Sec 3.1 -
Radian Measure
Radian Measure
what is (9pi)/10 in degrees?
162 degrees
What are the equations to find the area of a sector of a circle and length of an arc? (in radians)
Arc =(s)=rtheta
Area=1/2thetar^2
What do all connected gears have in common? What are the relationships between 2 gears and their central angles and radii given they aren't the same size?
The arc length is equal no matter what the angle of rotation is. The gear with the larger radius should always have the smaller angle of rotation.
What are the 5 reference angles and coordinate values of the unit circle?
What are the only types of angles that use 0 and 1 as outputs for sin and cos?
ref_theta=0,pi/6,pi/4,pi/3,pi/2
Outputs: 0,1/2, sqrt2/2, sqrt3/2, 1
The only angles that give 0 or 1 for sin and cos are quadrantal angles.
What part of a butterfly 'tastes'?
Its feet.
if the minute hand of a clock has rotated 40 minutes, what is that in radians?
(4pi)/3rad
if a circle has an arc length of 15pi and a central angle of (5pi)/6 , what is the radius?
15pi=r*(5pi)/6=>18
If city_1 is (3pi)/8 rad above the equator, and city_2 is (3pi)/8 rad below the equator:
How far apart are they if the earth has a radius of 6400 km?
(3pi)/8 rad+(3pi)/8 rad=(6pi)/8 rad
(6pi)/8 rad * 6400 km = 4800pi km
What is the sin, cos and tan of
theta=(3pi)/4, (3pi)/2, 4pi
Sin((3pi)/4)=sqrt2/2,cos((3pi)/4)=-sqrt2/2, Tan((3pi)/4)=-1
Sin((3pi)/2)=-1,cos((3pi)/2)=0, Tan((3pi)/2)=Und
Sin(4pi)=0, cos(4pi)=1, tan(4pi)=0
How many teeth does the average human have?
32
What are sin, cos and tan of (7pi)/4 ?
Sin((7pi)/4)=-sqrt2/2; Cos((7pi)/4)=sqrt2/2; tan((7pi)/4)=-1
If a circle has a diameter of 12 ft and a central angle of (5pi)/4 rad , what is the area of the sector?
Area=(45pi)/2 ft
Gear1 has a radius of 4 cm and rotates (5pi)/4 rad . If gear2 rotates (5pi)/3 rad , what is its radius? Explain how you know your answer makes sense.
(5pi)/4 rad* 4cm=5picm
5picm=(5pi)/3*r=>r=3cm
Because it has a larger degree of rotation, the radius should be smaller than 4.
Find the output:
a) sin((5pi)/6)=
b) cos(-(2pi)/3)=
c) tan((8pi)/3)
a. 1/2, b. -1/2, c. -sqrt3
What are the last 5 unique teams to win the superbowl?
Kansas City, Los Angeles, Tampa Bay, New England, Denver
Using pizza slices, explain how many slices there are in the pizza and find the reference angles in each quadrant to pi/7 .
There should be 14 slices since we cut each half into 7.
Q_1=pi/7; Q_2=(6pi)/7;Q_3=(8pi)/7;Q_4=(13pi)/7
A circle has an area of 33pi cm^2 and a central angle of (11pi)/6 rad . Find its radius.
33picm^2=(1/2)(11pi)/6r^2=>r=6cm
Gear1 has a radius of 8 cm and rotates (2pi)/3 and Gear2 has a radius of 6. How much does gear 2 rotate?
Gear1:
8cm*(2pi)/3=(16picm)/3
Gear2:
(16picm)/3=(6cm)/(1rad)theta=>theta=(8pi)/9 rad
Smaller radius for gear 2, therefore it must have a larger central angle.
Where does tan(theta)=-sqrt3/3, [0,2pi] ?
(5pi)/6 & (11pi)/6
Draw a complete unit circle from memory.
How many radians does the hour hand of a clock turn in 4 days? 10 days? half a day?
The hour hand rotates around the clock 2 times per day. So, 1 day is
(4pi rad)/(1 day)
Therefore: 4 days is 16pi rad, 10 days is 40pi rad and half a day is 2 pi rad.
Given the arc of a circle is 16pi and has a central angle of (4pi)/3 , find the area of the sector of the circle formed by the same arc.
Arc: 16pi=r*(4pi)/3=>r=12
Area=(1/2)(4pi)/3(12)^2=96pi
If city_1 is (pi)/3 rad above the equator, and city_2 is (pi)/6 rad above the equator:
How far apart are they if the earth has a radius of 6400 km?
(pi)/3 rad-(pi)/6 rad=(pi)/6 rad
(pi)/6 rad * 6400 km = (3200pi)/3 km
Where does Sin^2(theta)=1/2, [-pi,pi] ?
Theta= +-pi/4, +-3pi/4
Riddle: What vehicle is always too tired?
a motorcycle (a bicycle can work i guess)