2𝜋 to find a coterminal angle.
Recall your Quadrants from Geometry:
In Q1  0 < 𝜃 <
𝜋
2
𝜋
In Q2  < 𝜃 < 𝜋
2
In Q3  𝜋 < 𝜃 <
3𝜋
2
In Q4 
3𝜋
2
< 𝜃 < 2𝜋
Complementary – two angles whose sum is _______ radians or 90 degrees.
Supplementary – two angles whose sum is _____________ radians or 180 degrees.
𝜋
EX: Find the complement and supplement of .
6
Complement:
Supplement:
EX: Find the complement and supplement of
5𝜋
6
.
Complement:
Since the angle is negative, ______________________________.
Supplement:
*There are _______ degrees 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.
*We could write this as a positive angle.
𝜋
EX: Express as a degree measure.
6
EX: Express
5𝜋
3
as a degree measure.
EX: Express 3 radians as a degree measure.
*We could also have done ___________________________.
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.
Next, apply the formula for arc length, 𝑠 = 𝑟𝜃.
Linear Speed (v):
Linear Speed v =
𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ
𝑡𝑖𝑚𝑒
=? ?
Angular Speed 𝜔 (omega):
𝜔=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑎𝑛𝑔𝑙𝑒
=? ?
𝑡𝑖𝑚𝑒
*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.
Now find its linear speed.
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.
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
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.
Let’s first draw a picture of the situation.
HW
Pg.290
6-90 6ths, 102, 106, 117-120.