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7.01 The unit circle

Worksheet
Unit circle
1

Calculate the fraction of the circumference of the unit circle the following angle measures represent:

a

x = 60 \degree

b

x = 180 \degree

c

s = 450 \degree

2

State the quadrants in which the following angles are located:

a

299 \degree

b

5 \degree

c

160\degree

d

229\degree

e

40\degree

f

310\degree

g

- 138\degree

h

- 244\degree

3

Point P on the unit circle shows a rotation of 330 \degree. Find the related acute angle in the first quadrant.

-1
1
0 \degree
-1
1
90 \degree
4

Consider the point \left(x, y\right) on the following unit circles:

i

Find the value of x.

ii

Find the value of y.

a
b
5

Consider the point on the given unit circle. Find the value of \theta.

6

Write four different angles between 0 \degree and 360 \degree inclusive, that lie on the quadrant boundaries.

7

For each of the following circular functions:

i

State the quadrants in which the function is positive.

ii

State the quadrants in which the function is negative.

a

Sine

b

Cosine

c

Tangent

8

State the quadrant where the angle in each scenario is located:

a

\theta is an angle such that \sin \theta > 0 and \cos \theta < 0.

b

\theta is an angle such that \tan \theta < 0 and \sin \theta > 0.

c

\theta is an angle such that \tan \theta < 0 and \cos \theta < 0.

d

\theta is an angle such that \tan \theta > 0 and \sin \theta > 0.

9

State whether the values of the following are positive or negative:

a

\sin 31 \degree

b

\tan 31 \degree

c

\cos 267 \degree

d

\sin 267 \degree

e

\cos 180 \degree

f

\tan 296 \degree

g

\sin 120 \degree

h

\cos 91 \degree

i

\sin 296 \degree

j

\cos 120 \degree

k

\cos 296 \degree

l

\sin 90 \degree

m

\cos 51 \degree

n

\sin 51 \degree

o

\cos 233 \degree

p

\tan 233 \degree

10

For each of the following graphs, state:

i

\sin a

ii

\cos a

iii

\tan a

a
-1
1
x
-1
1
y
b
-1
1
x
-1
1
y
c
-1
1
x
-1
1
y
11

Determine whether the following statements are possible:

a

\sin \theta = \dfrac{7}{8}

b

\sin \theta = - 0.6

c

\sin \theta = 1.1

d

\sin \theta = - \dfrac{2}{3}

12

Consider the following angles:

w = 253 \degree, \quad x = 265 \degree, \quad y = 193 \degree, \quad z = -258 \degree, \quad a = -182 \degree, \quad b = -257 \degree

Determine whether the following statements are true or false:

a

Angles z, a, and b are in Quadrant 3.

b

\sin z and \sin b are both negative.

c

\tan \left(z - 90 \degree\right) and \tan \left(z + 90 \degree\right) are both positive.

d

The angles w, x, and y all have negative cosine values.

e

Angles w, x, and y are in Quadrant 3.

13

Consider the following angles:

w = 65 \degree, \quad x = 10 \degree, \quad y = 44 \degree, \quad z = -31 \degree , \quad a = -10 \degree , \quad b = -29 \degree

Determine whether the following statements are true or false:

a

\tan z, \cos \left(90 \degree + x\right) and \sin \left(180 \degree + y\right) are negative.

b

\sin x, \cos z, and \tan b are all positive.

c

Angles w, x, and y are in Quadrant 1.

d

\sin x, \cos y, and \tan w are all positive.

e

Angles z, a, and b are in Quadrant 1.

14

The terminal side of an angle, \theta passes through the point \left( - 4 , - 6 \right) as shown in the diagram:

a

Find the value of r, the distance between \left(0, 0\right) and \left( - 4 , - 6 \right).

b

Hence find the exact values of the following:

i

\sin \theta

ii

\cos \theta

iii

\tan \theta

-5
-4
-3
-2
-1
1
2
3
-7
-6
-5
-4
-3
-2
-1
1
15

Suppose that \cos \theta = \dfrac{3}{5}, where \\ 270 \degree < \theta < 360 \degree. Find the exact value of the following:

a

\sin \theta

b

\tan \theta

16

Suppose that \sin \theta = - \dfrac{\sqrt{7}}{4}. Find the exact value of the following:

a

\cos \theta

b

\tan \theta

17

Given angle \theta such that \sin \theta = 0.6 and \tan \theta < 0, find the exact value of the following:

a

\cos \theta

b

\tan \theta

18

Given angle \theta such that \tan \theta = - 1.3 and 270 \degree < \theta < 360 \degree, find the exact value of the following:

a

\cos \theta

b

\sin \theta

19

Given that \tan \theta = - \dfrac{15}{8} and \sin \theta > 0, find the exact value of \cos \theta.

20

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

a

\cos \theta = - \dfrac{60}{61} and 0 \degree \leq \theta \leq 180 \degree

b

\cos \theta = - \dfrac{6}{7} and \tan \theta < 0

21

Given the following, find the exact value of \tan \theta.

a

\cos \theta = \dfrac{3}{7} and \theta is acute

b

\sin \theta = \dfrac{1}{\sqrt{10}} and - 90 \degree \leq \theta \leq 90 \degree

Equivalent ratios
22

Consider the ratio \sin 150 \degree.

a

State the quadrant in which 150 \degree is located.

b

State whether \sin 150 \degree is positive or negative.

c

Find the positive acute angle that 150 \degree is related to.

d

Hence rewrite \sin 150 \degree in terms of a trigonometric ratio of a positive acute angle.

23

Consider the ratio \cos 150 \degree.

a

State the quadrant in which 150 \degree is located.

b

State whether \cos 150 \degree is positive or negative.

c

Find the positive acute angle that 150 \degree is related to.

d

Hence rewrite \cos 150 \degree in terms of a trigonometric ratio of a positive acute angle.

24

Consider the ratio \tan 330 \degree.

a

State the quadrant in which 330 \degree is located.

b

State whether \tan 330 \degree is positive or negative.

c

Find the positive acute angle 330 \degree is related to.

d

Hence rewrite \tan 330 \degree in terms of a trigonometric ratio of a positive acute angle.

25

For each of the following, rewrite the ratio as an equivalent trigonometric ratio of a positive acute angle:

a

\sin 93 \degree

b

\cos 195 \degree

c

\tan 299 \degree

d

\sin 240 \degree

e

\sin 147 \degree

f

\tan \left( - 50 \degree \right)

g

\sin 400 \degree

h

\cos 605 \degree

i

\tan 525 \degree

j
\sin 235 \degree
k
\cos 298 \degree
l
\tan 178 \degree
m

\sin \left( - 139 \degree \right)

n

\cos \left( - 222 \degree \right)

o

\tan \left( - 289 \degree \right)

p

\sin \left( - 221 \degree \right)

q
\cos \left( - 103 \degree \right)
r
\tan \left( - 40 \degree \right)
26

Consider the following diagram of a unit circle with centre O.

Express the following trigonometric ratios in terms of a and/or b:

a

\tan 65 \degree

b

\cos 475 \degree

c

\sin 65 \degree

d

\cos 65 \degree

e

\cos 295 \degree

f

\sin 245 \degree

27

The diagram shows points P, Q, R and S, which represent rotations of 49 \degree, 131 \degree, 229 \degree and 311 \degree respectively around the unit circle.

a

If point P has coordinates \left(a, b\right), write the coordinates of the following points in terms of a and b:

i

Q

ii

R

iii

S

b

State whether the following ratios are equivalent to \sin 49 \degree or - \sin 49 \degree:

i

\sin 131 \degree

ii
\sin 229 \degree
iii
\sin 311 \degree
c

Hence, express each of the following in terms of \sin x:

i

\sin \left(180 \degree - x\right)

ii

\sin \left(180 \degree + x\right)

iii

\sin \left(360 \degree - x\right)

-1
1
0 \degree
-1
1
90 \degree
28

The diagram shows points P, Q, R and S, which represent rotations of 63 \degree, 117 \degree, 243 \degree and 297 \degree respectively around the unit circle.

a

If point P has coordinates \left(a, b\right), write the coordinates of the following points in terms of a and b:

i

Q

ii

R

iii

S

b

State whether the following are equivalent to - \cos 63 \degree or \cos 63 \degree:

i

\cos 117 \degree

ii

\cos 243 \degree

iii

\cos 297 \degree

c

Hence, express each of the following in terms of \cos x:

i

\cos \left(180 \degree - x\right)

ii

\cos \left(180 \degree + x\right)

iii

\cos \left(360 \degree - x\right)

-1
1
0 \degree
-1
1
90 \degree
29

The diagram shows points P, Q, R and S, which represent rotations of 62 \degree, 118 \degree, 242 \degree and 298 \degree respectively around the unit circle.

a

If point P has coordinates \left(a, b\right), write the coordinates of the following points in terms of a and b:

i

Q

ii

R

iii

S

b

State whether the following are equivalent to - \tan 62 \degree or \tan 62 \degree:

i

\tan 118 \degree

ii

\tan 242 \degree

iii

\tan 298 \degree

c

Hence, express each of the following in terms of \tan x:

i

\tan \left(180 \degree - x\right)

ii

\tan \left(180 \degree + x\right)

iii

\tan \left(360 \degree - x\right)

-1
1
0 \degree
-1
1
90 \degree
30

Simplify:

a
\sin \left(360 \degree + \theta\right)
b
\cos \left(360 \degree - \theta\right)
c
\sin \left(180 \degree - \theta\right)
d
\cos \left(180 \degree + \theta\right)
e
\sin \left(180 \degree + \theta\right)
f
\sin \left(90 \degree - \theta\right)
g

\sin \left( - \theta \right)

h

\cos \left( - \theta \right)

31

Given the approximations \cos 21 \degree = 0.93 and \sin 21 \degree = 0.36, state the approximate values of the following, to two decimal places:

a

\cos 339 \degree

b

\sin \left( - 339 \degree \right)

c

\cos 159 \degree

d

\sin 159 \degree

e

\sin 201 \degree

f

\cos 201 \degree

g

\sin (-201) \degree

h

\cos (-201) \degree

Use technology
32

Find the following, correct to two decimal places:

a

\sin 146 \degree

b

\cos 126 \degree

c

\tan 161 \degree

d

\tan 386 \degree

e

\cos 621 \degree

f

\cos (-218) \degree

g

\sin (-42.5) \degree

h

-\tan (-89) \degree

i

\left(\sin 35 \degree \right)^{2}

j

\sin ^{2} 35 \degree

k

\sin \left( 35 \degree \right)^{2}

l

-\sin ^{2} 35 \degree

33

Find the following:

a

\sin ^{2} 35 \degree + \cos ^{2} 35 \degree

b

\sin ^{2} \left(-64 \right) \degree + \cos ^{2} \left(-64 \right) \degree

c

\sin ^{2} 275 \degree + \cos ^{2} 275 \degree

d

\sin ^{2} x \degree + \cos ^{2} x \degree

34

Consider the following diagram of a unit circle:

a

Given that point B, represents a rotation of 35 \degree around the unit circle, find the coordinates of B, correct to three decimal places.

b

Point C has coordinates \left(0.294, 0.956\right). Find the angle that point C represents to the nearest degree.

c

Given that the angle subtended by arc \angle CD = 54 \degree, find the coordinates of D, correct to three decimal places.

35

The diagram shows P, which represents a rotation of 66 \degree around the unit circle. Find the following, rounding your answers to two decimal places:

a

Coordinates of point P.

b

Coordinates of point R, that represents point P reflected horizontally about the y-axis.

c

Size of the rotational angle for R around the unit circle.

d

Coordinates of point Q, that represents point P reflected vertically about the \\ x-axis.

e

Size of the rotational angle for Q around the unit circle.

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