5.4 Investigation: Sketching the Graphs of
f(x) = sin x, f(x) = cos x, and f(x) = tan x
Graphing y = sin x
Copy and complete the following table of values for y = sin x,
0° ≤ x ≤ 360°. Include the exact value of sin x for each value of x. Also
include the decimal value of sin x, rounded to the nearest tenth, if necessary.
1.
Value of x
(radians)
Value of x
(degrees)
Exact Value of
sin x
Decimal Value
of sin x
0
π
6
0
30 60 90 120 150 180 210 240 270 300 330 360
0
1
2
0
π
3
π
2
2π 5π
3 6
π
7π 4π 3π 5π 11π 2π
6 3 2 3 6
3
2
0.5 0.9
Use the decimal values of sin x
and plot the ordered pairs
(x, sin x) on a grid, like the one
shown. Join the points with a
smooth, continuous curve.
2.
y
1
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
–1
60
120
3. a) What is the maximum value of y = sin x?
b) For what value of x does the maximum value
of y occur?
4. a) What is the minimum value of y = sin x?
b) For what value of x does the minimum value
of y occur?
5.
180
240
300
360
x
What is the amplitude of y = sin x?
Make a conjecture about the appearance of the graph of y = sin x for
the domain 0° ≤ x ≤ 720°.
b) Use your conjecture to sketch the graph of y = sin x for the domain
0° ≤ x ≤ 720°.
6. a)
5.4 Investigation: Sketching the Graphs of f(x) = sin x, f(x) = cos x, and f(x) = tan x • MHR 363
Test your conjecture from question 6a) by graphing y = sin x on a
graphing calculator. Using the mode settings, select the degree mode. Adjust
the window variables to include Xmin = 0, Xmax = 720, Ymin = −1.5, and
Ymax = 1.5. Display the graph. Compare the result with your sketch from
question 6b).
7.
8. a) Make a conjecture about the appearance of the graph of y = sin x for
the domain −360° ≤ x ≤ 720°.
b) Use your conjecture to sketch the graph of y = sin x for the domain
−360° ≤ x ≤ 720°.
Test your conjecture from question 8a) by graphing y = sin x on a
graphing calculator. In degree mode, adjust the window variables to include
Xmin = −360, Xmax = 720, Ymin = −1.5, and Ymax = 1.5. Display the
graph. Compare the result with your sketch from question 8b).
9.
10. a) Is the graph of y = sin x periodic? Explain.
b) If so, what is the period of the graph of y = sin x?
11.
How can you verify that y = sin x is a function?
12. For the function
a) the domain?
y = sin x, what is
b)
the range?
Graphing y = cos x
13. Copy and complete the following table of values for y = cos x,
0° ≤ x ≤ 360°. Include the exact value of cos x for each value of x. Also
include the decimal value of cos x, rounded to the nearest tenth, if necessary.
Value of x
(radians)
Value of x
(degrees)
Exact Value of
cos x
Decimal Value
of cos x
0
π
6
π
3
π
2
2π 5π
3 6
0
30 60 90 120 150 180 210 240 270 300 330 360
π
7π 4π 3π 5π 11π 2π
6 3 2 3 6
Use the decimal values of cos x and graph y = cos x, 0° ≤ x ≤ 360°, on
the same set of axes as y = sin x.
14.
15.
What is the amplitude of y = cos x?
364 MHR • Chapter 5
Make a conjecture about the appearance of the graph of y = cos x
for the domain 0° ≤ x ≤ 720°.
b) Use your conjecture to sketch the graph of y = cos x for the domain
0° ≤ x ≤ 720°.
16. a)
Test your conjecture from question 16a) by graphing y = cos x on a
graphing calculator. Compare the result with your sketch from
question 16b).
17.
18. a) Make a conjecture about the appearance of the graph of y = cos x
for the domain −360° ≤ x ≤ 720°.
b) Use your conjecture to sketch the graph of y = cos x for the domain
−360° ≤ x ≤ 720°.
Test your conjecture from question 18a) by graphing y = cos x,
−360° ≤ x ≤ 720°, on a graphing calculator. Compare the result with your
sketch from question 18b).
19.
20. a) Is the graph of y = cos x periodic? Explain.
b) If so, what is the period of the graph of y = cos x?
21.
How can you verify that y = cos x is a function?
22. For the function
a) the domain?
y = cos x, what is
b)
the range?
Compare the graphs of y = sin x and y = cos x. How are they the same?
How are they different?
23.
Graphing y = tan x
Copy and complete the following table of values for y = tan x.
Round decimal values of tan x to the nearest tenth, if necessary.
24.
Value of x
0 45 60 70 80 90 100 110 120 135 150 180
(degrees)
Decimal Value
of tan x
Value of x
225 240 250 260 270 280 290 300 315 330 360
(degrees)
Decimal Value
of tan x
5.4 Investigation: Sketching the Graphs of f(x) = sin x, f(x) = cos x, and f(x) = tan x • MHR 365
Graph y = tan x on a grid
like the one shown.
25.
y
8
6
4
2
0
–2
–4
–6
–8
26.
60
120
180
240
300
What happens to the value of tan x as x increases from 0° to 90°?
27. a) What is the value of tan x when x = 90°?
b) How is this value of tan x shown on the graph?
28.
What happens to the value of tan x as x increases from 90° to 270°?
29. a) What is the value of tan x when x = 270°?
b) How is this value of tan x shown on the graph?
Make a conjecture about the appearance of the graph of y = tan x for the
domain 0° ≤ x ≤ 720°.
b) Use your conjecture to sketch the graph of y = tan x for the domain 0° ≤ x ≤ 720°.
30. a)
31. Test your conjecture from question 30a) by graphing y = tan x, 0° ≤ x ≤ 720°,
on a graphing calculator. Compare the result with your sketch from question 30b).
32. a) Make a conjecture about the appearance of the graph of y = tan x for
the domain −360° ≤ x ≤ 720°.
b) Use your conjecture to sketch the graph of y = tan x for the domain
−360° ≤ x ≤ 720°.
Test your conjecture from question 32a) by graphing y = tan x, −360° ≤ x ≤ 720°,
on a graphing calculator. Compare the result with your sketch from question 32b).
33.
34. a) Is the graph of y = tan x periodic? Explain.
b) If so, what is the period of the graph of y = tan x?
35.
Does y = tan x have a maximum value? a minimum value? Explain.
36.
How can you verify that y = tan x is a function?
37. For the function
a) the domain?
366 MHR • Chapter 5
y = tan x, what is
b)
the range?
360
x