Download Unit 2 WS

Introduction to Trigonometric Graphs
The following graph shows the sine function from 0º to 360º.
1) Where does sin Θ = 0?
sin _______ = 0
sin ________ = 0
sin ________ = 0
2) Where does sin Θ = 1?
sin _______ = 1
3) Where does sin Θ = -1?
sin _______ = -1
The sine function (and the other five trigonometric functions) are periodic. This means the graph
repeats itself at certain intervals. The sine function repeats itself every 360º. The graph below
shows the sine function from -720º to 720º.
4) Find the values by referring to the graph above.
sin 450º = _______
sin -90º = _______
sin 720º = _______
sin -270º = ______
sin -540º = ______
sin 630º = _______
5) In the interval [-720˚, 720˚], find all values of Θ for which each equation is true.
sin Θ = 0
________________________
sin Θ = 1
________________________
sin Θ = -1 ________________________
6) Graph the sine curve in the interval 0º ≤ Θ ≤ 1440º
7) Approximate each value to 3 decimal places (use your calculator, f(x) = sinx) and plot the points
on the graph below. to 3 places. This will give you a more accurate graph of the sine curve.
sin 10º = ________ sin 20º = _______
sin 50º = _______
sin 30º = _______
sin 60º = _______ sin 70º = ______
sin 40º = _______
sin 80º = _______
sin 90º = _______ sin 180º = _______ sin 270º = _______ sin 360º = _______
WS 2.1
Refer to the Graphs of the Six Trigonometric Functions to answer the following questions.
1. What similarity do you notice between the sine and cosine curves?
2. Cosecant and secant are reciprocals of the sine and cosine ratios, respectively. What
similarities do you see between the cosecant and secant curves?
3. How do the sine and cosecant graphs compare?
4. How do the cosine and secant graphs compare?
5. How do the tangent and cotangent graphs compare?
A quadrantal angle is an angle in standard position whose terminal side coincides with one of the
coordinate axes. The measures of the quadrantal angles are…..-270º, -180º, -90º, 0º, 90º, 180º,
270º…. Knowing the characteristics of the basic trigonometric graphs can help you quickly determine
quadrantal values of the trigonometric function.
6. The value of the sine function at –360º, -180º, 0º, 180º, 360º and all other integral multiples of
180º is: a) zero
b) 1
c) –1
d) undefined
7. The value of the sine function at –270º and 90º and all other integral multiples of 360º is:
a) zero b) 1
c) –1
d) undefined
8.
The value of the sine function at -90º and 270º and all other integral multiples of 360º is:
a) zero b) 1
c) –1
d) undefined
9.
The maximum value for the sine function is _______ and occurs at _______º in the
interval [0, 360]
10. The minimum value for the sine function is ______ and occurs at _______º in the
interval [0, 360]
11.
The value of the cosine function at –360º, 0º, 360º and all other integral multiples of 180º
is: a) zero b) 1
c) –1
d) undefined
12. The value of the cosine function at –270º and 90º and all other integral multiples of 360º is:
a) zero b) 1
c) –1
d) undefined
13. The value of the cosine function at -90º and 270º and all other integral multiples of 360º is:
a) zero b) 1
c) –1
d) undefined
14. The maximum value for the cosine function is _______ and occurs at________º in the
interval [0, 360]
15. The minimum value for the cosine function is _______ and occurs at________º in the
interval [0, 360]
16. The vertical lines in the graphs of the tangent, cotangent, secant, and cosecant functions are
asymptotes. What do they indicate about the values of Θ where the asymptotes cross the Θ
axis?
The fact that the trigonometric functions are periodic allows us to find more function values easily.
The period of a trigonometric function is the number of degrees taken to complete one cycle (which is
the portion of the graph of the function from one point on the graph to the point at which the graph
starts repeating itself).
Since the graph of the sine function repeats itself every 360º, its period is 360º. This makes it easy
to find other zero, maximum, and minimum values. For example, since we know that sin 90º = 1, sin
(90º  360pº) = 1 for any integral value of p. The same method can be used to find special angles of
other functions.
17. State the period of each of the trigonometric functions. (How many degrees does it take for
the graph to complete one cycle?)
sine
______________
cosecant
___________
cosine
_____________
secant
___________
tangent ____________
cotangent ___________
18. Find the values of Θ for which each equation is true. [0, 360]
sin Θ = -1
cos Θ = 1
tan Θ = 0
csc Θ = undefined
19. Complete the table by finding the values of cosine Θ in the interval –540º≤ Θ ≤ 0º Plot the
ordered pairs. Connect these points with a smooth continuous curve.
Θ
-540º
-450º
-360º
-270º
-180º
-90º
0º
cos Θ
Find all values in the domain: [-360º, 360º]
20. cos Θ = 0
21. cos Θ = 1
22. cos Θ = -1
23. tan Θ = 0
24. tan Θ = undefined
25. csc Θ = 1
26. csc Θ = -1
27. csc Θ = undefined
28. sec Θ = 1
29. sec Θ = -1
30.. sec Θ = undefined
31. cot Θ = 0
Find each value.
32. sin (-720º)
33. cos (540º)
34. tan (-180º)
35. csc (720º)
36. sec (180º)
37. cot (180º)
38. sin (270º)
39. cos (-180º)
40. tan (270º)
41. csc (-90º)
42. sec (270º)
43. cot (-270º)
Graph each function on the given interval.
44. y = sin x; -90º ≤ Θ ≤ 90º
45. y = cos x; -180º ≤ Θ ≤ 180º
46. y = tan x, -90º ≤ Θ ≤ 270º
48. y = csc x, 0º ≤ Θ ≤ 360º
50. y = sin x, -720º ≤ Θ ≤ 720º
47. y = sec x, -270º ≤ Θ ≤ 270º
49. y = cot x, -270º ≤Θ ≤ 270º
WS 2.2
Find the amplitude, period, and phase shift. You may need to put in standard form first.
8. y = -4sin(6Θ – 180º)
1. y = 4sinΘ
9. y = 7cos(Θ – 45º)
2. y = 2sin5Θ
10. y = -5cos(3Θ + 60º)
3. y = 2cosΘ
11. y = 7sin5(Θ – 90º)
4. y = 3cos½Θ
12. y = -8cos(5Θ + 180º)
5. y = 6sin4Θ
13. y = 3sin(½Θ + 60º)
6. y = 10cos6Θ
14. y = 4cos (3Θ – 30º)
7. y = 5sin(Θ + 90º)
Write an equation of the sine function given the amplitude, period, and phase shift.
15. a: ¾
p: 360º
ps: 30º
____________________________
16. a: 4
p. 180º
ps: -30º
____________________________
17. a: 5
p: 45º
ps: -60º
____________________________
18. a: 7
p: 225º
ps: 90º
____________________________
19. a: 9
p: 30º
ps: -180º
____________________________
20. a: 1
p: 60º
ps: 120º
____________________________
Write an equation of the cosine function given the amplitude, period, and phase shift.
21. a: 1/3
p: 180º
ps: 0º
____________________________
22. a: 3
p: 180º
ps: 120º
____________________________
23. a: 100
p: 90º
ps: -45
____________________________
24. a: 1
p: 360º
ps: -90º
____________________________
25. a: 3.75
p: 90º
ps. 4º
____________________________
26. a: 12
p: 45º
ps: -180º
____________________________
Graph each function in the domain of the graph.
27. y = ½sin x
28. y = cos(2x)
30. y = sin(½x)
33. y = 4sinx
31. y = 3cos x
29. y = sin(x – 90º)
32. y = cos(x + 90º)
WS 2.3
Put in standard form, if necessary. Find the amplitude, period, phase shift, vertical shift, frequency,
and DBCV. Sketch the graph of one complete cycle on notebook paper clearly showing the critical
points.
1. y = 4 cos(3Θ – 30º) + 7
2. y = -2 sin(2Θ + 60º) - 10
3. y = 5 cos(½Θ + 45º) + 3
4. y = 6 sin(Θ – 30º) + 11
5. y = -2 - 3 cos(4Θ – 180º)
6. 4 + y = - 5 sin(10Θ + 300º)
7. y = 3 + 6cos(Θ – 15º)
3
Given the following graphs, write the equation of the sinusoid as a “+ sine” function” and as a “- sine”
function.
8.
9.
10.
WS 2.4
Sketch one cycle of the graph of the circular function.

2

4. f (x )  2  6 sin (x  1)
1. f (x )  4  5 sin  x  
4
3
2
2. f (x )  4  3cos
1
(x   )
4
5. f (x )  4  5 cos

3
(x  2)

 
6. 12  36sin  3x    4y
15 



4  8cos  2x  
9

3. f (x ) 
2
Use the graphs shown on the back of this worksheet to write an equation of the sinusoids shown
there for the following problems #7 – 12, using a circular function.
7. “+” cosine, “-” sine
10. “+” sine, “+” cosine
8. “-” cosine, “+” sine
11 “-” cosine, “-” sine
9. “+” cosine, “-” sine
12. “+” cosine, “-” sine
Write the equation of the given sinusoid.
13. sine: amplitude: 12, period:
2
5
14. cosine: amplitude: 8, period:
, phase shift: left

, vertical shift: down 5
2


, phase shift: right , vertical shift: up 2
4
6
15. sine: period: 10π, phase shift: left π, vertical shift: down 4; amplitude: 1
16. cosine: amplitude: 12, period: 8π, phase shift: right

, vertical shift up 10
4
17. sine: amplitude: 1, period : 30, phase displacement: right 4, vertical shift down 3
18. sine: amplitude: ½, period: 12, phase displacement: left 10, vertical shift down 2
19. cosine: amplitude: 7, period: 10, phase shift: left 2, vertical displacement: up 8
20. cosine: amplitude: 4, period: 6, phase shift: right 6, vertical displacement: none
Graphs for Worksheet 2.4 #7-12
WS 2.5
Sketch one cycle of the given function . (1 - 4: trigonometric, 5 - 8: circular)
1. y = tan2θ
2. y = 2cscθ
3. y = 4cot3θ
4. y = cot((π/3)x)
Radians
5. y = 3sec(2x)
Radians
6. y = -5 + 3cot4θ
7. y = -1 + 3tan2(θ - 30°)
8. y = 3 + 2sec4(θ + 10°)
9. y = -2 + 4csc(x + 4)
Radians
10. y = -3 + 6csc((π/2(x + 1)
Radians