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5.03 Reciprocal relationships

Worksheet
Reciprocal relationships
1

Write:

a

\text{cosec } \theta in terms of \sin \theta.

b

\text{sec } \theta in terms of \cos \theta.

c

\text{cot } \theta in terms of \tan \theta.

2

The point P on the unit circle has the coordinates \left(x, y\right):

Write the following in terms of x and/or y:

a

\text{cosec } \theta

b

\sec \theta

c

\cot \theta

-1
1
x
-1
1
y
3

Consider that \theta is an acute angle such that \tan \theta = x. Write an expression for the following in terms of x:

a

\sec \theta

b

\cot \theta

c

\cos \theta

d

\sin \theta

e

\text{cosec } \theta

4

Simplify:

a

\dfrac{\cos x}{\sin x}

b

\dfrac{\sin x }{\text{cosec } x }

c

\dfrac{\sec x}{\cos x}

d
\tan x \cot x
e
\cot x \sin x
f
\cos x \tan x
g
\cot x \sec x
h
\dfrac{\cot x}{\text{cosec }x}
5

Prove the following:

a
\tan x + \cot x = \text{cosec }x \sec x
b
\tan x + \sec x = \dfrac{1 + \sin x}{\cos x}
c
\cos x \text{cosec }x=\cot x
d
\sin x \cot x \sec x = 1
6

Find the exact value of \tan 70 \degree \cot 70 \degree.

7

Find \cot \theta given that \tan \theta = 11.

8

Consider the approximations: \cos 26 \degree = 0.8988 and \sin 26 \degree = 0.4384, find the value of the following, correct to four decimal places:

a

\text{cosec } 26 \degree

b

\sec 26 \degree

c

\cot26 \degree

Exact values of sec, cosec and cot
9

Consider the graph of the unit circle shown:

0 \degree
a

Find the value of:

i
\sin 30 \degree
ii

\text{cosec }30 \degree

iii

\cos 60 \degree

iv

\sec 60 \degree

v

\tan 30 \degree

vi

\cot 30 \degree

vii

\cot 90 \degree

viii

\text{cosec } 90 \degree

b

For what values of \theta, where 0 \leq \theta \leq 90, are the following ratios undefined:

i
\cot \theta
ii
\text{cosec } \theta
iii
\sec \theta
10

P\left(\dfrac{1}{2}, - \dfrac{\sqrt{3}}{2} \right) is a point on the unit circle that corresponds to the angle \theta, measured anticlockwise from the positive x-axis. Find the exact values of the following:

a

\sin \theta

b

\cos \theta

c

\tan \theta

d

\text{cosec } \theta

e

\sec \theta

f

\cot \theta

-1
1
x
-1
1
y
11

Find the exact values of the following:

a

\sec 0 \degree

b

\sec \left( - 30 \right) \degree

c

\sec 30 \degree

d

\cot 30 \degree

e

\sec 90 \degree

f

\text{cosec } 180 \degree

g

\cot 180 \degree

h

\cot 225 \degree

i

\text{cosec } 270 \degree

j

\text{cosec } 300 \degree

k

\sec 300 \degree

l

\sin 330 \degree

m

\cos 330 \degree

n

\tan 330 \degree

o

\text{cosec } 330 \degree

p

\sec 330 \degree

q

\cot 330 \degree

r

\cot 585 \degree

s

\sin 945 \degree

t
\cos 945 \degree
u

\tan 945 \degree

v

\text{cosec } 945 \degree

w

\sec 945 \degree

x

\cot 945 \degree

12

Find the exact value of the following trigonometric ratios. Write your answers with a rational denominators.

a

\text{cosec }60 \degree

b

\cot 45 \degree

13

State whether the following statements are true:

a

When \sin \theta = \dfrac{1}{3}, \text{cosec } \theta = 3.

b

When \sin \theta = \dfrac{1}{3}, \text{cosec } \theta = \dfrac{1}{3}.

c

When \sin \theta = 3, \text{cosec } \theta = \dfrac{1}{3}.

d

When \sin \theta = - \dfrac{1}{3}, \text{cosec } \theta = 3.

e

When \cos \theta \gt 0, \sec \theta \lt 0.

f

\cos \theta = \sec \theta

g

When \cos \theta \lt 0, \sec \theta \lt 0.

h

When \cos \theta \lt 0, \sec \theta \gt 0.

i

When \tan \theta = 3.5, \cot \theta = - 3.5.

14

If \sin \alpha = - \dfrac{4}{5} and \cos \alpha = \dfrac{3}{5}, find the exact values of:

a

\tan \alpha

b

\cot \alpha

c

\sec \alpha

d

\text{cosec } \alpha

15

If \sin \alpha = \dfrac{\sqrt{7}}{4} and \cos \alpha = \dfrac{3}{4}, find the exact values of:

a

\tan \alpha

b

\cot \alpha

c

\sec \alpha

d

\text{cosec } \alpha

16

If \cot \theta = 0.6 and \text{cosec } \theta \lt 0, find the exact values of:

a

\sin \theta

b

\text{cosec } \theta

c

\tan \theta

d

\sec \theta

17

If \sec \theta = - \dfrac{6}{5} and \sin \theta \gt 0, find the exact values of:

a

\tan \theta

b

\text{cosec }\theta

c

\cot \theta

18

Given the following, find the value of \sin \theta.

a

\text{cosec } \theta = \sqrt{2}

b

\text{cosec } \theta = - \dfrac{2 \sqrt{6}}{3}

19

Given the following, find the value of \text{cosec } \theta.

a

\sin \theta = \dfrac{2}{9}

b

\cot \theta = \dfrac{2}{\sqrt{3}} and - 90 \degree \leq \theta \leq 90 \degree

20

Given the following, find the value of \sec \theta.

a

\cos \theta = \dfrac{2}{9}

b

\text{cosec } \theta = \dfrac{3}{2}, \theta \gt 0 and obtuse

c

\cot \theta = \dfrac{9}{40}, \theta \gt 0 and reflex

21

Find \tan \theta given that \sec \theta = - \dfrac{17}{8} and 0 \degree \leq \theta \leq 180 \degree

22

Consider the angle 270 \degree:

a

State the coordinates of the point on the unit circle that corresponds to 270 \degree.

b

Find the value of \cos 270 \degree.

c

Find the value of \sin 270 \degree.

d

Determine whether the following are undefined:

i

\tan 270 \degree

ii

\cot 270 \degree

iii

\text{cosec }270 \degree

iv

\sec 270 \degree

23

Consider the angle 450 \degree:

a

State the coordinates of the point on the unit circle that corresponds to 450 \degree.

b

Find the value of \cos 450 \degree.

c

Find the value of \sin 450 \degree.

d

Determine whether the following are undefined:

i

\cot 450 \degree

ii

\tan 450 \degree

iii

\text{cosec } 450 \degree

iv

\sec 450 \degree

24

Consider the angle - 630 \degree:

a

State the coordinates of the point on the unit circle that corresponds to - 630 \degree.

b

Find the value of \cos \left( - 630 \degree\right).

c

Find the value of \sin \left( - 630 \degree\right).

d

Determine whether the following are undefined:

i

\sec \left( - 630 \degree\right)

ii

\text{cosec } \left( - 630 \degree\right)

iii

\tan \left( - 630 \degree\right)

iv

\cot \left( - 630 \degree\right)

25

Show that \dfrac{\sec 30 \degree - \text{cosec } 300 \degree}{\cot 480 \degree + \tan \left( - 225 \degree \right)} = 2 - 2 \sqrt{3}.

Complementary relationships
26

Simplify:

a
\sec \left(90 \degree - x \right)
b
\cot \left(90 \degree - x \right)
c
\text{cosec } \left(90 \degree - x \right)
d
\tan\left(90 \degree - x \right) \text{cosec } \left(90 \degree - x \right)
e

\dfrac{\tan \left(90 \degree - x\right)}{\cot \left( - x \right)}

f

\cos \left(90 \degree - \theta\right) \text{cosec } \left(180 \degree + \theta\right)

27

Find the value of \theta for the following equations:

a

\tan \left(15 \degree - \theta\right) = \cot \left( 2 \theta + 60 \degree\right) where \theta \gt 0 and acute

b

\text{cosec } 32 \degree = \sec \left(\theta + 27 \degree\right) where 0 \degree \leq \left(\theta + 27 \degree\right) \leq 90 \degree

c

\cot \left( 2 \theta + 5 \degree\right) = \tan \left( 3 \theta - 15 \degree\right) where \theta \gt 0and acute

d

\text{cosec } \theta = \sec 24 \degree where 0 \degree \leq \theta \leq 90 \degree

e

\tan 24 \degree = \cot \theta where 0 \degree \leq \theta \leq 90 \degree

f

\tan \left( 3 \theta - 5 \degree\right) = \dfrac{1}{\cot \left( 7 \theta - 53 \degree\right)}

Graphs of reciprocal trigonometrical functions
28

State whether the following values are positive or negative:

a

\sec 324 \degree

b

\text{cosec } 107 \degree

c

\tan 162 \degree

29

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

a

Complete the table of values, writing '-' if the value is undefined:

x0\degree45\degree90\degree135\degree180\degree225\degree270\degree315\degree360\degree
\cos x
\sec x
b

State the minimum positive value of \sec x.

c

State the maximum negative value of \sec x.

d

Sketch the graph of y = \sec x and y = \cos x for 0 \leq x \leq 360 \degree on the same number plane.

30

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

a

Complete the table of values, writing '-' if the value is undefined:

x0\degree45\degree90\degree135\degree180\degree225\degree270\degree315\degree360\degree
\sin x
\text{cosec } x
b

State the minimum positive value of \text{cosec } x.

c

State the maximum negative value of \text{cosec } x.

d

Sketch the graph of y = \text{cosec } x and y = \sin x for 0 \leq x \leq 360 \degree on the same number plane.

31

Consider the identity \cot x = \dfrac{\cos x}{\sin x}.

a

Complete the table of values, writing '-' if the value is undefined:

x0\degree45\degree90\degree135\degree180\degree225\degree270\degree315\degree360\degree
\cot x
b

Find the x-intercepts of the graph of y = \cot x in the interval \left[0 \degree, 360 \degree\right].

c

Sketch the graph of y = \cot x for 0 \leq x \leq 360 \degree.

32

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

a

Complete the table of values, rounding each value to two decimal places:

x57.30\degree85.94\degree89.38\degree89.95\degree90.53\degree91.67\degree114.59\degree
\sec x
b

What happens to the value of \sec x when x approaches 90 \degree from the left? Explain your answer.

33

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

a

Complete the table of values, rounding each value to two decimal places:

x171.89\degree177.62\degree179.34\degree179.91\degree180.48\degree183.35\degree229.18\degree
\text{cosec }x
b

What happens to the value of \text{cosec } x when x approaches 180 \degree from the left? Explain your answer.

34

Consider the function y = \sec x:

-360
-270
-180
-90
90
180
270
360
x
-2
-1
1
2
y
a

If x = 45 \degree, y = \sqrt{2}. Find the next positive x-value for which y = \sqrt{2}.

b

Find the period of the graph.

c

Find the smallest value of x greater than 360 \degree for which y = \sqrt{2}.

d

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

35

Consider the function y = \text{cosec } x:

-360
-300
-240
-180
-120
-60
60
120
180
240
300
360
x
-2
-1
1
2
y
a

If x = 30 \degree, y = 2. Find the the next positive x-value for which y = 2.

b

Find the period of the graph.

c

Find the smallest value of x greater than 360 \degree for which y = 2.

d

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

36

Consider the function y = \cot x:

-360
-300
-240
-180
-120
-60
60
120
180
240
300
360
x
-2
-1
1
2
y
a

If x = 30 \degree, y = \dfrac{1}{\sqrt{3}}.Find the next positive x-value for which y = \dfrac{1}{\sqrt{3}}.

b

Find the period of the graph.

c

Find the smallest value of x greater than 360 \degree for which y = \dfrac{1}{\sqrt{3}}.

d

Find the first x-value less than 0 \degree for which y = \dfrac{1}{\sqrt{3}}.

37

Consider the functions y=\text{cosec } x and y=\sec x shown below. State the domain for which \\ \text{cosec } x \lt 0, \, \sec x \gt 0 and 0 \leq x \leq 360.

45
90
135
180
225
270
315
360
405
x
-4
-3
-2
-1
1
2
3
4
y
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Outcomes

MA11-1

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

MA11-4

uses the concepts and techniques of periodic functions in the solutions of trigonometric equations or proof of trigonometric identities

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