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5.04 Angles with exact values

Worksheet
Exact values
1

Consider the equilateral triangle with side lengths of 2 \text{ cm}.

Find the exact values of the following:

a

Perpendicular height of the triangle.

b

Size of angle x.

c
i

\sin 60 \degree

ii

\cos 60 \degree

iii

\tan 60 \degree

d
i

\sin x

ii

\cos x

iii

\tan x

2

Consider the right-angled triangle where \angle ABC measures 45 \degree and AC = 1 unit. Find the exact value of the following:

a

Length of BC.

b

Exact length of AB.

c
i

\sin 45 \degree

ii

\cos 45 \degree

iii

\tan 45 \degree

3

Find the exact side length of an equilateral triangle with a perpendicular height of \sqrt{21} \text{ cm}.

4

Consider the following diagram of the unit circle:

Find the exact value of the following:

a

\cos \dfrac{\pi}{3}

b

\sin \dfrac{\pi}{3}

c

\sin \dfrac{\pi}{4}

d

\cos \dfrac{\pi}{6}

e

\cos \dfrac{\pi}{2}

f

\sin \dfrac{\pi}{2}

g

\cos 2\pi

h

\tan \dfrac{\pi}{6}

5

Find the exact value of the following:

a

\cos 0 \degree

b

\sin 90 \degree

c

\tan 90 \degree

d

\tan 30 \degree.

e

\sin 30 \degree

f

\cos 30 \degree

g

\sin 45 \degree

h

\tan 45 \degree

6

Evaluate the following, leaving your answers in exact form:

a

\dfrac{\sin 30 \degree}{\cos 60 \degree}

b

\sin 45 \degree + \cos 60 \degree

c

\sin \dfrac{\pi}{6} \cos \dfrac{\pi}{4}

d

\sin 45 \degree \cos 30 \degree + \tan 45 \degree

e

\cos^{2}\left(30 \degree\right)

f

\cos^{2}\left(\dfrac{\pi}{4}\right)

g

\sin ^{2}\left(\dfrac{\pi}{6}\right) - \cos ^{2}\left(\dfrac{\pi}{3}\right)

h

\sin ^{2}\left(30 \degree\right) + \cos ^{2}\left(30 \degree\right)

7

Prove the following equations are true:

a

\tan 30 \degree \times \cos 30 \degree = \cos 60 \degree.

b

\tan \dfrac{\pi}{3} \times \cos \dfrac{\pi}{6} =\tan \dfrac{\pi}{4}

Equivalent ratios
8

Find the values of the following:

a

\cos 720 \degree

b

\sin 630 \degree

c

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

d

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

e

\sin \dfrac{5\pi}{2}

f

\cos 5\pi

g

\tan \dfrac{7\pi}{2}

h

\cos \left(-\dfrac{5\pi}{2}\right)

9

The first diagram shows a unit circle with point P \left(\dfrac{1}{\sqrt2}, \dfrac{1}{\sqrt2}\right) marked on the circle. Point P represents a rotation of 45 \degree anticlockwise around the origin from the positive x-axis:

a

Find the exact values of the following:

i

\sin 45\degree

ii

\cos 45\degree

iii

\tan 45\degree

b

On the second diagram, the coordinate axes shows a 45 \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. State the exact coordinates of each point:

i

Q

ii

R

iii

S

d

Write the following in terms of an equivalent ratio of 45 \degree:

i

\sin 135\degree

ii

\cos 225 \degree

iii

\tan 315 \degree

iv

\sin \left(-45 \right) \degree

e

Hence find the exact value of the following:

i

\sin 135\degree

ii

\cos 225 \degree

iii

\tan 315 \degree

iv

\sin \left(-45 \right) \degree

10

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. State the exact coordinates of each point:

i

Q

ii

R

iii

S

d

Write the following in terms of an equivalent ratio of 60 \degree:

i

\sin 120\degree

ii

\cos 240 \degree

iii

\tan 300 \degree

iv

\cos \left(-60 \right) \degree

e

Hence find the exact value of the following:

i

\sin 120\degree

ii

\cos 240 \degree

iii

\tan 300 \degree

iv

\cos \left(-60 \right) \degree

11

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. State the exact coordinates of each point:

i

Q

ii

R

iii

S

d

Write the following in terms of an equivalent ratio of 30 \degree:

i

\cos 150\degree

ii

\sin 210 \degree

iii

\tan 330 \degree

iv

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

e

Hence find the exact value of the following:

i

\cos 150\degree

ii

\sin 210 \degree

iii

\tan 330\degree

iv

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

12

Find the exact values of the following:

a

\sin \left(\dfrac{7 \pi}{6}\right)

b

\cos \left(\dfrac{2 \pi}{3}\right)

c

\tan \left(\dfrac{7 \pi}{4}\right)

d

\sin \left(\dfrac{19 \pi}{6}\right)

e

\cos \left( - \dfrac{\pi}{6} \right)

f

\tan \left(\dfrac{11 \pi}{4}\right)

g

\tan \left(-\dfrac{ \pi}{6}\right)

h

\sin \left(-\dfrac{5 \pi}{6}\right)

13

Find the exact values of the following:

a

\sin \dfrac{\pi}{2}

b

\tan \dfrac{3 \pi}{2}

c

\sin \dfrac{5 \pi}{6}

d

\tan \dfrac{3 \pi}{4}

e

\sin \dfrac{7 \pi}{6}

f

\tan \dfrac{7 \pi}{6}

g

\cos \dfrac{5 \pi}{3}

h

\sin \left( - \dfrac{\pi}{6} \right)

14

For each of the following trigonometric ratios:

i

Write the ratio in terms of the related acute angle.

ii

Hence, find the exact value of the ratio.

a

\cos \left( - 210 \degree \right)

b

\sin 840 \degree

c

\cos \left( - \dfrac{2 \pi}{3} \right)

d

\tan \left( \dfrac{7 \pi}{6} \right)

15

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

a
\dfrac{\sin 120 \degree \cos 240 \degree \tan 330 \degree}{\tan \left( - 45 \right) \degree}
b
\dfrac{\left(\sin \dfrac{2 \pi}{3}\right) \left(\cos \dfrac{2 \pi}{3}\right) \left(\tan \dfrac{3 \pi}{4}\right)}{\tan \left( - \dfrac{\pi}{4} \right)}
c

\dfrac{\sin \left(\dfrac{2 \pi}{3}\right) + \cos \left(\dfrac{5 \pi}{6}\right) - \tan \left(\dfrac{7 \pi}{4}\right)}{\cos \left(\dfrac{4 \pi}{3}\right)}

d

\dfrac{- \sin \left(\dfrac{5 \pi}{3}\right) + \cos \left(\dfrac{7 \pi}{6}\right) + \tan \left(\dfrac{5 \pi}{3}\right)}{- \cos \left(\dfrac{2 \pi}{3}\right)}

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Outcomes

1.2.8

recognise the exact values of cos⁡θ, sin⁡θ and tan⁡θ at integer multiples of π/6 and π/4

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