Sine and Cosine Rule (SL) 1.
The following diagram shows triangle t riangle ABC.
diagram not to scale
ˆ AB = 7 cm, BC = 9 cm and ABC = 120°.
a !
"ind AC. (3)
#!
ˆ "ind BAC . (3) (Total 6 marks)
2.
There is a $ertical tower TA of height %& m at the #ase A of a hill. A straight 'ath goes (' the hill from A to a 'oint ). This information is re'resented # * the following diagram.
The 'ath ma+es a ° angle with the hori-ontal. The 'oint ) on the 'ath is 2 m awa * from the #ase of the tower. The to' of the tower is fi/ed to ) #* a wire of length x m.
a!
Com'le Com'lete te the the diagra diagram, m, show showing ing clea clearl* rl* all the the infor informa matio tion n a#o$e a#o$e..
IB Questionbank Maths SL
1
(3)
IB Questionbank Maths SL
2
#!
"ind x. (4) (Total 7 marks)
3.
ˆ ˆ The diagram #elow shows a (adrilateral ABC with o#t(se angles ABC and AC .
diagram not to scale
ˆ ˆ ˆ AB = cm, BC = cm, C = cm, A = cm, BAC = %0°, ABC = x°, AC = y°.
a!
)se the cosine r(le to show that AC =
,1 − ,0 cos x . (1)
#!
)se the sine r(le in triangle ABC to find another e/'ression for AC. (2)
c!
i!
ence, find x, gi$ing *o(r answer to two decimal 'laces.
ii!
"ind AC. (6)
IB Questionbank Maths SL
3
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.
diagram not to scale
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)
2
The area of sector 3C5 is cm . c!
"ind the si-e of angle C35. (2)
IB Questionbank Maths SL
4
d!
"ind 5". (5) (Total 15 marks)
5.
The following diagram shows the triangle ABC.
diagram not to scale 2
The angle at C is o#t(se, AC = cm, BC = 1%.& cm and the area is 20 cm .
a!
ˆ "ind ACB . (4)
#!
"ind AB. (3) (Total 7 marks)
IB Questionbank Maths SL
5
6.
The circle shown has centre 3 and radi(s %.9 cm.
diagram not to scale
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 = &. c m. et C #e a 'oint on the line B s(ch that BC = AC = 7 cm.
diagram not to scale
a!
"ind the si-e of angle ACB. (3)
IB Questionbank Maths SL
6
#!
"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.
diagram not to scale
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 l ength of 68; is ˆ 10 cm, and 8:; is 7°.
IB Questionbank Maths SL
7
a!
ˆ "ind 8:;. (3)
#!
"ind the area of triangle 8:;. (3) (Total 6 marks)
1.
A shi' lea$es 'ort A on a #earing of 0%0°.
diagram not to scale
A second shi' lea$es 'ort A and sails directl* to C.
a!
"ind the distance the second shi' will tra$el. (4)
IB Questionbank Maths SL
8
#!
"ind the #earing of the co(rse ta+en #* the second shi'. (3) (Total 7 marks)
11.
A farmer owns a triang(lar field ABC. 3ne side of the triangle, 6AC, is 10 m, a second side, 6AB, is & m and the angle #etween these two sides is &0°. a!
)se the cosine r(le to calc(late the length of the third side of the field. (3)
%
#!
i$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$ides the field into two 'arts A1 and A2 #* constr(cting a straight fence 6A of length x metres, as shown on the diagram #elow.
IB Questionbank Maths SL
9
& x
c!
, .
i!
>how that the area of A1 is gi$en #*
ii!
"ind a similar e/'ression for the area of A2.
iii!
q % , where q is an integer. !en"e, find the $al(e of x in the form (7)
d!
i!
ˆ ˆ 5/'lain wh* sinAC = sinAB .
B ii!
)se the res(lt of 'art i! and the sine r(le to show that C
=
4. (5) (Total 18 marks)
12.
ˆ the si-e of 8:; ?
#!
the area of triangle 8:;. (Total 6 marks)
13.
The following diagram shows a 'entagon ABC5, with AB = 9.2 cm, BC = %.2 cm, B = 7.1 ˆ ˆ ˆ cm, A5 =110°, A5 = 2° and AB = &0°.
a!
"ind A. (4)
#!
"ind 5.
IB Questionbank Maths SL
10
(4)
c!
ˆ 2 The area of triangle BC is .&4 cm . "ind BC . (4)
d!
"ind AC. (4)
e!
"ind the area of (adrilateral ABC. (5) (Total 21 marks)
14.
2
Answer @ .. (Total 6 marks)
IB Questionbank Maths SL
11
15.
The 'oints 8, :, ; are three mar+ers on le$el gro(nd, oined #* straight 'aths 8:, :;, 8; as ˆ ˆ: 8; shown in the diagram. :; = 9 +m, 8:; = %°, = 2°. 8
: a!
%.°
2.° 9 +m
;
iagram not to scale
"ind the length 8;. (3)
#!
1
Tom sets o(t to wal+ from : to 8 at a stead* s'eed of 4 +m h . At the same time, Alan 1
sets o(t to og from ; to 8 at a stead* s'eed of a +m h . 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)
IB Questionbank Maths SL
12
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.
A 12 cm
3
8
7.D
d ia $ r a m n o t to s"ale
B
a!
)sing the cosine r(le, show that the length of AB is 12
2(1 cos 7°)
. (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)
IB Questionbank Maths SL
13
17.
Two #oats A and B start mo$ing from the same 'oint 8. Boat A mo$es in a straight line at 1
1
20 +m h and #oat B mo$es in a straight line at %2 +m h . The angle #etween their 'aths is 70°. "ind the distance #etween the #oats after 2. ho(rs. Working:
Answer @ ...................................................................... (Total 6 marks)
IB Questionbank Maths SL
14
2 18.
ˆC 2 , BC = &, AB = °.
The diagram shows a triangle ABC in which AC = 7 A
2 7 2
% ia $ r a m n ot to s"ale
B
.°
C
&
& ˆ )se the fact that sin ° = 2 to show that sin BAC = 7 . 2
a!
(2)
& ˆ The 'oint is on AB!, #etween A and B, s(ch that sin BC = 7 . #!
i!
ˆ ˆ Erite down the $al(e of BC F BAC .
ii!
Calc(late the angle BC.
iii!
"ind the length of 6B. (6)
Area of ∆BC c!
B
>how that Area of ∆BAC = BA . (2) (Total 1 marks)
IB Questionbank Maths SL
15
19.
ˆ
.
A
4°
7
d i a $ r a m n o t t o s " a le
B
ˆ "ind B, gi$ing *o(r answer correct to the nearest degree. Working:
Answer @ ...................................................................... (Total 6 marks)
IB Questionbank Maths SL
16
2.
The following diagram shows a triangle with sides cm, 7 cm, 4 cm.
.
7
4
%ia$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)
IB Questionbank Maths SL
17
21.
The diagrams #elow show two triangles #oth satisf *ing the conditions ˆ AB = 20 cm, AC = 17 cm, ABC = 0°. %ia$rams not to s"ale
T ria n g le 1
T ria n g le 2
A
A
B
C
a!
ˆ Calc(late the si-e of ACB in Trian$le 2.
#!
Calc(late the area of Trian$le 1.
B
C
Working:
Answers@ a! .................................................................. #! .................................................................. (Total 4 marks)
IB Questionbank Maths SL
18
22.
ˆ
8 x , 1 0 ! y
=10
A 4 , & !
3 0, 0!
x
%ia$ram to s"ale
a!
A8 = x 2 1& x + 40.
i!
>how that
ii!
Erite down a similar e/'ression for 38 in terms of x. (2)
#!
ence, show that ˆA = cos 38
x 2 4 x + 0
√ I x 2 1& x + 40! x 2 + 100!H
, (3)
c!
ˆ "ind, in degrees, the angle 38A when x = 4. (2)
d!
ˆ "ind the 'ositi$e $al(e of x s(ch that 38A = &0° . (4)
et the f(nction f #e deGned #* ˆA = f x! = cos 38
IB Questionbank Maths SL
x 2 4 x + 0
√ I x 2 1& x + 40! x 2 + 100!H
, 0 ≤ x ≤ 1.
19
e!
Consider the e(ation f x! = 1. i!
5/'lain, in terms of the 'osition of the 'oints 3, A, and 8, wh* this e(ation has a sol(tion.
ii!
"ind the e&a"t sol(tion to the e(ation. (5) (Total 16 marks)
23.
Town A is 4 +m from town B and %2 +m from town C as shown in the diagram. C %2+m A 4+m
B
ˆ i$en that town B is & +m from town C, find the si -e of angle CAB to the nearest degree. Working:
Answer: .................................................................... (Total 4 marks)
IB Questionbank Maths SL
20
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
%& %0 70
:
°
B
°
°
A B:
= 0 m
ˆ: 8B
ˆ: BA
= %&° = 70°
ˆ: AB = %0°
IB Questionbank Maths SL
21
"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)
IB Questionbank Maths SL
22