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5.03 The unit circle and angles of any magnitude

Worksheet
The unit circle
1

State the quadrant 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

2

For the given functions, state the following:

i

The quadrants where the function is positive.

ii

The quadrants where the function is negative.

a

Sine function

b

Cosine function

c

Tangent function

3

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

a

\sin 310 \degree

b

\sin 50 \degree

c

\sin 130 \degree

d

\sin 230 \degree

e

\sin 31 \degree

f

\tan 31 \degree

g

\cos 267 \degree

h

\sin 267 \degree

i

\cos 180 \degree

4

Consider the given angles below and state whether the following are true or false:

w=253\degree,\quad x=265\degree,\quad y=193\degree, \quad z=-258\degree,\quad a=-182\degree,\quad b=-257\degree
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 have negative cosine values.

e

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

5

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

a

The point \left(x, y\right) is on the unit circle at a rotation of \theta such that \sin \theta = \dfrac{5}{13} and \cos \theta = \dfrac{12}{13}.

b

The point P\left(x, y\right) is on the unit circle at a rotation of \theta such that \sin \theta = - \dfrac{12}{37} and \cos \theta = \dfrac{35}{37}.

6

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.

Equivalent angles
7

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
8

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

a

Find the coordinates of the following points in terms of a and b:

i

Q

ii

R

iii

S

b

Find the equivalent trigonometric ratio in the first quadrant of the following:

i

\sin 131 \degree

ii

\sin 229 \degree

iii

\sin 311 \degree

c

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

c

\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
9

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

a

Find the coordinates of the following points in terms of a and b:

i

Q

ii

R

iii

S

b

Find the equivalent trigonometric ratio in the first quadrant of the following:

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
10

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

a

Find the coordinates of the following points in terms of a and b:

i

Q

ii

R

iii

S

b

Find the equivalent trigonometric ratio in the first quadrant of the following:

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
11

For each of the following graphs, find:

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
12

The following are angle rotations at point P on the unit circle. Find the acute angle in the first quadrant related to each rotation:

a

295 \degree

b

- 150 \degree

c

240 \degree

d

- 75 \degree

13

Rewrite each expression as a trigonometric ratio of a positive acute angle:

a

\sin 93 \degree

b

\cos 195 \degree

c

\tan 299 \degree

d

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

e

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

f

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

14

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)

15

Given the approximations \cos 21 \degree = 0.93 and \sin 21 \degree = 0.36, find the approximate values of the following and giving your answers in 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

16

Evaluate the following correct to two decimal places:

a

\sin 146 \degree

b

\tan 386 \degree

c

\cos 387 \degree

17

The diagram shows P, which represents a rotation of 66 \degree around the unit circle:

Find the following to two decimal places:

a

Coordinates of P(x,y)

b

The new coordinates of P if it was reflected about the y-axis.

c

\sin 114 \degree

d

\cos 114 \degree

e

\tan 114 \degree

18

Suppose that \cos \theta = \dfrac{3}{5}, where \\ 270 \degree < \theta < 360 \degree.

a

Find \sin \theta.

b

Find \tan \theta.

19

Suppose that \sin \theta = - \dfrac{\sqrt{7}}{4}.

a

Find \cos \theta.

b

Find \tan \theta.

20

Consider an angle \theta such that \sin \theta = 0.6 and \tan \theta < 0, find:

a

\sin \theta

b

\cos \theta

c

\tan \theta

21

Consider an angle \theta such that \tan \theta = - 1.3 and 270 \degree < \theta < 360 \degree, find:

a

\cos \theta

b

\sin \theta

22

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

23

Given the following, find \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

24

Given the following, find the value(s) 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

Exact trigonometric ratios
25

The first diagram shows a unit circle with point P \left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right) marked on the circle. Point P represents a rotation of 60 \degree anticlockwise around the origin from the positive x-axis:

a

Find the exact values of the following:

i

\sin 60\degree

ii

\cos 60\degree

iii

\tan 60\degree

b

On the second diagram, the coordinate axes shows a 60 \degree angle that has also been marked in the second, third, and fourth quadrants. For each quadrant, find the relative angle.

i

Quadrant 2

ii

Quadrant 3

iii

Quadrant 4

c

The points Q, R and S mark rotations of point P to the corresponding angles on the unit circle. Find the exact coordinates of each point:

i

Q

ii

R

iii

S

26

The first diagram shows a unit circle with point P \left(\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right) marked on the circle. Point P represents a rotation of 30 \degree anticlockwise around the origin from the positive x-axis:

a

Find the exact values of the following:

i

\sin 30\degree

ii

\cos 30\degree

iii

\tan 30\degree

b

On the second diagram, the coordinate axes shows a 30 \degree angle that has also been marked in the second, third and fourth quadrants. For each quadrant, find the relative angle:

i

Quadrant 2

ii

Quadrant 3

iii

Quadrant 4

c

The points Q, R and S mark rotations of point P to the corresponding angles on the unit circle. Find the exact coordinates of each point:

i

Q

ii

R

iii

S

27

Consider the trigonometric ratio \cos \left( - 210 \degree \right).

a

Find the value of the related acute angle.

b

Hence, find the value of \cos \left( - 210 \degree \right) using exact values.

28

Find the exact value of the following:

a

\sin 225 \degree

b

\cos 225 \degree

c

\tan 225 \degree

d

\sin 690 \degree

e

\sin 135 \degree

f

\cos 315 \degree

g

\tan 150 \degree

h

\sin 240 \degree

i

\cos 690 \degree

j

\tan 690 \degree

k

\sin 50 \degree + \sin 160 \degree + \sin 200 \degree + \sin 310 \degree

l

\tan 60 \degree + \tan 315 \degree

m

\tan ^{2}\left(240 \degree\right) - \sin ^{2}\left(210 \degree\right) + \cos ^{2}\left(180 \degree\right)

29

By rewriting each ratio in terms of the related acute angle, find the exact value of:

\dfrac{\sin 120 \degree \cos 240 \degree \tan 330 \degree}{\tan \left( - 45 \right) \degree}
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