Download Unit 2 Packet

AFM UNIT 2 PACKET
Angles and Their Measure
NAME ________________________
1. Ray: A part of a line that has only one endpoint
2. Angle: Formed by two rays that have a common endpoint
3. Initial Side: Ray that remains fixed; Angle of rotation begins from this ray
4. Terminal Side: Ray that rotates away from the initial side; where the angle ends.
5. Vertex: The common endpoint of an angle’s initial side and terminal side.
6. Positive Angle:
Angle that has counterclockwise rotation
9. Standard Position:
7. Negative Angle:
Angle that has a clockwise rotation
8. Label the Quadrants:
when an angle has its vertex at the origin and its initial side along the positive x-axis.
10. Sketch the following angles in standard position:
a. 𝜃 = 125°
b. 𝜃 = −60°
c. 𝜃 = 480°
11. Quadrantal Angle: when the terminal side of an angle in standard position coincides with one of the axes.
Sketch the following angles:
a. 𝜃 = 270°
b. 𝜃 = −900°
c. List 5 different measures of quadrantal angles(degrees): ________, _______, ________, _________, _________
1
14. Angle measured in radians(𝜃
𝑅
):
Therefore, s = _________.
For one rotation,
ratio of arc length(s) to radius(r)
𝜃𝑅 =
𝑠
𝑟
The arc length of a circle in one rotation is represented by __________.
𝜃 𝑅 = _________ radians. So, 360⁰ = _______.
To change from degrees to radians: Multiply by
𝜋
180
To change from Radians to degrees: Multiply by
15. Change to radians.
a) 45°
16. Change to degrees.
a)
7
8
b) 120°
b)
c) -30°
2
5
c) 
9
4
d) 5 radians
Co-terminal Angles: Same initial and terminal ray. (differ by 360º or multiple of 360º) or 2π if in radians.
17. Find one positive angle and one negative angle that are co-terminal with each given angle .
a) 395°
b) -45°
c)
2
3
d)
11
4
2
Sketch each angle in standard position and determine the quadrant in which its terminal side lies.
5.
15𝜋
4
7. −220°
9. −
4𝜋
Change each degree measure to radian measure in terms of 𝜋.
Change each radian measure to degree measure.
11.
3
13. 18°
14𝜋
3
15. 1°
17. 𝜋
19. −
7𝜋
6
State whether each pair of angles is co-terminal. Write yes or no.
𝜋 9𝜋
14𝜋
21. 4 , 4
23. 120°, 3
Sketch each angle in standard position and determine the quadrant in which its terminal side lies.
11𝜋
25. −167°
27. 227°
29. −730°
31. − 5
Change each degree measure to radian measure in terms of 𝜋.
33. −150°
35. 105°
37. −450°
39. −1250°
Change each radian measure to degree measure.
𝜋
41. −3.5
43. − 2
47.
7𝜋
45. − 12
17𝜋
6
Find one positive angle and one negative angle that are co-terminal with each angle.
5𝜋
49. 12
51. −310°
3
Practice Worksheet
Sketch each angle and determine the quadrant in which its terminal side lies.
1.
7𝜋
12
5. −156°
2. −
2𝜋
3
6. 1000°
14𝜋
3. 371°
4.
7. 332°
8. −240°
5
Change each degree measure to radian measure in terms of 𝜋. (EXACT VALUE)
9. 36°
10. −250°
11. −145°
12. 6°
13. 870°
16. 345°
14. 18°
15.−820°
Change each radian measure to degree measure. (EXACT VALUE)
17. −1
18. 4𝜋
19. −2.56
3𝜋
21. 16
22. −
7𝜋
9
23.
13𝜋
30
20. 12.85
24. −
17𝜋
3
Find one positive angle and one negative angle that are co-terminal with each angle.
25. 70°
26. −
2𝜋
5
27. −300°
28.
3𝜋
4
4
Day 2 notes Central Angles and Arcs
Central Angles: an angle whose vertex lies at the center of the circle
Length of an Arc: “s” also known as arc length: s   r (θ must be in radians)
1. Find the length of an arc that subtends a central angle of 38 in a circle of radius 5cm.
2. Find the length of an arc that subtends a central angle of

in a circle of diameter 20ft.
6
3. Find the length of an arc that subtends a central angle of
4
in a circle of radius 8mm.
7
4. Find the degree measure of the central angle in a circle that has an arc length of 87cm and a
radius of 16 cm.
5. Find the degree measure of the central angle in a circle that has an arc length of 5.6 cm
and a radius of 12 cm.
Sector of a circle: A region bounded by a central angle and the intercepted arc.
Area of a Circular Sector: A 
1 2
r  (𝜃 𝑚𝑢𝑠𝑡 𝑏𝑒 𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛𝑠)
2
or 𝐴 =
1 𝑅 2
𝜃 𝑟
2
6. Find the area of the sector of the circle that has a central angle measure of

and a radius of 14 cm. Exact value.
6
7. A sector has arc length 12 cm and a central angle measuring 1.25 radians.
Find the radius of the circle and the area of the sector. Round to nearest tenth.
8. A sector has central angle measuring 50⁰ and arc length measuring 10 cm. Find the length of the radius (exact value).
Find the area of the sector. Round final answer to nearest tenth.
5
HOMEWORK Leave answers in terms of pi unless otherwise noted
Given the radian measure of a central angle, find the measure of its intercepted are in terms of 𝜋 in a circle of
radius 10.
𝜋
2𝜋
5𝜋
2𝜋
16. 4
17. 3
18. 6
19. 5
Given the measurement of a central angle, find the measure of its intercepted arc in terms of 𝜋 in a circle of
diameter 30 in.
20. 30°
21. 5°
22. 77°
23. 57°18′
Given the measure of an arc, find the degree measure to the nearest tenth of the central angle it subtends in a
circle of radium 8 cm.
24. 5
25. 14
26. 24
27. 12.5
Find the area of each sector to the nearest tenth, given its central angle, and the radius of the circle.
5𝜋
2𝜋
28. 𝜃 = 12 , 𝑟 = 10 𝑓𝑡
29. 𝜃 = 54°, 𝑟 = 6 𝑖𝑛.
30. 𝜃 = 3 , 𝑟 = 1.36 𝑚
31. 𝜃 = 82°, 𝑟 = 7.3 𝑘𝑚
32. 𝜃 = 45°, 𝑟 = 9.75 𝑚𝑚
33. 𝜃 = 12°, 𝑟 = 14 𝑦𝑑
For Exercises 34-38, round answers to the nearest tenth.
34. An arc is 6.5 cm long and it subtends a central angle of 45°. Find the radius of the circle.
35. An arc is 70.7 , long and it subtends a central angle of
2𝜋
7
. Find the diameter of the circle.
36. A sector has arc length 6 ft and central angle of 1.2 radians. Find the radius and area of the circle.
37. A sector has area of 15 in2 and central angle of 0.2 radians. Find the radius of the circle and arc length of
the sector.
38. A sector has a central angle of 20° and arc length of 3.5 mm. Find the radius and area of the circle.
6
5.2 Practice Worksheet
I. Given the radian measure of a central angle, find the measure of its intercepted arc in terms of 𝜋 in a circle of
radius 10 cm.
1.
5.
𝜋
𝜋
2. 3
6
3𝜋
6.
5
4𝜋
7
𝜋
𝜋
3. 2
4. 5
𝜋
𝜋
7. 12
8. 24
II. Given the measurement of a central angle, find the measure of its intercepted arc in terms of 𝜋 in a circle of
diameter 60 in.
9. 10°
10. 60°
11. 42°
12. 50°
13. 72°
14. 110°
15. 35°
16. 65°
III. Given the measure of an arc, find the degree measure to the nearest tenth of the central angle if subtends in a
circle of radius 16 cm.
17. 87
18. 5.6
19. 12
20. 25
21. 10.24
22. 7.9
23. 11
24. 6
IV. Find the area of each sector to the nearest tenth, given its central angle, and the radius of the circle.
𝜋
25. 𝜃 = 6 , 𝑟 = 14 𝑐𝑚
26. 𝜃 = 36°, 𝑟 = 12 𝑓𝑡
7
Unit Circle
Center: ______ radius = ___
Equation: ____________
8
Circular Functions
I. Definition of Sine and Cosine
If the terminal side of an angle in standard position intersects the unit circle P(x,y), then cos  = x and sin  = y
●P(
,
)
SIGNS FOR EACH QUADRANT
cosθ = x
sinθ = y
y
● 𝜃
x
1. Sine and Cosine Functions of an Angle is Standard Position For any circle whose radius is r.
𝑠𝑖𝑛𝜃 =
𝜃
𝑐𝑜𝑠𝜃 =
𝑥 2 + 𝑦 2 = 𝑟 2 therefore 𝑟 = √
, 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑢𝑠 𝑖𝑠 𝐴𝐿𝑊𝐴𝑌𝑆 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒.
2. Find the values of the sine and cosine functions of an angle in standard position with measure  if the point with
coordinates (5, 12) lies on its terminal side. Sketch the reference triangle.
3. Find the sin  when cos  =
8
and the terminal side of  is in the fourth quadrant.
17
9
4. Trigonometric Functions of an Angle in Standard Position
For any angle in standard position with measure  , a point P(x, y) on its terminal side, and r 
x 2  y 2 , the
trigonometric functions of  are as follows:
sin  =
cos  =
tan  =
5. The terminal side of an angle  in standard position contains the point with coordinates (-3, -4).
Find sinθ, cosθ, tan  ,
6. The terminal side of an angle  in standard position contains the point with coordinates (0, -4).
Find sinθ, cosθ, tan  ,
7. The terminal side of an angle  in standard position contains the point with coordinates (7, 0).
Find sinθ, cosθ, tan  ,
8. If tan  = -2 and  lies in Quadrant IV, find sin  and cos  .
9. Exact Values:
(0, 1)
0º
sinθ
90º
180º
270º
360º
(1,0)
(−1,0)
cosθ
tanθ
(0, −1)
10
HOMEWORK: Only find sine, cosine, and tangent for each
Find the value for sine, cosine, and tangent of an angle in standard position if a point with the given coordinates
lies on its terminal side. Plot the point and sketch a reference triangle for each if possible.
25. (1, −8)
23. (15,8)
27. (−√2, √2)
29. (0,2)
Suppose 𝜃 is an angle in standard position whose terminal side lies in the given quadrant. For each functions,
find the values of the remaining two trigonometric functions of 𝜃. Sketch the reference triangles.
4
1
30. sin 𝜃 = − 5 ; 𝑄𝑢𝑎𝑑𝑟𝑎𝑛𝑡 𝐼𝑉
31. cos 𝜃 = − 2 ; 𝑄𝑢𝑎𝑑𝑟𝑎𝑛𝑡 𝐼𝐼
32. tan 𝜃 = 2; 𝑄𝑢𝑎𝑑𝑟𝑎𝑛𝑡 𝐼
Tell whether the value of each trigonometric function is positive, negative, zero, or undefined.
𝜋
5𝜋
34. sin 2𝜋
35. cos 4
36. tan 315°
37. cos 4
38. sin
11𝜋
39. tan 90°
4
42. Fill in the tables.
Ordered pair
40. cos 450°
Quadrant
sinθ
cosθ
41. sin(−45°)
tanθ
(-15,8)
QUADRANT
x
y
r
III
II
sinθ
cosθ
tanθ
21
20
2
7
11
Day 5
Name the quadrant(s) that satisfy the given conditions:
1. cosθ > 0
Fill in the chart with EXACT values.
𝜋
𝜋
𝜋
6
4
3
2. sinθ < 0 and tanθ > 0
3. cosθ and sinθ have opposite signs
sinθ
4. cosθ < 0 and tanθ < 0
5. tanθ and sinθ have the same sign
6. cosθ and tanθ have opposite signs
cos θ
tan θ
Unit Circle with EXACT Values
12
Given Angle
ordered pair
Quadrant
sin θ
cos θ
tan θ
1. 135°
2. 210°
3. 420°
4. 180°
5. 315°
6.
7.
8.
9.
7𝜋
6
3𝜋
2
9𝜋
4
10𝜋
3
10. −
15𝜋
4
11. -
12. sin 813o
13. cos 930
14. tan 2150
15. sin 17.430
16. cos (−
5𝜋
9
)
17. tan (10.4)
18. sin 6
19. cos (-13.58π)
13
Find each exact value. Do not use a calculator.
𝜋
9𝜋
16. cos 60°
17. sin 3
18. tan 4
22. cos
5𝜋
6
23. sin
7𝜋
25. tan 3𝜋
6
Use a calculator to approximate each value to four decimal places.
28. sin 710°
30. sin 7
33. tan 115°
34. cos 72°
Practice Worksheet
Trigonometric Functions of Special Angles
Find each exact value. Do not use calculator.
𝜋
𝜋
𝜋
1. sin 4
2. cos 4
3. tan 4
5. sin 300°
6. tan 330°
7. sin
10. sin 90°
11. tan 45°
9. tan
3𝜋
4
13. tan
3𝜋
2
14. sin
3𝜋
4
4. cos 210°
8. cos
3𝜋
4
12. cos
3𝜋
2
3𝜋
2
Use a calculator to approximate each value to four decimal places.
15. sin 634°
16. cos 235°
17. sin 2
14
Day 5 HW worksheet
EXACT VALUES without a Calculator!
Given Angle
ordered pair
Quadrant
Name ________________________
sin θ
cos θ
tan θ
1. 450°
2. 90°
3. 120°
4. 900°
5. 300°
6.
7𝜋
3
7. −
8.
9.
7𝜋
2
9𝜋
4
11𝜋
6
10. −
19𝜋
4
11. 3
Find the values USING CALCULATOR. Round to FOUR places. **Be careful of MODE***
_____________________12. sin 710°
_____________________13. cos 72°
_____________________14. tan 115°
_____________________15. sin 7.42°
_____________________16. 𝑐𝑜𝑠 (−
4𝜋
9
)
_____________________17. 𝑡𝑎𝑛(16.4)
_____________________18. sin 2
_____________________19. 𝑐𝑜𝑠(11.55𝜋)
15
AFM Day 6 Examples
Sinusoidal Functions
NAME ____________________
𝑦 = asin(𝑏(𝑥 − 𝑐)) + 𝑑 𝑜𝑟 𝑦 = 𝑎 cos(𝑏(𝑥 − 𝑐)) + 𝑑
A = _______________
b = _______________
c = _____________
d = _______________
Period = _________________
I. Determine the amplitude, b value, and period for each function.
Amplitude
b – value
period
1. 𝑦 = 3𝑠𝑖𝑛𝑥
1
2. 𝑦 = 𝑐𝑜𝑠2𝑥
2
3. 𝑦 = −5𝑠𝑖𝑛𝜋𝑥
4. 𝑦 = −8𝑐𝑜𝑠
5𝜋
4
1
𝑥
5. 𝑦 = 2.5𝑐𝑜𝑠 𝑥
2
6. 𝑦 = −𝑠𝑖𝑛3𝑥
7. Graph y = sinx
a = __________
b = _________
p = __________
16
8. Graph y = cosx
a = __________
b = _________
p = __________
9. y = -2cos3x
a = __________
b = _________
p = __________
a = __________
b = _________
p = __________
1
10. 𝑦 = −3𝑠𝑖𝑛 2 𝑥
17
HW AFM Worksheet
I. Determine the amplitude, b value, and period for each function.
Amplitude
NAME ____________________
b – value
period
1
1. 𝑦 = − 𝑐𝑜𝑠𝑥
2
2. 𝑦 = 2𝑐𝑜𝑠6𝑥
𝜋
3. 𝑦 = −4𝑠𝑖𝑛 𝑥
4
3
4. 𝑦 = 3𝑠𝑖𝑛 𝑥
2
5. 𝑦 = −5𝑐𝑜𝑠
5𝜋
3
𝑥
6. 𝑦 = 𝑐𝑜𝑠2𝑥
II. Find the following and then graph at least 2 cycles of each function on p. 22 and p. 23.
a
b
period
7. 𝑦 = 3𝑠𝑖𝑛𝑥
8. 𝑦 = 5𝑐𝑜𝑠𝑥
9. 𝑦 = 4𝑐𝑜𝑠2𝑥
1
10. 𝑦 = −2𝑠𝑖𝑛 𝑥
2
11. 𝑦 = −𝑐𝑜𝑠3𝑥
1
12. 𝑦 = 2𝑠𝑖𝑛 𝑥
3
III. Determine the amplitude, b value, period, phase shift and vertical shift for each function.
a
b
p
PS
VS
13. 𝑦 = 𝑐𝑜𝑠2𝑥 − 5
14. 𝑦 = sin(𝑥 − 𝜋)
15. 𝑦 = −2 sin(3𝑥 + 6𝜋) + 4
𝜋
16. 𝑦 = 4 cos (3𝑥 − ) − 7
3
1
17. 𝑦 = − 𝑐𝑜𝑠(4𝑥 − 2𝜋) − 6
3
18. 𝑦 = 3𝑠𝑖𝑛6𝑥 − 3
IV. Find the following and then graph at least 2 cycles on GRAPH paper.
a
b
p
19. 𝑦 = 𝑐𝑜𝑠2𝑥 − 1
20. 𝑦 = 2𝑠𝑖𝑛(𝑥 + 𝜋)
1
PS
VS
𝜋
21. 𝑦 = 3𝑐𝑜𝑠 (𝑥 − ) + 4
2
2
22. 𝑦 = −4𝑠𝑖𝑛(3𝑥 − 6𝜋) − 2
18
Day 7 Notes
a
b
p
PS
VS
1. 𝑦 = sin(𝑥 − 𝜋)
2. 𝑦 = 3𝑠𝑖𝑛6𝑥 − 3
3. 𝑦 = 2cos(3𝑥 + 3𝜋)
1
4. 𝑦 = −𝑐𝑜𝑠 𝑥 + 4
2
5. 𝑦 = −2 cos (3𝑥 −
4𝜋
3
)−3
6.
7. y = cos3x + 3
8. 𝑌 = −3 sin(2𝑥 + 2𝜋) − 2
19
3
9. 𝑦 = −5𝑐𝑜𝑠 (2 𝑥 + 3𝜋) + 4
10. Write a sine and cosine equation for the given graphs.
20
Day 8 notes AFM Class Examples
Write a sine and cosine equation for each graph
1.
2.
3.
4.
5.
6.
21
DRAW THE X AND Y AXIS SO THE GRAPH WILL FIT from -2π to 2π OR SHOWING 2 FULL CYCLES
22
DRAW THE X AND Y AXIS SO THE GRAPH WILL FIT from -2π to 2π OR SHOWING 2 FULL CYCLES
23
Day 8 HW: Write a sine and cosine equation for each graph.
1.
2.
_________________________________________ ____________________________________________
3.
4.
__________________________________________
______________________________________
5.
6.
__________________________________________
___________________________________________
24
Day 8 or 9 Trig Word Problems
1. The initial behavior of the vibrations of the note E above middle C can be modeled by 𝑦 = 0.5 sin 660𝜋𝑡.
a. What is the amplitude of this model?
b. What is the period of this mode?
𝜋
2. If the equilibrium point is 𝑦 = 0, then 𝑦 = −4 cos( 𝑡) models a buoy bobbing up and down in the water.
6
a. What is the period of the function?
b. What is the location of the buoy at 𝑡 = 10.
3. A rodeo performer spins a lasso in a circle perpendicular to the ground. The height of the knot from the ground is
modeled by y = −3 cos(
a.
b.
c.
d.
5𝜋
3
𝑡) + 3.5, where t is the time measured in seconds.
What is the highest point reached by the knot?
What is the lowest point reached by the knot?
What is the period of the model?
According to the model, find the height of the knot after 25 seconds.
𝜋
4. The function 𝑦 = 25 sin( 𝑡) + 60, where t is in months and 𝑡 = 0 corresponds to April 15, models the average
6
high temperature in degrees Fahrenheit in Centerville.
a.
b.
c.
d.
Find the period of the function.
What does the period represent?
What is the maximum high temperature?
When does the maximum occur?
5. The figure shows the depth of water at the end of a boat dock. The depth is 6 feet at low tide and 12 feet at high tide.
On a certain day, low tide occurs at 6 A.M. and high tide occurs at noon. If y represents the depth of the water x hours
after midnight, use a cosine function of the form 𝑦 = 𝐴𝑐𝑜𝑠 𝐵𝑥 + 𝐷 to model the water’s depth.
6. An average seated adult breathes in and out every 4 seconds. The average minimum amount of air in the lungs is 0.08
cubic liters, and the average maximum amount of air in the lunges is 0.82 cubic liters. Suppose the lunges have a
minimum amount of air at 𝑡 = 0, where t is the time in seconds.
a. Write a function that models the amount of air in the lungs.
b. Determine the amount of air in the lungs at 5.5 seconds.
25
7. A Ferris wheel at an amusement park has riders get on at position A, which is 3 m above the ground. The highest
point of the ride is 23 meters above the ground. The ride takes 40 seconds for one complete revolution.
Write the equation that models the height of the Ferris wheel over time.
23 m
8. A Ferris wheel has a diameter of 80 feet. Riders enter the Ferris wheel at its lowest point, 6 feet above the ground
at time t = 0 seconds. One complete rotation takes 67 seconds. Write a function modeling a riders height, h(t), at
t seconds.
9. Sam and Dan are being dared to ride the Ferris wheel The height h (in feet) above the ground at any time t (in
seconds) can be modeled by:
𝜋
y= 40𝑐𝑜𝑠 (
20
𝜋
𝑡 + 2 ) + 50
a. Find the amplitude and period.
b. The Ferris wheel turns for 160 seconds before it stops to let Sam and Dan get off.
How many times will they go around?
c. What are the minimum and maximum heights for Sam and Dan?
10. Suppose a Ferris wheel has a radius of 20 feet and operates at a speed of 3 revolutions per minute.
The bottom car is 4 feet above the ground. Write a model for the height of a person above the ground
whose height when t = 0 is 44 feet.
26
AFM Review Unit 2 Test
NAME _________________________
Determine the quadrant each angle lies in.
1. −
7𝜋
2.
4
8𝜋
3. -447º
3
4. 623º
5.
23𝜋
6
Change each degree to radian measure. (exact value)
6. 220º
7. -442º
8. 124º
9. -450º
10. 15º
Change each radian to degrees.
11. −
11𝜋
12.
7
9𝜋
13. −
4
5𝜋
2
14.
17𝜋
18
Find a positive and negative co-terminal angle for each given angle. Keep units the same.
15.
17𝜋
16. -108º
6
Find the values of each given the point on the terminal side of θ.
25. (-2, 7)
sinθ =
cosθ =
tanθ =
26. (1, -9)
sinθ =
cosθ =
tanθ =
27. If θ lies in quadrant II, and sinθ =
8
, then tan θ = __________ and cosθ = ________
17
28. Find the arc length of a circle given the central angle is 1.5 radians and the radius is 7 cm.
29. An arc has a measure of 6 cm intercepts a central angle of 75º. Find the radius of the circle.
Round to the nearest tenths.
30. Find the area of a sector if its central angle is 35º and the radius of the circle is 12.4 cm. Round
to nearest tenths.
Use a calculator and round to FOUR places. Beware of MODE.
3𝜋
31. sin 34º14’48”
32. tan(-85.3º)
33. 𝑐𝑜𝑠 (− 8 )
23𝜋
34. 𝑡𝑎𝑛 ( 25 )
27
EXACT VALUES NO CALC!!!
35. tan 180º
40. 𝑐𝑜𝑠
36. cos 315º
2𝜋
41. 𝑠𝑖𝑛 (−
3
7𝜋
4
)
37. sin 150º
42. 𝑡𝑎𝑛
3𝜋
2
38. tan 135º
43. 𝑐𝑜𝑠
39. sin 0º
11𝜋
6
44. sin 3𝜋
Graph on graph paper. Label the axes!
𝜋
45. 𝑦 = 2𝑠𝑖𝑛2 (𝑥 + 2 ) − 5
47. Given ℎ(𝑡) = −32𝑐𝑜𝑠
46. 𝑦 = −3𝑐𝑜𝑠(3𝑥 + 𝜋) + 4
𝜋
23
𝑡 + 48
___________________= max height
_________________= min height
_______________= period
48. A Ferris wheel with a radius of 17 feet is rotating at a rate of 4 revolutions per minute. The bottom chair is
8 feet above the ground. Write a model for the height of a person above the ground whose height when t = 0 is
42 feet.
49. Write a sine and cosine equation for the following graph.
28