Trigonometry Unit 3 Packet Name____________________ Hr___ Radian Measure EXAMPLE 1: Convert from radians to degrees. a) 3 4 b) 5 2 c) 4 3 d) 11 6 EXAMPLE 2: Convert from degrees to radians. a) 120° b) 270° c) 315° d) 210° EXAMPLE 3: Find the value of the functions: a) 2 tan 3 b) 3 sin 2 c) 5 cos 6 Radian Degree 6 4 3 2 1 radian = _________°
14
Embed
Radian Measure Radian Degree - Manhattan High · PDF file... A circle has a radius 15 cm. Find the length of ... How far does the tip of the minute hand move in 15 minutes? ... How
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Trigonometry Unit 3 Packet
Name____________________ Hr___
Radian Measure
EXAMPLE 1: Convert from radians to degrees.
a) 3
4
b)
5
2
c)
4
3
d)
11
6
EXAMPLE 2: Convert from degrees to radians.
a) 120°
b) 270°
c) 315°
d) 210°
EXAMPLE 3: Find the value of the functions:
a) 2
tan3
b)
3sin
2
c)
5cos
6
Radian Degree
6
4
3
2
1 radian = _________°
Trigonometry Unit 3 Packet
Section 3.2: Applications of Radian Measure
Converting degrees to radians:
EXAMPLE 1: Convert to radians.
a) 147° b) 37.9° c) 221°
Arc length:
EXAMPLE 1: A circle has a radius 15 cm. Find the length of the arc intercepted by a
central angle having the following measure:
a) 2
5
radians b)
EXAMPLE 2: Two gears are adjusted to that the smaller gear drives the larger one. If
the smaller gear rotates through 225°, through how many degrees will the larger gear
rotate?
Area of a Sector:
EXAMPLE 3: Find the area of the given sector.
Caution‼!
θ must be in
RADIANS
Trigonometry Unit 3 Packet
Linear and Angular Speed
Suppose that an object moves along a circle or radius at a constant speed. If is the
distance traveled along the circle in time , then:
As the object moves along the circle, let θ be the angle made in time . Then:
There is an interesting relationship between linear and angular speed: _______________
EXAMPLE 1: Suppose that point P is on a circle with radius 10 cm, and ray OP is rotating
with angular speed 18
radian per second.
a) Find the angle generated
by P in 6 sec.
b) Find the distance traveled
by P along the circle in 6
sec.
c) Find the linear speed of P.
EXAMPLE 2: A belt runs a pulley of radius 6 cm at 80 revolutions per minute.
a) Find the angular speed of the pulley
in radians per second.
b) Find the linear speed of the belt in
centimeters per second.
linear speed =
angular speed =
Trigonometry Unit 3 Packet
EXAMPLE 3: The shoulder joint can rotate at about 25 radians per second. If a golfer’s
arm is straight and the distance from the shoulder to the club head is 5 ft, estimate the
linear speed of the club head from shoulder rotation.
EXAMPLE 4: Suppose you have rented a paddle boat at Tuttle Creek Lake. The current in
the lake causes the circular paddle wheel with radius 4 feet to rotate at a speed of 10
revolutions per minute. What is the speed of Tuttle Creek’s current in miles per hour?
Trigonometry Unit 3 Packet
PRACTICE:
Convert from radians to degrees:
1. 8
3
2.
7
4
3.
5
6
Convert from degrees to radians:
4. 300° 5. 390° 6. 225°
Find the exact value of the expression WITHOUT a calculator.
7. sin3
8. cos
6
9. tan
4
10. 2
cot3
11. csc
2
12.
5tan
6
13. 5
tan3
14.
15. 3
sec4
16. Through how many radians will the hour hand on a clock rotate in:
a) 24 hours? b) 4 hours?
Trigonometry Unit 3 Packet
17. Find the exact length of the arc
shown.
18. Find the radius of the circle.
Find the length of the arc intercepted by the central angle in a circle of radius .
19.
20.
21. If the radius of a circle is doubled, how is the length of the arc intercepted by a
fixed central angle changed?
22. Radian measure simplifies many formulas, including the one for arc length, .
Give the corresponding formula when is measured in degrees instead of radians.
23. Find the distance in kilometers between Wichita, KS, , and Fort Worth, TX,
, assuming they lie on the same north-south line. (The radius of the earth is
6400 km.)
3
12 in 3π
3π
Trigonometry Unit 3 Packet
Find the area of a sector having radius and central angle . Round to the nearest tenth.
24.
25.
26. Find the measure (in radians) of a central angle of a sector with area 16 in² inside a
circle with radius 3 inches
27. Find the radius of a circle in which a central angle of 6
radian determines a sector
of area 64 m².
28. Suppose that a point on a circle with radius and ray is rotating with angular
speed . Find each of the following:
i. the angle generated by in time .
ii. the distance traveled by along the circle in time .
iii. the linear speed of
a)
b)
Trigonometry Unit 3 Packet
29. Use the formula for angular velocity t
to find the value of the missing
variable.
a)
b)
c)
30. Use the formula for linear velocity ( ) to find the value of the missing
variable.
a)
b)
31. Find the angular velocity (in radians per second) of a line from the center to the
edge of a DVD revolving 300 times per minute.
32. Find the angular velocity of the minute hand of a clock.
Trigonometry Unit 3 Packet
33. Watch the video found at https://www.youtube.com/watch?v=WehHFJki9yQ.
Name five of the cool ways trigonometry helps us in the real world.
EXPANSION 1: Engineers use the term grade to represent
of a right angle and
express it as a percent. For example, an angle of would be referred to as a 1% grade.
a) By what number should you multiply a grade (disregarding the % symbol) to convert
it to radians?
b) In a rapid-transit rail system, the maximum grade allowed between two stations is
3.5%. Express this angle in degrees and radians.
EXPANSION 2: Find the linear velocity of the tip of an airplane propeller long,
rotating 500 times per minute.
EXPANSION 3: A circular pulley is rotating about its center. Through how many radians
would it turn in:
a) 8 rotations
b) 30 rotations
EXPANSION 4: The phase of the moon is modeled by ( )
( ) and gives the
fraction of the moon’s face that is illuminated by the sun. Evaluate each expression and