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7.025 Trigonometric functions

Worksheet
Trigonometric functions
1

Consider the equation y = \sin x.

a

Complete the table with values in exact form:

x0\dfrac{\pi}{6}\dfrac{\pi}{2}\dfrac{5 \pi}{6}\pi\dfrac{7 \pi}{6}\dfrac{3 \pi}{2}\dfrac{11 \pi}{6}2 \pi
\sin x
b

Sketch a graph for y = \sin x on the domain -2\pi \leq 0 \leq 2\pi.

c

State the value of \sin \left(-2 \pi\right).

d

State the sign of \sin \left( \dfrac{- \pi}{12} \right).

e

State the sign of \sin \dfrac{13 \pi}{12}.

f

Which quadrant of a unit circle does an angle with measure \dfrac{13 \pi}{12} lie in?

2

Consider the equation y = \cos x.

a

Complete the table with values in exact form:

x0\dfrac{\pi}{3}\dfrac{\pi}{2}\dfrac{2 \pi}{3}\pi\dfrac{4 \pi}{3}\dfrac{3 \pi}{2}\dfrac{5 \pi}{3}2 \pi
\cos x
b

Sketch a graph for y = \cos x on the domain -2\pi \leq 0 \leq 2\pi.

c

State the value of \cos \pi.

d

State the sign of \cos \left( \dfrac{- \pi}{4} \right).

e

State the sign of \cos \dfrac{11 \pi}{6}.

f

Which quadrant of a unit circle does an angle with measure \dfrac{11 \pi}{6} lie in?

3

Consider the equation y = \tan x.

a

Complete the table with values in exact form:

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

Sketch the graph of y = \tan x on the domain -2\pi \leq 0 \leq 2\pi.

c

Graph the line y = 1 on the same coordinate plane.

d

Hence, state the exact solutions to the equation \tan x = 1 over this domain.

e

State the value of \tan \left(-2 \pi\right).

f

State the sign of \tan \left( \dfrac{- \pi}{6} \right).

g

State the sign of \tan \dfrac{9 \pi}{5}.

h

Which quadrant of a unit circle does an angle with measure \dfrac{9 \pi}{5} lie in?

4

Consider the function y = 5 \sin x.

a

Sketch the graph of the given function over the domain \left[ - 2 \pi , 2 \pi\right].

b

Graph the line y = x on the same coordinate plane.

c

Hence, state the number of solutions to the equation 5 \sin x = x.

5

Consider the graph of y = - \tan x and the plotted points A, B, C, D and E shown:

a

At which point is the graph of \\ y = -\tan x equal to zero?

b

At which point will the corresponding graph of y = -\cot x be undefined? Explain your answer.

\frac{1}{2}π
\frac{3}{2}π
x
-2
-1
1
2
y
6

Consider the identity \sec x = \dfrac{1}{\cos x} and the following table of values:

x0\dfrac{\pi}{4}\dfrac{\pi}{2}\dfrac{3 \pi}{4}\pi\dfrac{5 \pi}{4}\dfrac{3 \pi}{2}\dfrac{7 \pi}{4}2 \pi
\cos x1\dfrac{1}{\sqrt{2}}0- \dfrac{1}{\sqrt{2}}- 1- \dfrac{1}{\sqrt{2}}0\dfrac{1}{\sqrt{2}}1
\sec x
a

Complete the table for the function y=\sec x.

b

State the range for y=\sec x.

c

Sketch the graph of y = \sec x on the domain \left[-2\pi, 2 \pi\right].

7

Consider the identity \text{cosec } x = \dfrac{1}{\sin x} and the following table of values:

x0\dfrac{\pi}{4}\dfrac{\pi}{2}\dfrac{3 \pi}{4}\pi\dfrac{5 \pi}{4}\dfrac{3 \pi}{2}\dfrac{7 \pi}{4}2 \pi
\sin x0\dfrac{1}{\sqrt{2}}1\dfrac{1}{\sqrt{2}}0- \dfrac{1}{\sqrt{2}}- 1- \dfrac{1}{\sqrt{2}}0
\text{cosec } x
a

State the values of x in the interval \left[0, 2 \pi\right] for which \text{cosec } x is not defined.

b

Complete the table for the function y=\text{cosec } x.

c

State the range for y=\text{cosec } x.

d

Sketch the graph of y = \text{cosec } x on the interval \left[-2\pi, 2 \pi\right].

8

Consider the identity \cot x = \dfrac{\cos x}{\sin x} and the following table of values:

x0\dfrac{\pi}{4}\dfrac{\pi}{2}\dfrac{3 \pi}{4}\pi\dfrac{5 \pi}{4}\dfrac{3 \pi}{2}\dfrac{7 \pi}{4}2 \pi
\cos x1\dfrac{1}{\sqrt{2}}0- \dfrac{1}{\sqrt{2}}- 1- \dfrac{1}{\sqrt{2}}0\dfrac{1}{\sqrt{2}}1
\sin x0\dfrac{1}{\sqrt{2}}1\dfrac{1}{\sqrt{2}}0- \dfrac{1}{\sqrt{2}}- 1- \dfrac{1}{\sqrt{2}}0
\cot x
a

State the values of x in the interval \left[0, 2 \pi\right] for which \cot x is not defined.

b

Complete the table for the function y = \cot x.

c

State the x-intercepts of the graph of y = \cot x in the interval \left[0, 2 \pi\right].

d

Sketch the graph of y = \cot x on the interval \left[-2\pi, 2 \pi\right].

e

Describe the graph of y = \cot x.

9

Consider the identity \sec x = \dfrac{1}{\cos x}.

a

Complete the table for x-values close to x = \dfrac{\pi}{2} \approx 1.57, where x is given in radians. Round each value to two decimal places.

x11.51.561.571.581.62
\sec x
b

Describe happens to the value of \sec x as x approaches \dfrac{\pi}{2} from the left. Explain your answer.

10

Consider the identity \text{cosec } x = \dfrac{1}{\sin x}.

a

Complete the table for x-values close to x = \pi \approx 3.14, where x is given in radians. Round each value to two decimal places.

x33.13.133.143.153.24
\text{cosec } x
b

Describe what happens to the value of \text{cosec } x as x approaches \pi from the left. Explain your answer.

11

Consider the table of values for \text{cosec } x over the domain \left(0, 2 \pi\right):

x\dfrac{\pi}{4}\dfrac{\pi}{2}\dfrac{3 \pi}{4}\dfrac{5 \pi}{4}\dfrac{3 \pi}{2}\dfrac{7 \pi}{4}
\text{cosec } x\sqrt{2}1\sqrt{2}- \sqrt{2}- 1- \sqrt{2}
a

Given that the period of \text{cosec } x is 2 \pi, complete the table of values over the domain \left( - 2 \pi , 0\right):

x- \dfrac{7 \pi}{4}- \dfrac{3 \pi}{2}- \dfrac{5 \pi}{4}- \dfrac{3 \pi}{4}- \dfrac{\pi}{2}- \dfrac{\pi}{4}
\text{cosec } x
b

Sketch the graph of y = \text{cosec } x on the interval \left( - 4 \pi , 4 \pi\right).

12

Consider the table of values for \sec x in the interval \left[0, 2 \pi\right]:

x0\dfrac{\pi}{4}\dfrac{3 \pi}{4}\pi\dfrac{5 \pi}{4}\dfrac{7 \pi}{4}2 \pi
\sec x1\sqrt{2}- \sqrt{2}- 1- \sqrt{2}\sqrt{2}1
a

Given that the period of \sec x is 2 \pi, complete the table of values over the domain \left[ - 2 \pi , 0\right]:

x- 2 \pi- \dfrac{7 \pi}{4}- \dfrac{5 \pi}{4}- \pi- \dfrac{3 \pi}{4}- \dfrac{\pi}{4}0
\sec x1
b

Sketch the graph of y = \sec x on the interval \left[ - 4 \pi , 4 \pi\right].

13

Consider the graph of y = \text{cosec } x and the line y=\sqrt{2}:

a

When x = \dfrac{\pi}{4}, y = \sqrt{2}. Find the next positive x-value for which y = \sqrt{2}.

b

State the period of y = \text{cosec } x.

c

Find the smallest value of x greater than 2 \pi for which y = \sqrt{2}.

d

Find the first x-value less than 0 for which y = \sqrt{2}.

\frac{1}{4}π
\frac{1}{2}π
\frac{3}{4}π
\frac{5}{4}π
\frac{3}{2}π
\frac{7}{4}π
\frac{9}{4}π
x
-2
-1
1
2
y
14

Consider the graph of y = \sec x:

a

When x = \dfrac{\pi}{3}, y = 2. Find the next positive x-value for which y = 2.

b

State the period of y = \sec x.

c

Find the smallest value of x greater than 2 \pi for which y = 2.

d

Find the first x-value less than 0 for which y = 2.

\frac{1}{3}π
\frac{2}{3}π
\frac{4}{3}π
\frac{5}{3}π
x
-2
-1
1
2
y
15

Consider the graph of y = \cot x and the line y=\sqrt{3}:

a

When x = \dfrac{\pi}{6}, y = \sqrt{3}. Find the next positive x-value for which y = \sqrt{3}.

b

State the period of y = \cot x.

c

Find the smallest value of x greater than 2 \pi for which y = \sqrt{3}.

d

Find the first x-value less than 0 for which y = \sqrt{3}.

\frac{1}{2}π
\frac{3}{2}π
x
-2
-1
1
2
y
16

Consider the graphs of \text{cosec } x and \sec x in the interval \left[-\dfrac{\pi}{2}, 2 \pi\right]:

State the interval where \text{cosec } x \gt 0 and \sec x \lt 0.

\frac{1}{2}π
\frac{3}{2}π
x
-3
-2
-1
1
2
3
y
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Outcomes

MA11-1

uses algebraic and graphical techniques to solve, and where appropriate, compare alternative solutions to problems

MA11-3

uses the concepts and techniques of trigonometry in the solution of equations and problems involving geometric shapes

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