3 17.1–17.2 Trigonometry in three dimensions › 2014 › 10 › ... · 3 17.3 Trigonometric ratios for any angle 345C Guided practice worksheet 4 Solve the equations in the given

�3 17.1–17.2 Trigonometry in three dimensions

343A

1 The diagram shows the back of a truck used to carry scaffold poles.

a Use Pythagoras’ Theorem to find the length of the scaffold pole

i AC ...................................................................................................................................................................................................................

..........................................................................................................................................................................................................................

ii AF ...................................................................................................................................................................................................................

..........................................................................................................................................................................................................................

iii AG ..................................................................................................................................................................................................................

..........................................................................................................................................................................................................................

b Use trigonometry to find the angle

i CAB ................................................................................................................................................................................................................

..........................................................................................................................................................................................................................

ii BAF ................................................................................................................................................................................................................

..........................................................................................................................................................................................................................

iii GAF ................................................................................................................................................................................................................

..........................................................................................................................................................................................................................

c The scaffold pole EL is 8 m long. How far does it extend beyond the line JK?

..................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................

d What angle does it make with the floor of the truck?

..................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................

Guided practice worksheet

K

L

G1m

6.5 m

3 m

0.8 m

D

C

F

A

B

J

E

H

A

Questions are targeted at the grades indicated


343B


2 A rocket was launched from this plant pot.

What angle does it make with the ground?

........................................................................................................................................................

........................................................................................................................................................

........................................................................................................................................................

3 A cube of stone of side 6 mm is made into a bead by drilling a hole through two opposite corners.

a Calculate the length of the hole.

..................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................

b Find the angle the hole makes with the face ABCD.

..................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................

4 The diagram shows a camera tripod with four legs of equal length.

a Calculate the lengths

i AC ............................................................................................................................

ii EF ...............................................................................................................................

b Use your answers to part a to find the angle the leg AE makes with the floor.

.....................................................................................................................................................

.....................................................................................................................................................

.....................................................................................................................................................

The legs are adjusted to a length of 1.2 m.

c Calculate the new angle each leg makes with the floor.

.................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................

8 cm

15 cm

20 cm

H

D

B

6 mmA

C

GF

E

FA

B

C

E

1.5 m

0.9 m 0.9 m

Hint Calculate the length of diagonal AC � rst.

A

A*


343C


5 a Calculate the angle ABF of this ramp.

.................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................

A skateboarder travels from A to C.

b How far does she travel?

.................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................

c Calculate the distance FC.

.................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................

d Find the angle to the horizontal at which the skateboarder travels.

.................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................

Hint Find angle ACF.

D

EA

F B

C

2 m

5 m

7 m

A*

�3 17.3 Trigonometric ratios for any angle

345A


Give your answers correct to 1 decimal place, where necessary.

1 The sketch below helps you to solve the equation sin θ = 0.8 for values of θ in the range –360° to 360°.

a Use your calculator to find θ. Mark the value on the sketch.

.................................................................................................................................................................................................................................

b Use the symmetry of the graph to find the other values of θ between –360° and 360°.

.................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................

2 Use the sketch below to solve the equation cos θ = –0.2 for values of θ in the range 0° to 720°.

......................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................

3 Use the sketches to solve the equations in the given range.

a sin x = 0.3 for values of x between 0° and 360°.

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

0° 180° 360°–180°–360°

0.8

–1

1

Sin θ

θ

0° 180° 360° 540° 720°–0.2

–1

1

Cos θ

θ

Hint press the [sin–1] key.

A*

0° 180° 360°

–1

1

Sin x

x



345B


b cos x = 0.5 for values of x between –180° and 180°.

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

c sin x = –0.7 for values of x between –360° and 360°.

.................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................

d cos x = –0.5 for values of x between –540° and 180°.

.................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................

0° 180°–180°

–1

1

Cos x

x

0° 180° 360°–180°–360°

–1

1

Sin x

x

0° 180°–180°–360°–540°

–1

1

Cos x

x

A*


345C


4 Solve the equations in the given range. Make a sketch to show your solutions.

a sin x = –0.62 for values of x between 0° and 360°.

.................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................

b cos x = 0.44 for values of x between –360° and 0°.

.................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................

c sin x = 0.05 for values of x between –180° and 360°.

.................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................

d cos x = –0.707 for values of x between 0° and 270°.

.................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................

(a) Show that one solution of 8 sin x = 1 is 7.2°.(b) Hence solve the equation for values of x in the range –360° to 360°.(a) 8 sin x = 1 1 sin x = – (divide both sides by 8) 8 sin x = 0.125 x = 7.2° (to 1 d.p.)(b) Solutions between –360° and 360° are x = 180° – 7.2° = 172.8° x = –360° + 7.2° = –352.8° x = –180° – 7.2° = –187.2°

0°–360°

–1

1

Cos x

x

0.125

7.2°–180° 180° 360°

A*


345D


5 a Show that one solution of 8 cos x = 5 is 51.3°.

.....................................................................................................

b Hence solve the equation for values of x in the range 0° to 720°.

.....................................................................................................

6 a Show that one solution of 50 sin x = –49 is –78.5°.

.....................................................................................................

b Hence solve the equation for values of x in the range –360° to 360°.

.....................................................................................................

7 a Show that one solution of cos x = 32

is 30°.

.....................................................................................................

b Hence solve the equation for values of x in the range –720° to 0°.

.....................................................................................................

A*

�3 17.4 Finding the area of a triangle using 1–2 ab sin C

347A


Calculate lengths and areas correct to 3 significant figures and angles correct to 1 decimal place.

1 Calculate the area of each triangle.

a b c d

............................................... ............................................... ............................................... ...............................................

............................................... ............................................... ............................................... ...............................................

2 Calculate the area of each shape.

a b c

....................................................... ....................................................... .......................................................

....................................................... ....................................................... .......................................................

....................................................... ....................................................... .......................................................

1Remember: Area of a triangle ABC = – ab sin C 2

b a

C

BA c

10 cm70°

8 cm10°

150 mm

200 mm

130°

1.4 m

1.6 m

25°

5 cm

5 cm

10 cm70°

8 cm10°

150 mm

200 mm

130°

1.4 m

1.6 m

25°

5 cm

5 cm

10 cm70°

8 cm10°

150 mm

200 mm

130°

1.4 m

1.6 m

25°

5 cm

5 cm

10 cm70°

8 cm10°

150 mm

200 mm

130°

1.4 m

1.6 m

25°

5 cm

5 cm

Parallelogram Dart

Kite

141°

2.7 m

1.2 m22°

17 cm

20 cm

70°

30°

1.1 m

1.5 m

Parallelogram Dart

Kite

141°

2.7 m

1.2 m22°

17 cm

20 cm

70°

30°

1.1 m

1.5 m

Hint Split the shape into two triangles.


A*

A


347B


3 Calculate the area of each regular polygon.

a b

........................................................................................................ ........................................................................................................

........................................................................................................ ........................................................................................................

........................................................................................................ ........................................................................................................

c Regular polygon with 9 sides and radius 1.2 m ..............................................................................................................................

.................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................

d Regular polygon with 10 sides and radius 200 mm ......................................................................................................................

.................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................

4 The area of each triangle is given. Calculate the marked angle.

a b c

....................................................... ....................................................... .......................................................

....................................................... ....................................................... .......................................................

....................................................... ....................................................... .......................................................

7 cm 20 mmx x

7 cm 20 mmx x

Hint Find the angle x at the centre � rst.

A*

1Area = – ab sin C 2

21 = 0.5 × 6.1 × 7 sin x21 = 21.35 sin x21 ÷ 21.35 = sin x (divide both sides by 21.35)sin x = 0.9836… (press [sin–1] on your calculator)x = 79.6° (1 d.p.)

6.1 mm 7 mm

Area = 21 mm2

A

x

12 mm 9 mm

Area = 27 mm2

x

30 cm

39 cm

Area = 210 cm2

x

1.7 m

2.4 mArea = 1.2 m2

x


347C


5 Calculate the area of each segment.

a b c

....................................................... ....................................................... .......................................................

....................................................... ....................................................... .......................................................

....................................................... ....................................................... .......................................................

Angle of sector 50Area of sector = –––––––––––––– × πr2 = –––– × π × 82 = 27.925… cm2

360 360 1Area of triangle = – ab sin C = 0.5 × 8 × 8 × sin 50° = 24.513… cm2 2

So area of segment = 27.925… cm2 – 24.513… cm2 = 3.41 cm2 (3 s.f.)

8 cm

B

A

C 50°

A*

20 mm

13 mm

12°

1.3 cm

O

O

A B

BA

110°

85°

20 mm

13 mm

12°

1.3 cm

O

O

A B

BA

110°

85°

20 mm

13 mm

12°

1.3 cm

O

O

A B

BA

110°

85°

�3 17.5–17.6 The sine rule and calculating an angle

349A


Use asinA = b

sinB to find a side, and sinAa

= sinBb

to find an angle.

Use this sine rule when you know a side and the opposite angle.

Calculate lengths correct to 3 significant figures and angles correct to 1 decimal place.

1 Find the length of the marked side.

a b c d

............................................... ............................................... ............................................... ...............................................

2 Calculate the missing angle. Then use the sine rule to � nd the marked side.

a b c

....................................................... ....................................................... .......................................................

b

A

C

B

a

x 30–––––– = ––––––––sin 25° sin 100° 30 x sin 25°x = ––––––––––– (multiply both sides by sin 25°) sin 100°x = 12.9 mm (3 s.f.)

30 mm

x 100°

25°

1.2 m

x32°

53°

50 mm

a23°

135°8 cm y

68°62°

37 cm

t

45°

52°

9 cm

a

30° 125°

100 mm

b

41° 72°

17 mm

70°c

A



349B


3 Use the sine rule to � nd the size of the marked angle. Then calculate the third angle.

a b c d

............................................... ............................................... ............................................... ...............................................

sin x sin 100–––– = ––––––– 22 38 22 x sin 100sin x = ––––––––––– 38sin x = 0.57015…

x = 34.8° (1 d.p.)

Third angle = 180° – 34.8° – 100° = 45.2°

38 mm

22 mm

x

100°

5 mm

7 mm

a

40°

1.5 cm

2.4 cm

b

23°300 mm

200 mm

c

140°

6.7 cm

10 cm

54°

d

A


349C


4 For each triangle � nd i the marked angle ii the third angle iii the marked side iv the area of the triangle (use 1

2 ab sin C).

a b c

i ............................................... i ............................................... i ...............................................

ii ............................................... ii ............................................... ii ...............................................

iii ............................................... iii ............................................... iii ...............................................

iv ............................................... iv ............................................... iv ...............................................

5 a Use the sine rule to calculate the length of DB.

...........................................................................................................................................

b Find the angle ADB and use it to calculate the length of AB.

............................................................................................................................................

c Use the sine rule to calculate angle DCB.

.................................................................................................................................................................................................................................

30 mm

a

20 mm

x 35°

2.5 cm

1.8 cm

b

c

x

121°140 m

200 m

x

95°

270 m

200 m

A B

D

C

46° 60°

31°

A


349D


d Find the angle DBC and use it to calculate the length of DC.

.................................................................................................................................................................................................................................

e Use the formula 12 ab sin C to calculate the areas of the two triangles and hence the area of the

quadrilateral ABCD.

.................................................................................................................................................................................................................................

6 The map shows some of the ancient monuments of Egypt.

Use the sine rule to find

a angle x .............................................................................................

b angle y .............................................................................................

c the distance from the Sphinx to Saqqara ........................................................................................................................................

................................................................................................................................................................................................................................

Use your answer to part a to find the bearing of

d Khum from the Sphinx ...............................................................................................................................................................................

...............................................................................................................................................................................................................................

e the Sphinx from Khum ..............................................................................................................................................................................

...............................................................................................................................................................................................................................

N

N

10.3 km

Sphinx

Khum Saqqara9.6 km

x

y

151°

57°

A

�3 17.7–17.8 The cosine rule and calculating an angle

351A


Use a2 = b2 + c2 – 2bc cos A to find a side,

and cos A = b c abc

2 2 2

2+ −

to find an angle.

Use the cosine rule when you know: all three sides or two sides and their included angle.

Calculate lengths correct to 3 significant figures and angles correct to 1 decimal place.

1 Find the length of the marked side.

a b c

....................................................... ....................................................... .......................................................

b

A

C

B

a

c

a2 = 2.32 + 0.82 – 2 × 2.3 × 0.8 × cos 145°a2 = 5.29 + 0.64 – 3.68 × (–0.819…)a2 = 5.29 + 0.64 + 3.014.. (–× – = +)a2 = 8.944…a = 2.99 cm (3 s.f.)

145°0.8 cm

2.3 cm

a

85°

1.3 m

2.8 m

a

59°

250 mm

300 mm

b

142°

40°

1.28 cm

0.31 cm

20 cm

d

c

A



351B


2 Use the cosine rule to � nd side x. Then use the sine rule to � nd angle y.

a b c

....................................................... ....................................................... .......................................................

3 Use the cosine rule to � nd angle x.

a b c

....................................................... ....................................................... .......................................................

24°

74°

26°

12 cm

9.1 cm50 mm

25 mm

11 km14 km

y

yy

x

x

x

2002 + 2302 – 1202 78 500cos x = ––––––––––––––––––––– = –––––––– = 0.853… 2 x 200 x 230 92 000x = 31.4° (1 d.p.) 200 mm

230 mm

120 mm

x

x

6.5 cm

8 cm

4 cm

x

17 m

19 m

31 m

x1.5 cm

5 cm

4 cm

70 mm

100 mm

60 mm

x

A


351C


4 Use the cosine rule to � nd angle x. Then use the sine rule to � nd angle y.

a b c

....................................................... ....................................................... .......................................................

5 PQR is a triangle where PQ = 30 mm, QR = 40 mm and angle PQR = 65°.

a Sketch the triangle.

......................................................................................................

b Use the cosine rule to find the length of PR.

......................................................................................................

c Use the sine rule to find angle QPR.

......................................................................................................

d Calculate angle PRQ.

......................................................................................................

72 mm

51 mm

40 mm

y

0.8 m 0.65 m

0.6 m

x

y

x

110 cm150 cm

90 cmx

y

A

�3 17.9 Using trigonometry to solve problems

353A


Use the sine rule when you know a side and the angle opposite. Otherwise, use the cosine rule.

Calculate lengths and areas correct to 3 significant figures and angles correct to 1 decimal place.

1 Which rule can be used to � nd a missing side or angle?

a ............................................... b ............................................... c ............................................... d ...............................................

2 Use the sine and cosine rules to calculate the unknown sides and angles in Question 1.

a ............................................... b ............................................... c ............................................... d ...............................................

............................................... ............................................... ............................................... ...............................................

............................................... ............................................... ............................................... ...............................................

3 Use the sine and cosine rules to calculate the marked angles and sides.

a b c d

............................................... ............................................... ............................................... ...............................................

............................................... ............................................... ............................................... ...............................................

............................................... ............................................... ............................................... ...............................................

23 mm

15 mm28°

2.3 cm2.1 cm

1.2 cm

23 cm

15 cm

20°

10 mm

35 mm

120°

50 mm

70 mm26°9.2 cm a

a

b

x

52° 80°

7 m

5 m

8 m

x

yx

y

x

2.2 m

1.3 m

2.7 m

112°

A


�3 17.9 Using trigonometry to solve problems

353B


4 ABC is a triangle where AB = 24 mm, AC = 20 mm and angle CAB = 28°.

a Sketch the triangle.

b Calculate the length of side BC.

.................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................

c Calculate angle

i ABC .................................................................................................................................................................................................................................

ii BCA .................................................................................................................................................................................................................................

d Calculate the area of the triangle

.................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................

1Hint use the formula – ab sin C. 2

A

Documents

3 17.1–17.2 Trigonometry in three dimensions › 2014 › 10 › ... · 3 17.3 Trigonometric ratios for any angle 345C Guided practice worksheet 4 Solve the equations in the given