Sine and Cosine Rule Problems(SL)

8/17/2019 Sine and Cosine Rule Problems(SL)

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Sine and Cosine Rule (SL)

1. The following diagram shows triangle ABC.

diagram not to scale

AB = 7 cm, BC = 9 cm and CB̂A = 120°.

a! "ind AC.(3)

#! "ind CÂB .(3)

(Total 6 marks)

2. There is a $ertical tower TA of height %& m at the #ase A of a hill. A straight 'ath goes (' the

hill from A to a 'oint ). This information is re'resented #* the following diagram.

The 'ath ma+es a ° angle with the hori-ontal.

The 'oint ) on the 'ath is 2 m awa* from the #ase of the tower.

The to' of the tower is fi/ed to ) #* a wire of length x m.

a! Com'lete the diagram, showing clearl* all the information a#o$e.

IB Questionbank Maths SL 1


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(3)



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#! "ind x.(4)

(Total 7 marks)

3. The diagram #elow shows a (adrilateral ABC with o#t(se angles CB̂A and ĈA .


AB = cm, BC = cm, C = cm, A = cm, CÂB = %0°, CB̂A = x°, ĈA = y°.

a! )se the cosine r(le to show that AC = xcos,0,1− .(1)

#! )se the sine r(le in triangle ABC to find another e/'ression for AC.(2)

c! i! ence, find x, gi$ing *o(r answer to two decimal 'laces.

ii! "ind AC.(6)



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d! i! "ind y.

ii! ence, or otherwise, find the area of triangle AC.(5)

(Total 14 marks)

4. The diagram #elow shows a circle with centre 3 and radi(s 4 cm.


The 'oints A, B, C, , 5 and " are on the circle, and 6A" is a diameter. The length of arc ABC

is & cm.

a! "ind the si-e of angle A3C.(2)

#! ence find the area of the shaded region.(6)

The area of sector 3C5 is cm2.

c! "ind the si-e of angle C35.(2)



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d! "ind 5".(5)

(Total 15 marks)

5. The following diagram shows the triangle ABC.


The angle at C is o#t(se, AC = cm, BC = 1%.& cm and the area is 20 cm2.

a! "ind BĈA .(4)

#! "ind AB.(3)

(Total 7 marks)



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6. The circle shown has centre 3 and radi(s %.9 cm.


8oints A and B lie on the circle and angle A3B is 1.4 radians.

a! "ind AB.(3)

#! "ind the area of the shaded region.

(4)(Total 7 marks)

7. The diagram #elow shows a triangle AB with AB = 1% cm and A = &. cm.

et C #e a 'oint on the line B s(ch that BC = AC = 7 cm.


a! "ind the si-e of angle ACB.(3)



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#! "ind the si-e of angle CA.(5)

(Total 8 marks)

8. The following diagram shows a circle with centre 3 and radi(s cm.


The 'oints A, B and C lie on the circle. The 'oint is o(tside the circle, on 3C!.

Angle AC = 0.% radians and angle A3C = 0.4 radians.

a! "ind A.(3)

#! "ind 3.(4)

c! "ind the area of sector 3ABC.(2)

d! "ind the area of region ABC.(4)

(Total 13 marks)

9. The diagram #elow shows triangle 8:;. The length of 68: is 7 cm, the length of 68; is

10 cm, and ; :̂8 is 7°.



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a! "ind;.:̂8

(3)

#! "ind the area of triangle 8:;.(3)

(Total 6 marks)

1. A shi' lea$es 'ort A on a #earing of 0%0°.


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#! "ind the #earing of the co(rse ta+en #* the second shi'.(3)

(Total 7 marks)

11. A farmer owns a triang(lar field ABC. 3ne side of the triangle, 6AC, is 10 m, a second side,

6AB, is & m and the angle #etween these two sides is &0°.

a! )se the cosine r(le to calc(late the length of the third side of the field.(3)

#! i$en that sin &0°= 2

%

, find the area of the field in the form %% p where p is an

integer.(3)

et #e a 'oint on 6BC s(ch that 6A #isects the &0° angle. The farmer di$ides the field into

two 'arts A1 and A2 #* constr(cting a straight fence 6A of length x metres, as shown on the

diagram #elow.



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c! i! >how that the area of A1 is gi$en #* ,

& x

.

ii! "ind a similar e/'ression for the area of A2.

iii! !en"e, find the $al(e of x in the form%q , where q is an integer.

(7)

d! i! 5/'lain wh* B̂sinAĈsinA = .

ii! )se the res(lt of 'art i! and the sine r(le to show that 4

C

B=

.(5)

(Total 18 marks)

12.


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(4)

c! The area of triangle BC is .&4 cm2. "ind CB̂ .

(4)

d! "ind AC.(4)

e! "ind the area of (adrilateral ABC.(5)

(Total 21 marks)

14.


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15. The 'oints 8, :, ; are three mar+ers on le$el gro(nd, oined #* straight 'aths 8:, :;, 8; as

shown in the diagram. :; = 9 +m,; :̂8

= %°,:; ˆ8

= 2°.

% . ° 2 . °9 + m

8

: ;iagram not to scale

a! "ind the length 8;.(3)

#! Tom sets o(t to wal+ from : to 8 at a stead* s'eed of 4 +m h 1

. At the same time, Alan

sets o(t to og from ; to 8 at a stead* s'eed of a +m h 1

. The* reach 8 at the same time.

Calc(late the $al(e of a.(7)

c! The 'oint > is on 68:, s(ch that ;> = 2:>, as shown in the diagram.

8

: ;

>

"ind the length :>.(6)

(Total 16 marks)



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16. The diagram #elow shows a circle, centre 3, with a radi(s 12 cm. The chord AB s(#tends at an

angle of 7° at the centre. The tangents to the circle at A and at B meet at 8.

1 2 c m

A

8

B

7 . D3d i a $ r a m n o t t o

s " a l e

a! )sing the cosine r(le, show that the length of AB is 12 ( )°7cos 12

.(2)

#! "ind the length of B8.(3)

c! ence find

i! the area of triangle 3B8?

ii! the area of triangle AB8.(4)

d! "ind the area of se"tor 3AB.(2)

e! "ind the area of the shaded region.(2)

(Total 13 marks)



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17. Two #oats A and B start mo$ing from the same 'oint 8. Boat A mo$es in a straight line at

20 +m h 1

and #oat B mo$es in a straight line at %2 +m h 1

. The angle #etween their 'aths is

70°.

"ind the distance #etween the #oats after 2. ho(rs.

Working:

Answer @

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

(Total 6 marks)



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18. The diagram shows a triangle ABC in which AC = 7 2

2

, BC = &, CB̂A = °.

A

B C&

. °

722

% i a $ r a mn o t t o s " a l e

a! )se the fact that sin ° = 2

2

to show that sin CÂB = 7

&

.(2)

The 'oint is on AB!, #etween A and B, s(ch that sin ĈB = 7&

.

#! i! Erite down the $al(e of ĈB F CÂB .

ii! Calc(late the angle BC.

iii! "ind the length of 6B.(6)

c! >how that BACof Area

BCof Area

∆

∆

= BA

B

.(2)

(Total 1 marks)



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19.


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2. The following diagram shows a triangle with sides cm, 7 cm, 4 cm.

. 7

4 %ia$ram not to s"ale

"ind

a! the si-e of the smallest angle, in degrees?

#! the area of the triangle.

Working:

Answers@

a! ..................................................................

#! ..................................................................

(Total 4 marks)



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21. The diagrams #elow show two triangles #oth satisf*ing the conditions

AB = 20 cm, AC = 17 cm, CB̂A = 0°.

%ia$rams notto s"ale

A

B C

A

B C

T r i a n g l e 1 T r i a n g l e 2

a! Calc(late the si-e of BĈA in Trian$le 2.

#! Calc(late the area of Trian$le 1.

Working:

Answers@

a! ..................................................................

#! ..................................................................

(Total 4 marks)



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22.


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e! Consider the e(ation f x! = 1.

i! 5/'lain, in terms of the 'osition of the 'oints 3, A, and 8, wh* this

e(ation has a sol(tion.

ii! "ind the e&a"t sol(tion to the e(ation.(5)

(Total 16 marks)

23. Town A is 4 +m from town B and %2 +m from town C as shown in the diagram.

AB

C

% 2 + m

4 + m

i$en that town B is & +m from town C, find the si-e of angle BÂC to the nearest degree.

Working:

Answer:

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

(Total 4 marks)



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24. The diagram shows a $ertical 'ole 8:, which is s(''orted #* two wires fi/ed to the hori-ontal

gro(nd at A and B.

:

8

A

B% & °

% 0 °

7 0 °

B:

= 0 m:B̂8

= %&°

:ÂB = 70°

:B̂A = %0°



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"ind

a! the height of the 'ole, 8:?

#! the distance #etween A and B.

Working:

Answers@

a! ..................................................................

#! ..................................................................

(Total 4 marks)

25. A triangle has sides of length , , 7 (nits. "ind, to the nearest tenth of a degree, the si-e of the

largest angle.Working:

Answer:

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

(Total 4 marks)

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Sine and Cosine Rule Problems(SL)