chapter 9 - trigonometry - exercise

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s copy Lesson 93: Quadrants and Angles in the Unit Circle

Complete all the answer boxes with the correct angles.

For each of the following, state the quadrant the angle lies in:

a) 88o Answer : b) 156o

Answer :

c) 320o Answer : d) 190o

Answer :

Question 1.

≤ θ ≤ 90o

0o 360o 180o O

≤ θ ≤ o

180o ≤ θ ≤ 270o 270o ≤ θ ≤ 360o

Question 2.

Mathematics Form 4

2

The above diagram shows a unit circle with four points L, M, N and P on it’s circumference. Find the x–coordinate, y–coordinate and the ratio of y–coordinate to x–coordinate for each point. a) For point L,

The x–coordinate =

The y–coordinates =

The ratio = b) For point M,

The x–coordinate =

The y–coordinate =

The ratio =

Question 3.

O 1

1

−1

−1

L M

N

P

x

y

Mathematics Form 4

3

c) For point N,

The x–coordinate =

The y–coordinate =

The ratio = d) For point P,

The x–coordinate =

The y–coordinate =

The ratio =

Mathematics Form 4

4

ACTIVITY SHEET Teacher’s copy Lesson 93: Quadrants and Angles in the Unit Circle

Complete all the answer boxes with the correct angles.

For each of the following, state the quadrant the angle lies in:

e) 88o Answer : Quadrant I f) 156o

Answer : Quadrant II

g) 320o Answer : Quadrant IV h) 190o

Answer : Quadrant III

Question 1.

0o ≤ θ ≤ 90o

0o 360o 180o O

90o ≤ θ ≤ 180o

180o ≤ θ ≤ 270o 270o ≤ θ ≤ 360o

Question 2.

Mathematics Form 4

5

The above diagram shows a unit circle with four points L, M, N and P on it’s circumference. Find the x–coordinate, y–coordinate and the ratio of y–coordinate to x–coordinate for each point. e) For point L,

The x–coordinate = 0.7

The y–coordinates = 0.7

The ratio = 1 f) For point M,

The x–coordinate = −0.4

The y–coordinate = 0.9

The ratio = −2.25

Question 3.

O 1

1

−1

−1

L

M

N

P

x

y

Mathematics Form 4

6

g) For point N,

The x–coordinate = −0.8

The y–coordinate = −0.6

The ratio = 0.75 h) For point P,

The x–coordinate = 0.5

The y–coordinate = −0.85

The ratio = −1.7

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy LESSON 94: Angle in Quadrant I

1. Based on the unit circle given, complete the table given by determining the values of sine,

cosine and tangent of the following angles . Your answers should be rounded off to two decimal places .

No. Trigonometrical ratios Answer

1 sin 20°

2 sin 50°

3 sin 70°

4 cos 20°

5 cos 50°

6 cos 70°

7 tan 20°

8 tan 50°

9 tan 70°

y2

0.2

0.4

0.6

0.8

1.0

A2

B2

C

20°

50° 70°

x0.2 0.4 0.6 0.8 1.0

o

Mathematics Form 4

2

2. From the answers above, make an inference about the values of sine, cosine and

tangent as θ increases from 0° to 90°, by completing the table below.

sine ? ( 0° to 90° )

cosine ? ( 0° to 90° )

tangent ? ( 0° to 90° )

Mathematics Form 4

3

ACTIVITY SHEET Teacher’s Copy LESSON 94: Angle in Quadrant I

1. Based on the unit circle given, complete the table given by determining the values of sine,

cosine and tangent of the following angles . Your answers should be rounded off to two decimal places.

No. Trigonometrical ratios Answer

1 sin 20° 0.34

2 sin 50° 0.77

3 sin 70° 0.94

4 cos 20° 0.94

5 cos 50° 0.64

6 cos 70° 0.35

7 tan 20° 0.36

8 tan 50° 1.20

9 tan 70° 2.69

y2

0.2

0.4

0.6

0.8

1.0

A2

B2

C2

20°

50° 70°

x0.2 0.4 0.6 0.8 1.0

o

Mathematics Form 4

4

2. From the answers above, make an inference about the values of sine, cosine and tangent as θ increases from 0° to 90°, by completing the table below.

sine ? ( 0° to 90° )

The value of sin ? increases as ?

increases from 0° to 90°.

cosine ? ( 0° to 90° )

The value of cos ? decreases as ?

increases from 0° to 90°.

tangent ? ( 0° to 90° )

The value of tan ? increases as ?

increases from 0° to 90°.

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy LESSON 95: Angle in Quadrant II, III and IV

1. Based on the unit circle given, complete the following table by determining the values of

sine, cosine and tangent for the stated angles. Your answers should be rounded off to two decimal places.

No. Trigonometrical Ratios Answer

1 sin 120°

2 sin 220°

3 sin 310°

4 cos 120°

5 cos 220°

6 cos 310°

7 tan 120°

8 tan 220°

9 tan 310°

1

1

- 1

- 1

y

x

A

B C

0.5

−0.5

−0.5

0.5

120° 220°

310°

Mathematics Form 4

2

2.

Based on the unit circle shown, determine the values of

(a) sin 135° Answer:

(b) tan 285° Answer:

(c) cos 210° Answer:

(d) cos 135° Answer:

(e) y if tan 210° = 0.57 Answer:

O x

y

135°

M(−0.71, 0.71)

210°

N(−0.87, y)

75°

P(0.23, −0.97)

Mathematics Form 4

3

ACTIVITY SHEET Teacher’s Copy LESSON 95: Angle in Quadrant II, III and IV

1. Based on the unit circle given, complete the following table by determining the values of

sine, cosine and tangent for the stated angles. Your answers should be rounded off to two decimal places.

No. Trigonometrical Ratios Answer

1 sin 120° 0.87 or 0.86

2 sin 220° −0.64 or −0.65

3 sin 310° −0.77 or −0.76

4 cos 120° −0.5

5 cos 220° −0.77 or −0.76

6 cos 310° 0.64 or 0.65

7 tan 120° −1.73 or −1.74

8 tan 220° 0.84 or 0.83

9 tan 310° −1.19 or –1.2

1

1

- 1

- 1

y

x

A

B C

0.5

−0.5

−0.5

0.5

120° 220°

310°

Mathematics Form 4

4

22.423.097.0

285tan

−=

−=°

5.0)87.0(57.0

)87.0(210tan87.0

210tan

−=−×=

−×°=−

=°

y

y

2.

Based on the unit circle shown, determine the values of

(a) sin 135° Answer: sin 135° = 0.71

(b) tan 285° Answer:

(c) cos 210° Answer: cos 210° = −0.87

(d) cos 135° Answer: cos 135° = −0.71

(e) y if tan 210° = 0.57 Answer:

O x

y

135°

M(−0.71, 0.71)

210°

N(−0.87, y)

75°

P(0.23, −0.97)

Mathematics Form 4

Page 1 of 1

ACTIVITY SHEET

Pupil’s Copy

LESSON 96: Values of Sine, Cosine and Tangent.

1 State the quadrant of each angle and the determine whether the value of each of

the following is positive or negative.

Angle Quadrant Trigonometric ratio Positive or Negative

56° cos 56°

145° sin 145°

280° tan 280°

96° cos 96°

265° sin 265°

175° sin 175°

315° tan 315°

185° tan 185°

323° cos 323°

2 Calculate the value of each of the following:

(a) 4 sin 30° − 3 cos 60° =

(b) 6 tan 45 + 2 cos 180° =

(c) 5 sin 270° − 7 cos 90° =

(d) 3 tan 180° + 9 cos 360° =

(e) 10 cos 60° − 6 sin 90° =

Mathematics Form 4

Page 2 of 2

(a) Given cos θ = 1 and 0 ° ≤ θ ≤ 360°, state the values of θ.

(b) Given tan θ = 0 and 180° ≤ θ ≤ 360°, state the values of θ.

4 Find the value of x in each of the following:

(a)

(b)

(c) cos x = 0 and 180° ≤ x ≤ 360°.

(d) sin x = 0.5 and 0° ≤ x ≤ 180°.

4.5 cm 9 cm

10 cm

10 cm

x

x

Mathematics Form 4

Page 3 of 3

ACTIVITY SHEET

Teacher’s Copy

LESSON 96: Values of Sine, Cosine and Tangent.

1 State the quadrant of each angle and the determine whether the value of each of

the following is positive or negative.

Angle Quadrant Trigonometric ratio Positive or Negative

56° 1 cos 56° Positive

145° 2 sin 145° Positive

280° 4 tan 280° Negative

96° 2 cos 96° Negative

265° 3 sin 265° Negative

175° 2 sin 175° Positive

315° 4 tan 315° Negative

185° 3 tan 185° Positive

323° 4 cos 323° Positive

2 Calculate the value of each of the following:

(a) 4 sin 30° − 3 cos 60° = 4(0.5) − 3(0.5) = 2 − 1.5 = 0.5

(b) 6 tan 45 + 2 cos 180° = 6(1) + 2(− 1) = 6 − 2 = 4

(c) 5 sin 270° − 7 cos 90° = 5(−1) − 7(0) = − 5 − 0 = − 5

(d) 3 tan 180° + 9 cos 360° = 3(0) + 9(1) = 0 + 9 = 9

(e) 10 cos 60° − 6 sin 90° = 10(0.5) − 6(1) = 5 − 6 = − 1

Mathematics Form 4

Page 4 of 4

3 (a) Given cos θ = 1 and 0 ° ≤ θ ≤ 360°, state the values of θ.

If cos θ = 1 and 0° ≤ θ ≤ 360°, then θ = 0° or 360°.

(b) Given tan θ = 0 and 180° ≤ θ ≤ 360°, state the values of θ.

If tan θ = 0 and 180° ≤ θ ≤ 360°, then θ = 180° or 360°.

4 Find the value of x in each of the following:

(a) cos x = 95.4

= 21

x = 60°

(b) tan x = 1010

= 1 x = 45°

(c) cos x = 0 and 180° ≤ x ≤ 360°.

If cos x = 0, then x = 90° or 270°

but since 180° ≤ x ≤ 360°,

x = 270°

(d) sin x = 0.5 and 0° ≤ x ≤ 180°.

If sin x = 0.5, then x = 30° or 150°.

4.5 cm 9 cm

10 cm

10 cm

x

x

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s copy Lesson 97: Relationships between the Values of Sine, Cosine and Tangent. Find the angle in quadrant I which corresponds to each of the following. a) 110o b) 94o c) 144o d) 168o Solution: a) Angle in quadrant I which corresponds to 110° =

b) Angle in quadrant I which corresponds to 94° =

c) Angle in quadrant I which corresponds to 144° =

d) Angle in quadrant I which corresponds to 168° =

Question 1.

x

y

x

y

x

y

x

y

O

O

O

O

Mathematics Form 4

2

Find the angle in quadrant I which corresponds to each of the following. a) 220o b) 186o c) 209o d) 256o Solution: a) Angle in quadrant I which corresponds to 220° =

b) Angle in quadrant I which corresponds to 186° =

c) Angle in quadrant I which corresponds to 209° =

d) Angle in quadrant I which corresponds to 256° =

Question 2.

x

y

y

x

O

O

x

y

O

x

y

O

Mathematics Form 4

3

Find the angle in quadrant I which corresponds to each of the following. a) 280o b) 345o c) 309o Solution: a) Angle in quadrant I which corresponds to 280° =

b) Angle in quadrant I which corresponds to 345° =

c) Angle in quadrant I which corresponds to 309° =

Question 3.

x

y

x

y

O

x

y

O

O

Mathematics Form 4

4

Express the following in terms of trigonometrical functions of acute angles.

a) tan 170o b) sin 133o

c) cos 190o

d) tan 340o

e) sin 200o

f) cos 120o

Solution: a) tan 170o =

b) sin 133o =

c) cos 190o =

d) tan 340o =

e) sin 200o =

f) cos 120o =

Question 4.

Mathematics Form 4

5

ACTIVITY SHEET Teacher’s copy Lesson 97: Relationships between the Values of Sine, Cosine and Tangent. Find the angle in quadrant I which corresponds to each of the following. a) 110o b) 94o c) 144o d) 168o Solution: a) Angle in quadrant I which corresponds to 110° = 180° − 110°

= 70o

b) Angle in quadrant I which corresponds to 94° = 180° − 94°

= 86o

c) Angle in quadrant I which corresponds to 144° = 180° − 144°

= 36o

d) Angle in quadrant I which corresponds to 168° = 180° − 168°

= 12o

Question 1.

x

y

110° 70°

x 94° 86°

y

x 144°

36°

y

x 168°

12°

y

O

O

O

O

Mathematics Form 4

6

Find the angle in quadrant I which corresponds to each of the following. a) 220o b) 186o c) 209o d) 256o Solution: a) Angle in quadrant I which corresponds to 220° = 220° − 180°

= 40o

b) Angle in quadrant I which corresponds to 186° = 186° − 180°

= 6o

c) Angle in quadrant I which corresponds to 209° = 209° − 180°

= 29°

d) Angle in quadrant I which corresponds to 256° = 256° − 180°

= 76°

Question 2.

x

y

220°

40°

y

x

186°

6°

O

O

x

y

209°

29° O

x

y

256°

76°

y

O

Mathematics Form 4

7

Find the angle in quadrant I which corresponds to each of the following. a) 280o b) 345o c) 309o Solution: a) Angle in quadrant I which corresponds to 280° = 360° − 280°

= 80°

b) Angle in quadrant I which corresponds to 345° = 360° − 345°

= 15°

c) Angle in quadrant I which corresponds to 309° = 360° − 309°

= 51°

Question 3.

x

y

280°

80° O

x

y

345°

15° O

x

y

309°

51° O

Mathematics Form 4

8

Express the following in terms of trigonometrical functions of acute angles.

a) tan 170o b) sin 133o

c) cos 190o

d) tan 340o

e) sin 200o

f) cos 120o

Solution: a) tan 170o = −tan ( 180o – 170o ) = −tan 10o b) sin 133o = sin ( 180o – 133o ) = sin 47o c) cos 190o = −cos ( 190o – 180o ) = −cos 10o d) tan 340o = −tan ( 360o – 340o ) = −tan 20o e) sin 200o = −sin ( 200o – 180o ) = −sin 20o f) cos 120o = −cos ( 180o – 120o ) = −cos 60o

Question 4.

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy Problems Solving on Sine, Cosine and Tangent.

1 Calculate each of the following values based on the table below:

θ 27° 46° 55° 64°

sin θ 0.45 0.72 0.82 0.90

cos θ 0.89 0.69 0.57 0.44

tan θ 0.51 1.04 1.43 2.05

(a) tan 226° =

(b) cos 116° =

(c) sin 305° =

(d) 2 cos 207° + 5 sin 116° =

(e) 3 tan 296° + 4 cos 125° =

2 (a) Given that cos θ = −0.1736 and 180°≤ θ ≤ 360°, find the value of θ . Solution:

(b) Given that sin x = 0.4695 and 0°≤ x ≤ 360°, find the values of x.

Solution:

Mathematics Form 4

2

(c) Given that tan y = −1.9626 and 90°≤ y ≤ 360°, find the values of y.

Solution:

3 In the diagram, KLM is a straight line. Find the values of cos θ and θ .

Solution:

4 In the diagram, EFG is a straight line. Find the values of z and tan z.

Solution:

θ K

L M

J

15 cm

11 cm

z

E

G

D

F

18.2 cm

7 cm

Mathematics Form 4

3

5 The above diagram shows the plan of a garden. PQT is an isosceles triangle.

Given that PQ = 2.25 m and QT = 3.6 m, calculate, (a) cos ∠TPQ. (b) the perpendicular distance between P and the line SR.

Solution:

(a)

(b)

P

Q

RS

T

2.5 m

Mathematics Form 4

4

ACTIVITY SHEET

Teacher’s Copy

LESSON 98: Problems Solving on Sine, Cosine and Tangent.

1 Calculate each of the following values based on the table below:

θ 27° 46° 55° 64°

sin θ 0.45 0.72 0.82 0.90

cos θ 0.89 0.69 0.57 0.44

tan θ 0.51 1.04 1.43 2.05

(a) tan 226° = tan(226° − 180°) = tan 46° = 1.04

(b) cos 116° = − cos(180° − 116°) = − cos 64° = − 0.44

(c) sin 305° = −sin(360° − 305°)

= −sin 55° = −0.82

(d) 2 cos 207° + 5 sin 116° = 2(− cos(207° − 180°)) + 5 sin(180° − 116°) = −2 cos 27° + 5 sin 64° = −2(0.89) + 5(0.90) = 2.72

(e) 3 tan 296° + 4 cos 125° = 3(− tan(360° − 296°)) + 4(−cos(180° − 125°)) = − 3 tan 64° − 4 cos 55° = − 3(2.05) − 4(0.57) = − 8.43

2 (a) Given that cos θ = −0.1736 and 180°≤ θ ≤ 360°, find the value of θ . Solution: cos θ = −cos 80° = cos(180° + 80°) = cos 260 ° Therefore, θ = 260°

(b) Given that sin x = 0.4695 and 0°≤ x ≤ 360°, find the values of x.

Solution: sin x = sin 28° or sin(180° − 28°) = sin 28° or sin 152° Therefore, x = 28° or 152°

Mathematics Form 4

5

(c) Given that tan y = −1.9626 and 90°≤ y ≤ 360°, find the values of y.

Solution: tan y = tan(180° − 63°) or tan(360° − 63°) = tan 117° or tan 297° Therefore, y = 117° or 297°

3 In the diagram, KLM is a straight line. Find the values of cos θ and θ .

Solution:

cos θ = −cos(180° − θ )

= −

= −

θ = 180° − 42° 50' = 137° 10'

4 In the diagram, EFG is a straight line. Find the values of z and tan z.

Solution:

tan z = −tan(180° − z )

= −

= 22

7

DEEF −−

= 22 72.18

7

−−

= 8.16

7−

= −0.4167

z = 180° − 22° 37'

= 157° 23'

θ K

L M

J

15 cm

11 cm

JL KL 11 15

DE DF z

E

G

D

F

18.2 cm

7 cm

Mathematics Form 4

6

5 The above diagram shows the plan of a garden. PQT is an isosceles triangle.

Given that PQ = 2.25 m and QT = 3.6 m, calculate,

(a) cos ∠TPQ. (b) the perpendicular distance between P and the line SR.

Solution:

(a) PQ

QTTPQ

×=∠ 2

1

21

sin

= 25.2

6.321

×

= 25.28.1

°=∠ 13.5321

TPQ

∠TPQ = 53.13° × 2

= 106.26°

Therefore, cos ∠ TPQ = cos 106.26°

= −cos(180° − 106.26°)

= −cos 73.74°

= −0.2800

(b) The height of ∆PQT = 22 )21

( QTPQ −

= 22 8.125.2 −

= 1.35 m Therefore, the perpendicular distance between P and the line SR = 1.35 + 2.5 = 3.85 m

P

Q

RS

T

2.5 m

Mathematics Form 4

1

?

00

300

600

900

120 0

1500

1800

2100

2400

2700

3000

3300

3600

y=sin?

0

-0.87

0

ACTIVITY SHEET

Pupil’s Copy

LESSON 99: Graphs of sine, cosine and tangent.

1. Based from the graph of sine given, complete the table of value below.

y

1.0

0.5

- 0.5

- 1.0

? 300 600 900 120 0 1500 180 0 210 0 2400 2700 300 0 3300 3600

×

×

× ×

×

×

×

×

×

× ×

×

×

0.87 1 0

Mathematics Form 4

2

?

300

600

900

1500

1800

240 0

270 0

3300

y=cos?

1

-0.5

-0.87

0.5

1

2. Based from the table of value, draw the graph of cosine ? where 00 < ? < 3600

y

1.0

0.5

- 0.5

- 1.0

? 900 1200 150 0 1800 2100 240 0 2700 3000 3300 3600

00 1200 210 0 3000 3600

0.87 0.5 0 - 0.87 - 1 - 0.5 0 0.87

Mathematics Form 4

3

? 00 300 450 750 900 105 0 135 0 150 0 180 0 2100 2250 2550 2700 y=tan? 0 0.58 1 3.73 8 -3.73 -1 -0.58 0 0.58 1 3.73 8 285 0 3150 3300 3600 -3.73 -1 -0.58 0

3. Based from the table of value given, dram the graph of tangent ? where

00 < ? < 360 0

y

1

? 900 1800 2700 3600

2

3

4

-4

-3

-2

-1

Mathematics Form 4

4

4. Complete the empty boxes with the properties of each graph given.

Sin ? 1

-1

1800 3600

Cos ? 1

-1

1800 3600

Tan ? 4

-4

1800 3600 900 2700

y

?

y

?

y

?

The graph of Tan ? The graph of sin ? The graph of cos ?

Mathematics Form 4

5

?

00

300

600

900

120 0

1500

1800

2100

2400

2700

3000

3300

3600

y=sin?

0

0.87

- 0.5

-0.87

0

ACTIVITY SHEET

Teacher’s Copy

LESSON 99: Graphs of sine, cosine and tangent.

1. Based from the graph of sine given, complete the table of value below.

y

1.0

0.5

- 0.5

- 1.0

? 300 600 900 120 0 1500 180 0 210 0 2400 2700 300 0 3300 3600

×

×

× ×

×

×

×

×

×

× ×

×

×

0.5 0.87 1 0.5 0 - 0.87 - 1 - 0.5

Mathematics Form 4

6

?

300

600

900

1500

1800

240 0

270 0

3300

y=cos?

1

-0.5

-0.87

0.5

1

2. Based from the table of value, draw the graph of cosine ? where 00 < ? < 3600

y

1.0

0.5

- 0.5

- 1.0

? 900 1200 150 0 1800 2100 240 0 2700 3000 3300 3600

×

×

×

×

××

× ×

×

×

×

× ×

00 1200 210 0 3000 3600

0.87 0.5 0 - 0.87 - 1 - 0.5 0 0.87

Mathematics Form 4

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? 00 300 450 750 900 105 0 135 0 150 0 180 0 2100 2250 2550 2700 y=tan? 0 0.58 1 3.73 8 -3.73 -1 -0.58 0 0.58 1 3.73 8 285 0 3150 3300 3600 -3.73 -1 -0.58 0

4. Based from the table of value given, dram the graph of tangent ? where 00 < ? < 360 0

y

1

? 900 1800 2700 3600

×

×

×

×

×

××

×

×

×

×

×

×

2

3

4

-4

×

-3

-2

-1

×

Mathematics Form 4

8

4. Complete the empty boxes with the properties of each graph given.

Sin ? 1

-1 1800 3600

Cos ? 1

-1 1800 3600

Tan ? 4

-4

1800 3600 900 2700

y

?

y

?

y

?

The graph of Tan ? • Increasing when 00 < ? < 900 and 900 < ? < 2700 • Decreasing at none of

the intervals • Tan ? is undefined

when? = 900 and 2700

• Tan ? are equal to zero when ? = 00, 1800 and 3600

• There are no minimum or maximum points on the graph

• dotted line drawn through these value of ? are called asymtotes.

The graph of sin ? • Increasing when

00 < ? < 900 and 2700 < ? < 3600

• Decreasing when 900 < ? < 2700

• Maximum point = ( 900, 1 ) • Minimum point = ( 2700 , -1 ) • Sin ? = 0 when ? = 00 , 1800, 3600 • The graph of sin ?

repeat itself at 3600 interval

The graph of cos ? • Increasing when 1800

< ? < 3600 • Decreasing when 00 < ? < 1800 • Maximum point

= ( 00, 1 ) and ( 3600, 1 )

• Minimum point = ( 1800 , -1 ) • Cos ? = 0 when ? = 900 , 2700 • The graph of cos ?

repeat itself at 3600 interval

Mathematics Form 4

1

?

00

300

600

900

120 0

1500

1800

2100

2400

2700

3000

3300

3600

y=sin?

0

-0.87

0

ACTIVITY SHEET

Pupil’s Copy

LESSON 99: Graphs of sine, cosine and tangent.

1. Based from the graph of sine given, complete the table of value below.

y

1.0

0.5

- 0.5

- 1.0

? 300 600 900 120 0 1500 180 0 210 0 2400 2700 300 0 3300 3600

×

×

× ×

×

×

×

×

×

× ×

×

×

0.87 1 0

Mathematics Form 4

2

?

300

600

900

1500

1800

240 0

270 0

3300

y=cos?

1

-0.5

-0.87

0.5

1

2. Based from the table of value, draw the graph of cosine ? where 00 < ? < 3600

y

1.0

0.5

- 0.5

- 1.0

? 900 1200 150 0 1800 2100 240 0 2700 3000 3300 3600

00 1200 210 0 3000 3600

0.87 0.5 0 - 0.87 - 1 - 0.5 0 0.87

Mathematics Form 4

3

? 00 300 450 750 900 105 0 135 0 150 0 180 0 2100 2250 2550 2700 y=tan? 0 0.58 1 3.73 8 -3.73 -1 -0.58 0 0.58 1 3.73 8 285 0 3150 3300 3600 -3.73 -1 -0.58 0

3. Based from the table of value given, dram the graph of tangent ? where

00 < ? < 360 0

y

1

? 900 1800 2700 3600

2

3

4

-4

-3

-2

-1

Mathematics Form 4

4

4. Complete the empty boxes with the properties of each graph given.

Sin ? 1

-1

1800 3600

Cos ? 1

-1

1800 3600

Tan ? 4

-4

1800 3600 900 2700

y

?

y

?

y

?

The graph of Tan ? The graph of sin ? The graph of cos ?

Mathematics Form 4

5

?

00

300

600

900

120 0

1500

1800

2100

2400

2700

3000

3300

3600

y=sin?

0

0.87

- 0.5

-0.87

0

ACTIVITY SHEET

Teacher’s Copy

LESSON 99: Graphs of sine, cosine and tangent.

1. Based from the graph of sine given, complete the table of value below.

y

1.0

0.5

- 0.5

- 1.0

? 300 600 900 120 0 1500 180 0 210 0 2400 2700 300 0 3300 3600

×

×

× ×

×

×

×

×

×

× ×

×

×

0.5 0.87 1 0.5 0 - 0.87 - 1 - 0.5

Mathematics Form 4

6

?

300

600

900

1500

1800

240 0

270 0

3300

y=cos?

1

-0.5

-0.87

0.5

1

2. Based from the table of value, draw the graph of cosine ? where 00 < ? < 3600

y

1.0

0.5

- 0.5

- 1.0

? 900 1200 150 0 1800 2100 240 0 2700 3000 3300 3600

×

×

×

×

××

× ×

×

×

×

× ×

00 1200 210 0 3000 3600

0.87 0.5 0 - 0.87 - 1 - 0.5 0 0.87

Mathematics Form 4

7

? 00 300 450 750 900 105 0 135 0 150 0 180 0 2100 2250 2550 2700 y=tan? 0 0.58 1 3.73 8 -3.73 -1 -0.58 0 0.58 1 3.73 8 285 0 3150 3300 3600 -3.73 -1 -0.58 0

4. Based from the table of value given, dram the graph of tangent ? where 00 < ? < 360 0

y

1

? 900 1800 2700 3600

×

×

×

×

×

××

×

×

×

×

×

×

2

3

4

-4

×

-3

-2

-1

×

Mathematics Form 4

8

4. Complete the empty boxes with the properties of each graph given.

Sin ? 1

-1 1800 3600

Cos ? 1

-1 1800 3600

Tan ? 4

-4

1800 3600 900 2700

y

?

y

?

y

?

The graph of Tan ? • Increasing when 00 < ? < 900 and 900 < ? < 2700 • Decreasing at none of

the intervals • Tan ? is undefined

when? = 900 and 2700

• Tan ? are equal to zero when ? = 00, 1800 and 3600

• There are no minimum or maximum points on the graph

• dotted line drawn through these value of ? are called asymtotes.

The graph of sin ? • Increasing when

00 < ? < 900 and 2700 < ? < 3600

• Decreasing when 900 < ? < 2700

• Maximum point = ( 900, 1 ) • Minimum point = ( 2700 , -1 ) • Sin ? = 0 when ? = 00 , 1800, 3600 • The graph of sin ?

repeat itself at 3600 interval

The graph of cos ? • Increasing when 1800

< ? < 3600 • Decreasing when 00 < ? < 1800 • Maximum point

= ( 00, 1 ) and ( 3600, 1 )

• Minimum point = ( 1800 , -1 ) • Cos ? = 0 when ? = 900 , 2700 • The graph of cos ?

repeat itself at 3600 interval