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5.08 Circular functions: tangent

Worksheet
Graph of tan x
1

Which of the following is the correct domain of y = \tan x?

A

All real numbers, except integer multiples of 90 \degree.

B

All real numbers, except odd integer multiples of 90 \degree.

C

All real numbers, except even integer multiples of 90 \degree.

D

All real numbers.

2

Consider the graph of y = \tan x for - 2 \pi \leq x \leq 2 \pi.

a

State the y-intercept of the graph.

b

State the period of the function.

c

State the equations of the vertical asymptotes on the domain 0 \leq x \leq 2\pi.

d

Does the graph of y=\tan x increase or decrease between any two successive vertical asymptotes?

e

If x \gt 0, find the least value of x for which \tan x = 0.

-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-4
-3
-2
-1
1
2
3
4
y
3

Consider the graph of y = \tan x for - 2 \pi \leq x \leq 2 \pi.

a

Select the word that best describes the graph:

A

Periodic

B

Decreasing

C

Even

D

Linear

b

Determine the range of y = \tan x.

c

As x increases, determine the equation of the next asymptote of the graph after x = \dfrac{7 \pi}{2}.

-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-4
-3
-2
-1
1
2
3
4
y
4

Consider the graph of y = \tan x for - 2 \pi \leq x \leq 2 \pi.

a

Determine the sign of \tan x for \\ \pi \leq x < \dfrac{3 \pi}{2}.

b
In which quadrant of a unit circle is angle x if \pi \leq x < \dfrac{3 \pi}{2}.
c

Determine the sign of \tan x for \\- \dfrac{\pi}{2} < x \leq 0.

d
In which quadrant of a unit circle is angle x if - \dfrac{\pi}{2} < x \leq 0.
e

Describe the function y = \tan x as odd, even or neither.

-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-4
-3
-2
-1
1
2
3
4
y
5

Consider the right triangle containing angle \theta and the graph of y=\cos \theta.

-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
\theta
-1
1
y
a
As angle \theta increases from 0 to \dfrac{\pi}{2}, explain what happens to the value of the opposite side in the triangle.
b

Hence explain what happens to the value of \tan \theta as angle \theta increases from 0 to \dfrac{\pi}{2}, given that \tan \theta is defined as \dfrac{\text{opposite }}{\text{adjacent }}.

c

Express \tan \theta in terms of \sin \theta and \cos \theta.

d

Determine the values of \theta for which \cos \theta = 0, given - 2 \pi \leq \theta \leq 2 \pi.

e

Hence, state the values of \theta between - 2 \pi and 2 \pi for which \tan \theta is undefined.

f

Complete the table below.

\theta-2\pi-\dfrac{7\pi}{4}-\dfrac{5\pi}{4}-\pi-\dfrac{3\pi}{4}-\dfrac{\pi}{4}
\tan \theta
\theta0\dfrac{\pi}{4}\dfrac{3\pi}{4}\pi\dfrac{5\pi}{4}\dfrac{7\pi}{4}2\pi
\tan \theta
g

Hence sketch the graph of y = \tan \theta on the domain - 2 \pi \leq \theta \leq 2 \pi.

Vertical dilations
6

How has the graph y = 3 \tan x been transformed from y = \tan x?

7

Consider the graph of y = a \tan x.

a

From the graph, determine the value of y when x=\dfrac{\pi}{4}.

b

If y=\tan x, determine the value of y when x=\dfrac{\pi}{4}.

c

Find the vertical dilation factor that must be applied to y = \tan x to obtain this graph.

d

Hence state the value of a.

-1\pi
-\frac{3}{4}\pi
-\frac{1}{2}\pi
-\frac{1}{4}\pi
\frac{1}{4}\pi
\frac{1}{2}\pi
\frac{3}{4}\pi
1\pi
x
-4
-3
-2
-1
1
2
3
4
y
8

Determine the equation for each of the following functions, given the equation is in the form y = a \tan x:

a
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
x
-6
-4
-2
2
4
6
y
b
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
x
-6
-4
-2
2
4
6
y
c
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
x
-6
-4
-2
2
4
6
y
d
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
x
-6
-4
-2
2
4
6
y
e
-1\pi
-\frac{3}{4}\pi
-\frac{1}{2}\pi
-\frac{1}{4}\pi
\frac{1}{4}\pi
\frac{1}{2}\pi
\frac{3}{4}\pi
1\pi
x
-6
-4
-2
2
4
6
y
9

On the same set of axes, sketch the graphs of y = \tan x and y = \dfrac{1}{2} \tan x, on the domain -2\pi \leq x \leq 2\pi.

10

On the same set of axes, sketch the graphs of y = 5 \tan x and y = - 4 \tan x, on the domain -\pi \leq x \leq \pi.

11

On the same set of axes, sketch the graphs of the functions f \left( x \right) = - \dfrac{1}{2} \tan x and \\g \left( x \right) = 2 \tan x, on the domain -\pi \leq x \leq \pi.

Horizontal dilations
12

Consider functions of the form y=\tan bx.

a

Complete the table identifying the period of the function when b = 1, 2, 3, 4.

b

State the period of y = \tan b x.

c

As the value of b increases, describe the effect on the period of y=\tan b x.

FunctionPeriod
\tan x\pi
\tan 2x
\tan 3x
\tan 4x
13

Consider the function y = \tan 3x.

a

Complete the tables of values:

x-\dfrac{2\pi}{3}-\dfrac{7\pi}{12}-\dfrac{\pi}{2}-\dfrac{5\pi}{12}-\dfrac{\pi}{3}-\dfrac{\pi}{4}-\dfrac{\pi}{6}-\dfrac{\pi}{12}
\tan 3x
x0\dfrac{\pi}{12}\dfrac{\pi}{6}\dfrac{\pi}{4}\dfrac{\pi}{3}\dfrac{5\pi}{12}\dfrac{\pi}{2}\dfrac{7\pi}{12}\dfrac{2\pi}{3}
\tan 3x
b

State the equations of the vertical asymptotes on the domain -\pi \leq x \leq \pi.

c

Find the interval between the asymptotes of y = \tan 3 x.

d

Hence, determine the period of y = \tan 3 x.

e

Write an expression for the period of y = \tan n x.

f

Sketch the graph of the function y = \tan 3 x on the domain -\pi \leq x \leq \pi.

14

Consider the function y = \tan 2 x.

a

Complete the table of values:

x-\pi-\dfrac{3\pi}{4}-\dfrac{\pi}{4}0\dfrac{\pi}{4}\dfrac{3\pi}{4}\pi
2x - \dfrac{3\pi}{2}
\tan 2x \text{Und}
b

Find the interval between the asymptotes of y = \tan 2 x.

c

Hence, determine the period of y = \tan 2 x.

d

Sketch the graph of y = \tan 2 x on the domain - \pi \leq x \leq \pi.

15

The graph of f \left( x \right) = \tan x is shown. On the same set of axes, plot g \left( x \right) = \tan \left( \dfrac{1}{2} x\right).

-90
90
180
270
360
450
540
x
-4
-3
-2
-1
1
2
3
4
y
16

Consider the graph of a function in the form f \left( x \right) = \tan b x:

a

State the period of the function.

b

Hence, state the equation of the function.

-30
30
60
90
120
150
x
-4
-3
-2
-1
1
2
3
4
y
17

Consider the function f \left( x \right) = \tan 7 x.

a

Find the period of the function.

b

Find the equation of the first four asymptotes to the right of the origin.

18

If an asymptote of the function in the form g \left( x \right) = \tan b x is known to be x = \dfrac{\pi}{8}, find the equation of g \left( x \right).

19

Consider the graph of a function in the form y = \tan b x.

a

State the period of the function.

b

Hence, state the equation of the function.

-\frac{1}{6}\pi
\frac{1}{6}\pi
x
-4
-3
-2
-1
1
2
3
4
y
20

Consider the graph of f \left( x \right) = \tan \left( \alpha x\right), and the graph of g \left( x \right) = \tan \left( \beta x\right) displayed on the same coordinate axes:

-\frac{1}{2}π
-\frac{1}{4}π
\frac{1}{4}π
\frac{1}{2}π
x
-2
-1
1
2
y

Which is greater: \alpha or \beta? Explain your answer.

21

On the same set of axes, sketch the graphs of the functions f \left( x \right) = \tan \dfrac{1}{4} x and \\g \left( x \right) = \tan 4 x, on the domain - 2\pi \leq x \leq 2\pi.

22

The function f \left( x \right) has the form f \left( x \right) = \tan b x. If two neighbouring asymptotes of this function are known to have equations x = \dfrac{\pi}{12} and x = \dfrac{\pi}{4}, find the exact value of the \\x-intercept between the asymptotes.

23

A function in the form f \left( x \right) = \tan b x has adjacent x-intercepts at x = \dfrac{13 \pi}{6} and x = \dfrac{9 \pi}{4}.

a

Find the equation of the asymptote lying between the two x-intercepts.

b

Find the period of the function.

c

Determine the equation of the function.

24

The function f \left( x \right) = \tan 6 x is to be graphed on the interval \left[\dfrac{\pi}{12}, \dfrac{5 \pi}{12}\right].

a

Find the period of the function f \left( x \right) = \tan 6 x.

b

Find the equations of the asymptotes of the function that occur on this interval.

c

Find the x-intercepts of the function that occur on this interval.

d

Hence sketch the function f \left( x \right) = \tan 6 x on the given interval.

Horizontal translations
25

Consider the graph of f \left( x \right) = \tan x and three points A\left(0, 0\right), B\left(\dfrac{\pi}{4}, 1\right) and C\left(\dfrac{\pi}{2}, 0\right).

-1\pi
-\frac{5}{6}\pi
-\frac{2}{3}\pi
-\frac{1}{2}\pi
-\frac{1}{3}\pi
-\frac{1}{6}\pi
\frac{1}{6}\pi
\frac{1}{3}\pi
\frac{1}{2}\pi
\frac{2}{3}\pi
\frac{5}{6}\pi
1\pi
x
-3
-2
-1
1
2
3
y
a

If f \left( x \right) undergoes a transformation to g \left( x \right) = \tan \left(x - \dfrac{\pi}{3}\right), state the coordinates of the following points after the transformation:

i

A

ii

B

iii

C

b

Describe the transformation of f \left( x \right) to g \left( x \right).

c

Hence, sketch the graph of g \left( x \right) = \tan \left(x - \dfrac{\pi}{3}\right) on the domain - \pi \leq x \leq \pi.

26

Consider the graph shown:

-180
-120
-60
60
120
180
x
-2
2
y
a

Determine whether the following could be the equation of the graph:

i
p \left( x \right) = \tan \left(x - 120 \degree\right)
ii
q \left( x \right) = \tan \left(x + 60 \degree\right)
iii
r \left( x \right) = \tan \left(x - 1020 \degree\right)
iv
s \left( x \right) = \tan \left(x - 680 \degree\right)
b

Hence, which of the following statements is true?

A

The graph of y = \tan \left(x - h\right) is the same as the graph of y = \tan \left(x - h - 180 n\right), for any integer n.

B

The graph of y = \tan \left(x - h\right) can be reflected in the y-axis to make the graph of y = \tan \left(x - h - 180 n\right), for any integer n.

C

The graph of y = \tan \left(x - h\right) is different for every value of h.

D

The graph of y = \tan \left(x - h\right) is the same as the graph of y = \tan \left(x - h - 180 n\right) only when n = \pm 1.

27

Let f \left( x \right) = \tan x and g \left( x \right) = \tan \left(x - 45 \degree\right).

What can be said of the domain and range of both functions? Explain your answer.

28

Consider the function y = \tan \left(x - 90 \degree\right).

a

Does the function have y-intercept?

b

Determine the period of the function.

c

How far apart are the asymptotes of the function?

d

State the first asymptote of the function for x > 0.

e

State the first asymptote of the function for x \leq 0.

f

Sketch the graph of the function.

29

Consider the function y = \tan \left(x - 60 \degree\right).

a

Determine the value of y at the y-intercept.

b

Determine the period of the function.

c

How far apart are the asymptotes of the function?

d

State the first asymptote of the function for x \geq 0.

e

State the first asymptote of the function for x \leq 0.

f

Sketch the graph the function.

30

Consider the function y = \tan \left(x - \dfrac{\pi}{4}\right).

a

Complete the table with exact values for \tan \left(x - \dfrac{\pi}{4}\right):

x0\dfrac{\pi}{4}\dfrac{5\pi}{12}\dfrac{\pi}{2}\dfrac{7\pi}{12}\dfrac{11\pi}{12}\pi\dfrac{13\pi}{12}\dfrac{5\pi}{4}
\tan \left( x - \dfrac{\pi}{4} \right)
b

Sketch the graph of y = \tan \left(x - \dfrac{\pi}{4}\right) on the domain - 2\pi \leq x \leq 2\pi.

c

Describe the transformation that turns the graph of y = \tan x into the graph of \\y = \tan \left(x - h\right).

31

Consider the graph of a function in the form f \left( x \right) = \tan \left(x - h\right), where 0 \leq h < \pi.

a

If g\left(x\right) = \tan x, describe the transformation of g \left( x \right) to f \left( x \right).

b

State the equation of f \left( x \right).

-1\pi
-\frac{3}{4}\pi
-\frac{1}{2}\pi
-\frac{1}{4}\pi
\frac{1}{4}\pi
\frac{1}{2}\pi
\frac{3}{4}\pi
1\pi
x
-4
-3
-2
-1
1
2
3
4
y
32

Consider the function f \left( x \right) = \tan \left(x - \dfrac{\pi}{7}\right).

a

If g\left(x\right) = \tan x, describe the transformation of g \left( x \right) to f \left( x \right).

b

State the equations of the first four asymptotes of f \left( x \right) to the right of the origin.

33

On the same set of axes, sketch the graphs of f \left( x \right) = \tan x and g \left( x \right) = \tan \left(x - \dfrac{\pi}{2}\right) on the domain -2\pi \leq x \leq 2\pi.

34

The function f \left( x \right) = \tan \left(x - \dfrac{\pi}{6}\right) is to be graphed on the interval \left[\dfrac{2 \pi}{3}, \dfrac{8 \pi}{3}\right].

a

Find the equations of the asymptotes of the function that occur on this interval.

b

Find the x-intercepts of the function that occur on this interval.

c

Sketch the function f \left( x \right) = \tan \left(x - \dfrac{\pi}{6}\right) on the interval .

35

Consider the following functions of the form f\left(x\right)=\tan \left( x-h\right):

a

Sketch a graph of each of the functions on the domain -\pi \leq x \leq \pi.

i

p \left( x \right) = \tan \left(x - \dfrac{2 \pi}{3}\right)

ii

r \left( x \right) = \tan \left(x + \dfrac{4\pi}{3}\right)

iii

s \left( x \right) = \tan \left(x + \dfrac{\pi}{3}\right)

iv

r \left( x \right) = \tan \left(x - \dfrac{53 \pi}{3}\right)

b

Compare the graphs in part (a) and explain your answer.

36

Consider the graph of f \left( x \right) = \tan \left(x - \beta\right), where 0 \leq \beta < \pi, and g \left( x \right) = \tan \left(x - \alpha\right), where 0 \leq \alpha < \pi.

Determine whether the following statements are true or false:

a

\beta must be greater than \alpha as the function f(x) has a greater horizontal translation from the graph of y = \tan x.

b

\alpha must be greater than \beta as the function f(x) has a greater horizontal translation from the graph of y = \tan x.

-\frac{1}{4}\pi
\frac{1}{4}\pi
\frac{1}{2}\pi
\frac{3}{4}\pi
1\pi
x
-4
-3
-2
-1
1
2
3
4
y
37

Consider the graphs of the following functions and state whether or not the graph is the same as y = - \tan x:

a

y = - \tan \left(x + \dfrac{3 \pi}{4}\right)

b

y = - \tan \left(x + \pi\right)

c

y = - \tan \left(x + \dfrac{\pi}{2}\right)

d

y = - \tan \left(x + 2 \pi\right)

Mixed transformations
38

How has the graph y = \tan x + 4 been transformed from y = \tan x?

39

Determine whether the following statements regarding the graph of y = \tan x, are true or false:

a

Altering the period will alter the position of the vertical asymptotes.

b

A phase shift has the same effect as a horizontal translation.

40

Consider the graph of the given \tan function.

-1\pi
-\frac{3}{4}\pi
-\frac{1}{2}\pi
-\frac{1}{4}\pi
\frac{1}{4}\pi
\frac{1}{2}\pi
\frac{3}{4}\pi
1\pi
x
-6
-4
-2
2
4
6
y
a

The given graph is the result of what transformation from y = \tan x?

b

State the equation of the graphed function.

41

The graph of y = \tan x is shown. Find the equation of the new graph after the following transformations:

a

Reflected over the y-axis and then translated vertically 3 units down.

b

Dilated vertically by a scale factor of 2 and translated horizontally \dfrac{\pi}{3} units to the left.

c

Translated horizontally by \dfrac{\pi}{3} units right and then dilated horizontally by a scale factor of \dfrac{1}{2}.

-1\pi
-\frac{3}{4}\pi
-\frac{1}{2}\pi
-\frac{1}{4}\pi
\frac{1}{4}\pi
\frac{1}{2}\pi
\frac{3}{4}\pi
1\pi
x
-4
-3
-2
-1
1
2
3
4
y
42

Sketch the graph of the following functions:

a

y = \tan \left( 2 \left(x - 45\right)\right)

b

y = \tan \left( 2 x + 90 \degree\right)

43

Sketch the graph of the function y = \tan \left( 2 \left(x - \dfrac{\pi}{4}\right)\right).

44

Consider the function y = \tan \left( 2 x + \dfrac{\pi}{3}\right):

a

Rewrite the function so that it is in the form y = \tan \left( a \left(x + b\right)\right).

b

Hence, sketch the graph of y = \tan \left( 2 \left(x + \dfrac{\pi}{6}\right)\right).

45

The graph y = \tan x, as shown, is reflected over the y-axis and then translated vertically 3 units down.

Write the equation of the new graph.

-\frac{1}{2}\pi
\frac{1}{2}\pi
x
-2
-1
1
2
y
46

Consider the graph of the given \tan function.

-\frac{1}{2}\pi
-\frac{3}{8}\pi
-\frac{1}{4}\pi
-\frac{1}{8}\pi
\frac{1}{8}\pi
\frac{1}{4}\pi
\frac{3}{8}\pi
\frac{1}{2}\pi
x
-2
2
4
6
y
a

The given graph is the result of what transformation from y = \tan x?

b

State the equation of the graphed function.

47

For each of the following functions:

i

Determine the y-intercept.

ii

Determine the interval between the vertical asymptotes of the function.

iii

Hence, state the period of the function.

iv

State the equation of the first asymptote of the function for x \geq 0.

v

State the equation of the first asymptote of the function for x \leq 0.

vi

Sketch the graph the function on the domain -\pi \leq x \leq \pi.

a

y = - \tan x

b

y = \tan \left(x + \dfrac{\pi}{3}\right)

c

y = \tan \left(\dfrac{x}{2}\right)

48

For each of the following functions:

i

Find the value of y when x = \dfrac{\pi}{4}.

ii

Determine the period of the function.

iii

Hence, state the interval between the asymptotes of the function.

iv

State the equation of the first asymptote of the function for x \geq 0.

v

State the equation of the first asymptote of the function for x \leq 0.

vi

Sketch the graph the function on the domain -\pi \leq x \leq \pi.

a

y = 5 \tan x + 3

b

y = 4 \tan 3 x

49

Describe how the graph of each of the following functions has been transformed from the function y = \tan x:

a

y = - 5 \tan x

b

y = 3\tan x + 2

c

y = \tan \left(3\left(x + \dfrac{\pi}{4}\right)\right)

d

y = \tan \left( 2 x - \dfrac{\pi}{4}\right)

50

Determine the following features for each of the given functions:

i

Period

ii

Phase shift

iii

Range

iv

Midline

a

y = 6 - 3 \tan \left(x + \dfrac{\pi}{3}\right)

b

y = - 4 \tan \left(\dfrac{1}{5}x + \dfrac{\pi}{20}\right)

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Outcomes

1.2.7

understand the unit circle definition of cos⁡θ, sin⁡θ and tan⁡θ and periodicity using radians

1.1.26

examine translations and the graphs of y=f(x)+a and y=f(x-b)

1.1.27

examine dilations and the graphs of y=c f(x) and y=f(dx)

1.2.9

recognise the graphs of y=sin⁡x,y=cos⁡x, and y=tan⁡x on extended domains

1.2.11

examine period changes and the graphs of y=sin ⁡bx,y=cos bx, and y=tan ⁡bx

1.2.12

examine phase changes and the graphs of y=sin⁡(x-c), y=cos⁡(x-c) and y=tan⁡(x-c)

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