HW Chapter 6

 

1

6   TRIGONOMETRY TASK   6.1

In each below, note the side is intriangle the correct position orangle not. given and state whether the identified 1. 

 

   i  t  e   s   o   p   o  p

2. 

3. 

adjacent 58

°

4. 

°

63

32

°

adjacent

°

41

5. 

6. 

7. 

53

°

8. 

h  y    p  34 o t   t  e n  n u  u s e 

°

   t 68   c  e  n   a    j     d   a °

 pposite te o pposi

     e        t         i      s      o      p      p      o

h      y    p  o  t      e  n  u  s   e 

82

°

TASK   6.2

For each triangle below, find the sides marked with letters, correct to 3 significant figures. All lengths are in cm. 1. 

34 18

°

a

2. 

c

3. 

b

4. 

39

°

d 

8·6 43

°

37 23

°

5. 

14

6. 

7. 

17·3 64  e

15

°

53

°

 g 

28°

h 6·8

 f  



8. 

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74°

7·9

 

Trigono rigonomet metry ry

9. 

10. 

11. 

3·9 27°

 

23

 j

i

12. 

19°

 

k 

2

14·8

l  56°

82°

16 13.   Find the lengths lengths of  B a   BD b   CD



58°

14.   Find the the length length of PS.

Q



C

14 cm 53°

28°

P A

S

R 

49°

D

8 cm

15.   Find the length length of KN. 

M

16.   Find the length of of TU.  S

N

°

24°

64

18 cm 28 cm 73°

K 

R

U

T

16 cm L TASK   6.3

For each triangle below, find the angles marked, correct to 1 decimal place. All lengths are in cm. 5 1. 

2.  15

 

3. 

9

c

4.  28

b

7

12·4

19

d 

a 3·8

e

3·12

 g 

5. 

19·6

6. 

7. 

46

8. 

h

15·2 7·5

 f   13·7



8·2

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6

 

Trigono rigonomet metry ry

9.   

10.   Find the the angle ADB. 

Find the angle angle RSP.

R

6 cm

S

A

B

32

°

C

18 cm

8 cm 26 cm Q

D 7·3 cm P

11.   Find the angle ABC. 

A

 

12.   If PS PS : SR = 2 : 5, fin find d the the angl anglee SQR. SQR. 

5·2 cm D

23 cm

P

13

°

Q

S 9·4 cm

B

6·7 cm R 

C

TASK   6.4

In this task, give each answer to 3 significant figures or 1 decimal place for angles. 1.   

Find the angle angle ABC.

B

2.   

Find PR.

8·4 cm

P

Q

3.   

Find LM.

K 

38

°

20 cm

13 cm

L 27

°

A

C R 



17·6 cm

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M

3

 

Trigono rigonomet metry ry

4.   A builder wants wants a metal support to be inclined inclined at an angle of 40 to the horizontal. If the vertical height from the bottom to the top of the support is 3·8 m, what is the length of the metal support?

P

5.   Find the the length length of PQ. PQ.

S 12 cm

7 cm

Q

63

°

R 

6.   A wire is fixed fixed to the ground and and attache attached d to the top of a pole of height height 9 m. If the wire on the ground is 15 m from the base of the pole, at what angle to the horizontal is the wire?

7.   AC = BC BC Find the area of triangle ABC.

9 m

M

A

B

°

59

7 cm

C 8.   O is the centr centree of a circle circle of radius radius 6 cm. If angle AOB = 50, find the area of  triangle AOB.

A

O



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B

4

 

Trigono rigonomet metry ry

Q

9.   PQRS is a kite. Find the length of PS.

5 cm

82°

5 cm

P

R 

52° S

10.   Find the the length length of AD. AD.

C

B

17 cm D 144°

14° A

11.   Do not use a calculator calculator in this this question. question. O is the centre of the circle. AC is a tangent to the circle.

cos OAB =   12

O

C 7·5 cm

13

sin OAB =   5

B

13

tan OAB =   5

D

12

Find the length of AD.



A

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5

 

Trigono rigonomet metry ry

TASK   6.5

Give answers in this task to 1 decimal place when appropriate. 1.   Bao Yu is standing standing 90 m from from the foot of a flagpole flagpole.. The flagpole flagpole is 11 m tall ta ll.. What What is th thee angl anglee of elev elevat atio ion n of th thee to top p of th thee flag flagpo pole le fr from om Bao Bao Yu? Yu? 2.   Rosa is standing standing on the top of a hill looking down at her villa village. ge. The angle of  depression of the top of her house is 7 . If she is 163 m higher higher than the top of her house, what is the horizontal distance of Rosa from her house? 3.   Geir Geir is stan standi ding ng on to top p of a ca cast stle le,, 29 m ta tall ll.. A fr frie iend nd is st stan andi ding ng on th thee grou ground nd belo below, w, 70 m from the base of the castle. castle. What is the angle of depression depression of the friend from Geir? 4.   A camera camera is placed placed on a table table to film a church. The angle of elevation from the camera to the top of a flagpole on the church tower is 5 . Using the measurements measurements shown, what is the horizontal distance from the flagpole to the camera?

3 m

25 m 1·2 m

5.   The The angl anglee of el elev evat atio ion n of a bird bird from from an obse observ rvat atio ion n poin pointt on th thee grou ground nd is 6·7 6·7. If the horizontal distance distance of the bird from the observation point is 230 m, how high is the bird above the observation point? 6.   The angles angles of elevation elevation of the front of two tall buildings from a point A is shown in the diagram opposite. What is the difference in A the heights of the two buildings?

3

°

8

°

405 m

7.   A telegraph telegraph pole stands on a line joining joining two points P and Q on the ground. Thee te Th tellegra egraph ph po pole le is 9 m ta talll. The The an angl glee of de depr preessi ssion of the the top top of the the po pole le to po poiint P is 4. The angle of depression of the top of the pole to point Q is 7 . What is the horizontal distance between P and Q?



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107 m

6

 

Trigono rigonomet metry ry

TASK   6.6

Draw an accurate scale drawing of each shape below using the scale shown.



4 m



1.

2.

6 m

2 m 4 m 8 m

4 m 4 m 2 m

9 m Scale: 1 cm for every 3 m

4 m Scale: 1 cm for every 2 m

3.   Make a scale scale drawing drawing of the front front of this house house using a scale of 1 cm for every every 2 m.

2 m 2 m

5 m

2 m 6 m

4.   The model model of a statue statue is made made using using a scale scale of 1 : 40. If the the statue statue is 3·2 m tall, how tall is the model (give your answer in cm)? 5.   A park park is 5 cm long long on a map map whose whose scale scale is is 1 : 40 000. 000. Find Find the actual length (in km) of the park. 6.   Copy and complete complete the table table below. below. map length 7 cm

scale 1 : 60

real length m

5 cm

1 : 2000

m

8 cm

1 : 50 000

km

cm

1 : 100 000

3 km

cm

1 : 4000

320 m

cm

1 : 5 00 000 00 000

125 km km

7.   The distance distance between between two towns is 25 km. How far apart apart will they be on a map map of scale scale 1 : 50 500 0 00 000? 0? 8.   A pl plan an of a ho hous usee is made made us usin ing g a scal scalee of 1 : 30 30.. The The widt width h of the the ho hous usee on the plan is 40 cm. What is the real width width of the house? (give your answer in metres)



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7

 

Trigono rigonomet metry ry

TASK   6.7

Use a calculator and give all answers to 3 significant figures when appropriate.

V

1.   This pyramid pyramid has a square square base PQRS PQR S of side side length 9 cm.

VM Findis vertical. M is the midpoint of PR. a   the length length of of PM b   the height height VM of the pyramid. pyramid.

12 cm Q

P

M S K 

2.   Find the following following lengths lengths in this cuboid. a   QS b   LS c   KR

R  L

 N

5 cm

M P

Q 7 cm

S

20 cm

R 

3.   A man walks walks 30 m due west west from the the foot of an aeria aeriall mast which which is 40 m high. He then walks walks 50 m north. How far is he from the top of the mast?

P 4.   This pyramid pyramid has a rectangular rectangular base base ABCD which is horizontal. The vertex P is directly

above the centre of the rectangular base. The perpendicular height of the pyramid is 27 cm. Find a   AQ b   AP c   the area of the face face PCD

B Q A

5.   A cuboid is is such that that its height height is 1 m longer than than its width and its length length is 2 m longer than its height. height. If the longest diagonal diagonal in the cuboid cuboid is 6 m then find the width of the cuboid.



C

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16 cm

13 cm D

8

 

Trigono rigonomet metry ry

TASK   6.8

Use a calculator and give all answers to 3 significant figures.

P

1.   PQRSTUVW PQRSTUVW is a cuboid. Find a c    QV WU e   QW

    

b d    angle angle QVU WUV f    angle QWU

Q

S 14 cm

R  T

U 17 cm 30 cm

W 2.   BC is perpendicul perpendicular ar to CD. Find a   AC b   angle BAC c   CD d   the angle angle between between BD and the the

V B 25 cm

7 cm C

A

plane ACD

31 cm

D V 3.   This pyramid pyramid has a rectangular base base ABCD. VM is vertical. vertical. M is the midpoint of AC. Find a   BM b   VM c   the angle angle between between VB and the the plane ABCD ABCD

21 cm A

d   angle BVC

M D

4.   This diagram diagram represents represents the roof of a house. house.

ABCD is horizontal horizontal with with AB = 10 m and BC = 26 m. EF is horizontal. horizontal. EF is 20 m long and 2 m above the horizont horizontal al plane plane ABCD. AEB and DFC make the same angle with ABCD. BCFE and ADFE make the same angle with ABCD. N is in the plane ABCD and is vertically below E. M is the midpoint of  AB. X is the point on BC such that the angle NXC is a right angle.



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B

28 cm

18 cm C

9

 

Trigono rigonomet metry ry

E

F

D A

 N

M a b c d

       

B

C

X

Write down down the lengths lengths of EN and and MN. Find the angle angle between between EM and the plane ABCD. ABCD. Calculate Calculate the length length NC. Find the angle which which the line EC makes with the plane plane ABCD.

TASK   6.9

A

Remember

c

B

a   b   c sinA   = si sin n B   = si sin nC   sinB   =   sinC or   sinA  = c a b

b

C

a

Use a calculator and give all answers to 3 significant figures.



Find the value of each letter in Questions 1   to 6 1. 

2. 

74

°

a

3. 

b

c 6·4 cm

12 cm

83

°

59

°

88

°

19 cm

4·9 cm

42

°

4. 

5. 

°

39

7 cm

°

 f  

9·4 cm

81

°

72

°

63

°



6. 

e

42

5·4 cm

d 

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h

16 cm  g 

10

 

Trigono rigonomet metry ry

7.   In triangle triangle KLM, KM = 6·8 cm, cm, the angle K KLM LM = 78 and the angle MKL = 34. Find the perimeter of triangle KLM. 8.   Maddlestone Maddlestone is 28 km due north of Kenton. Howick Howick is on a bearing of 137 from Maddlestone and Howick is on a bearing of 067  from Kenton. Find the shortest distance from Kenton to Howick.

TASK   6.10

Remember

area =   1   ab sin —C

B c A

2

a C

b

Use a calculator and give all answers to 3 significant figures when appropriate.

 2. 

Find the value of each letter in Questions 1   to 3 1. 

5·9 cm 36

°

b

 

3. 

d 

 g 

5·3 cm a

7·4 cm

8·2 cm e

c  f  

i

h °

76

6·2 cm

°

69

12·8 cm 4.   In triangle ABC, ABC, AB = 15·1 cm, BC = 29·7 cm and AC = 19·6 cm. Find the area of triangle ABC. 5.   Balloon Balloon A travels 15 km on a bearing bearing of 029. Balloon B travels 39 km on a bearing of 043. a   How far apart apart are the balloons balloons now? b   On what bearing bearing would balloon balloon A have to travel travel in order to reach the position of balloon B?

Q

6.   If the area of triangle triangle PRS PRS is 127 cm2, find the area of triangle PQR.

11·9 cm 24° R 

P 12 cm

23 cm S



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11

 

Trigono rigonomet metry ry

7.   Hanif walks walks 6 km on a bearing bearing of 159 159 then 9 km on a bearing of 252. On what bearing must Hanif walk to return to his starting point and how much further must he walk?

TASK   6.11

Remember

A

 2bc a2 =   b2 + c2  2 bc cos —A

b

c B

a

2 2 cos —A =   b + c

or



 a2

2bc

C

Use a calculator and give all answers to 3 significant figures figures..



Find the value of each letter in Questions 1   to 3 1.



8 cm

75

°

2.

14 cm



a

3.

25 cm 23

b

 17 cm

°

11 cm

9 cm

c 9 cm 4.   In triangle triangle PQR, PQR, PQ = 32 cm, PR = 28 cm and QR = 21 cm. Find the size of angle QPR. 5.   A boat boat sa sail ilss du duee north north for 41 km th then en tr trav avel elss on a be bear arin ing g of 21 218 8 for 23 km. The The boat then sails back to its starting position. Find the total distance travelled by the boat. 6.   Yac Yaco is 14 km du duee east east of Davi Davina na.. Yac Yaco walk walkss on a be bear arin ing g of x and Davi Davina na walk walkss on a bearing of  y  y . They meet at a more southerly point when Yaco has walked 7 km and Davina has walked 10 km. Find x Find  x and  and y  y..

B

7.   Find the perimete perimeterr of triangle ABC. ABC.

°

52 9 cm

8 cm °

109

A D



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5 cm

C

12