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Trigonometry (IB SL)

1 (IB/s1/2017/November/Paper2/q10)
[Maximum mark: 17]
Note: In this question, distance is in millimetres.
Let f(x)=x+asin(xπ2)+a, for x0
(a) Show that f(2π)=2π.

The graph of f passes through the origin. Let Pt be any point on the graph of f with x-coordinate 2kπ, where kN. A straight line L passes through all the points Pk.
(b) (i) Find the coordinates of P0 and of P1.
(ii) Find the equation of L.
(c) Show that the distance between the x-coordinates of Pk and Pk+1 is 2π.

Diagram 1 shows a saw. The length of the toothed edge is the distance AB.


2 (Continue: 1)
The toothed edge of the saw can be modelled using the graph of f and the line L. Diagram 2 represents this model.

The shaded part on the graph is called a tooth. A tooth is represented by the region enclosed by the graph of f and the line L, between Pt and Pz+1
(d) A saw has a toothed edge which is 300 mm long. Find the number of complete teeth on this saw.


3 (IB/s1/2017/May/paper2tz1/q7)
[Maximum mark: 7]
A particle P moves along a straight line. Its velocity vpms1 after t seconds is given by vP=tsin(π2t), for 0t8. The following diagram shows the graph of vp.
(a) (i) Write down the first value of t at which P changes direction.
(ii) Find the total distance travelled by P, for 0t8.
(b) A second particle Q also moves along a straight line, Its velocity, vQms1 after t seconds is given by vQ=t for 0t8. After k seconds Q has travelled the same total distance as P. Find k, [4]


4 (IB/s1/2017/May/paper2tz1/q8)
[Maximum mark: 14]
At Grande Anse Beach the height of the water in metres is modelled by the function h(t)=pcos(q×t)+r, where t is the number of hours after 21:00 hours on 10 December 2017 . The following diagram shows the graph of h, for 0t72.

The point A(6.25,0.6) represents the first low tide and B(12.5,1.5) represents the next high tide.
(a) (i) How much time is there between the first low tide and the next high tide?
(ii) Find the difference in height between low tide and high tide.
(b) Find the value of
(i) p;
(ii) q :
(iii) r. [7]
(c) There are two high tides on 12 December 2017. At what time does the sccond high tide occur?


5 (IB/sl/2017/May/paper2tz2/q4)
[Maximum mark: 6]
The depth of water in a port is modelled by the function d(t)=pcosqt+7.5, for 0t12, where t is the number of hours after high tide. At high tide, the depth is 9.7 metres. At low tide, which is 7 hours later, the depth is 5.3 metres.
(a) Find the value of p.
(b) Find the value of q.
(c) Use the model to find the depth of the water 10 hours after high tide.


6 (IB/s1/2016/November/Paper2/q10)
[Maximum mark: 15]
The following diagram shows the graph of f(x)=asinbx+c, for 0x12.

The graph of f has a minimum point at (3,5) and a maximum point at (9,17).
(a) (i) Find the value of c.
(ii) Show that b=π6,
(iii) Find the value of a.

The graph of g is obtained from the graph of f by a translation of (k0). The maximum point on the graph of g has coordinates (11.5,17).
(b) (i) Write down the value of k.
(ii) Find g(x)

The graph of g changes from concave-up to concave-down when x=w.
(c) (i) Find w.
(ii) Hence or otherwise, find the maximum positive rate of change of g. [6]


7 (IB/s1/2016/May/paper1tz1/q3)
[Maximum mark: 7]
Let f(x)=3sin(π2x), for 0x4.
(a) (i) Write down the amplitude of f.
(ii) Find the period of f. [3]
(b) On the following grid sketch the graph of f. [4]


8 (IB/s1/2016/May/paper2tz1/q9)
[Maximum mark: 14]
A particle P moves along a straight line so that its velocity, v ms1, after t seconds, is given by r=cos3t2sint0.5, for 0t5. The initial displacement of P from a fixed point O is 4 metres.
(a) Find the displacement of P from O after 5 seconds.

The following sketch shows the graph of v.
(b) Find when P is first at rest. [2]
(c) Write down the number of times P changes direction.
(d) Find the acceleration of P after 3 seconds.
(e) Find the maximum speed of P. [3]


9 (IB/sl/2016/May/paper2tz2/q4)
[Maximum mark: 8]
The height, h metres, of a seat on a Ferris wheel after t minutes is given by h(t)=15cos1.2t+17, for t0
(a) Find the height of the seat when t=0.
(b) The seat first reaches a height of 20 m after k minutes. Find k.
(c) Calculate the time needed for the seat to complete a full rotation, giving your answer correct to one decimal place.


10 (IB/sl/2019/May/paper1tz2/q7)
[Maximum mark: 7]
Consider the graph of the function f(x)=2sinx,0x<2π. The graph of f intersects the line y=1 exactly twice, at point A and point B. This is shown in the following diagram.
(a) Find the x-coordinate of A and of B.

Consider the graph of g(x)=2sinpx,0x<2π, where p>0.
(b) Find the greatest value of p such that the graph of g does not intersect the line y=1.


11 (IB/s1/2018/November/Paper1/q7)
[Maximum mark: 6]
Given that sinx=13, where 0<x<π2, find the value of cos4x.


12 (IB/s1/2018/May/paper2tz1/q10)
[Maximum mark: 15]
Let f(x)=12cosx5sinx,πx2π, be a periodic function with f(x)=f(x+2π). The following diagram shows the graph of f. There is a maximum point at A. The minimum value of f is 13.
(a) Find the coordinates of A.
(b) For the graph of f, write down
(i) the amplitude;
(ii) the period.
(c) Hence, write f(x) in the form pcos(x+r).


13 (IB/s1/2018/May/paper2tz1/q10b)
A ball on a spring is attached to a fixed point O. The ball is then pulled down and released, so that it moves back and forth vertically.

The distance, d centimetres, of the centre of the ball from O at time t seconds, is given by d(t)=f(t)+17,0t5
(d) Find the maximum speed of the ball.
(e) Find the first time when the ball's speed is changing at a rate of 2 cm s2. [5]


14 (IB/s1/2018/May/paper2tz2/q6)
[Maximum mark: 8]
At an amusement park, a Ferris wheel with diameter 111 metres rotates at a constant speed. The bottom of the wheel is k metres above the ground. A seat starts at the bottom of the wheel.

The wheel completes one revolution in 16 minutes.
(a) After 8 minutes, the seat is 117 m above the ground. Find k.

After t minutes, the height of the seat above ground is given by h(t)=61.5+acos(π8t). for 0t32,
(b) Find the value of a.
(c) Find when the seat is 30 m above the ground for the third time.


15 (IB/s1/2016/November/Paper1/q2)
[Maximum mark: 5]
Let sinθ=53, where θ is acute.
(a) Find cosθ.
(b) Find cos2θ. [2]


16 (IB/sl/2019/November/Paper2/q6)
[Maximum mark: 6]
The diagram below shows a triangular-based pyramid with base ADC. Edge BD is perpendicular to the edges AD and CD.

AC=28.4 cm,AB=x cm,BC=x+2 cm,ABC=0.667,BAD=0.611 Calculate AD.


17 (IB/s1/2019/November/Paper1/q6)
[Maximum mark: 8]
Let f(x)=4cos(x2)+1, for 0x6π. Find the values of x for which f(x)>22+1


18 (IB/sl/2019/May/paper1tz1/q3)
[Maximum mark: 7]
The following diagram shows a right triangle ABC. Point D lies on AB such that CD bisects ACB.
(a) Given that sinθ=35, find the value of cosθ.
(b) Find the value of cos2θ. [2]
(c) Hence or otherwise, find BC. [2]


19 (IB/s1/2019/ May / paper 2tz2/q3)
[Maximum mark: 7]
The following diagram shows the quadrilateral ABCD.

AB=6.73 cm,BC=4.83 cm,BCD=78.2 and CD=3.80 cm
(a) Find BD.
(b) The area of triangle ABD is 18.5 cm2. Find the possible values of θ.


20 (IB/s1/2018/November/Paper2/q7)
[Maximum mark: 6]
A communication tower, T, produces a signal that can reach cellular phones within a radius of 32 km. A straight road passes through the area covered by the tower's signal.
The following diagram shows a line representing the road and a circle representing the area covered by the tower's signal. Point R is on the circumference of the circle and points S and R are on the road. Point $ is 38 km from the tower and RST =43,
(a) Let SR =x. Use the cosine rule to show that x2(76cos43)x+420=0.
(b) Hence or otherwise, find the total distance along the road where the signal from the tower can reach cellular phones.


21 (IB/sl/2018/May/paper2tz1/q6)
[Maximum mark: 7]
Triangle ABC has a=8.1 cm,b=12.3 cm and area 15 cm2. Find the largest possible perimeter of triangle ABC.


22 (IB/s1/2018/May/paper2tz2/q2)
[Maximum mark: 6]
The following diagram shows quadrilateral ABCD.

AB=11 cm,BC=6 cm,BAD=59,ADB=100, and CBD=82
(a) Find DB.
(b) Find DC.


23 (IB/s1/2017/November/Paper2/q1)
[Maximum mark: 6]
The following diagram shows a triangle ABC.
AB=5 cm,CAB=50 and ACB=112
(a) Find BC.
(b) Find the area of triangle ABC.


24 (IB/sl/2017/ May / paper 1tz1/q3)
[Maximum mark; 6]
The following diagram shows triangle PQR.

PQR=30,QRP=45 and PQ=13 cm
Find PR.


25
(IB/sl/2017/May/paper2tz2/q9)
[Maximum mark: 15] A ship is sailing north from a point A towards point D. Point C is 175 km north of A. Point D is 60 km north of C. There is an island at E. The bearing of E from A is 055 . The bearing of E from C is 134. This is shown in the following diagram.
(a) Find the bearing of A from E.
(b) Find CE.
(c) Find DE.
(d) When the ship reaches D, it changes direction and travels directly to the island at 50 km per hour. At the same time as the ship changes direction, a boat starts travelling to the island from a point B. This point B lies on (AC), between A and C, and is the closest point to the island. The ship and the boat arrive at the island at the same time. Find the speed of the boat.


26 (IB/s1/2016/May/paper1tz1/q6)
[Maximum mark: 7]
The following diagram shows triangle ABC. The point D lies on [BC] so that [AD] bisects BAC
AB=25 cm, AC=x cm, and DˆAC=θ, where sinθ=23.
The area of triangle ABC is 5 cm2. Find the value of x.


27 (IB/s1/2016/May/paper2tz1/q3)
[Maximum mark: 7 ]
The following diagram shows three towns A,B and C. Town B is 5 km from Town A, on a bearing of 070. Town C is 8 km from Town B, on a bearing of 115.
(a) Find ABC.
(b) Find the distance from Town A to Town C.
(c) Use the sine rule to find A^CB.


28 (IB/s1/2016/May/paper2tz2/q2)
[Maximum mark: 6]
The following diagram shows a quadrilateral ABCD.

AD=7 cm,BC=8 cm,CD=12 cm,DˆAB=1.75 radians, AˆBD=0.82 radians.
(a) Find BD. [3]
(b) Find DˆBC. [3]


29 (IB/s1/2015/May/paper1tz1/q5)
[Maximum mark: 7 ]
Given that sinx=34, where x is an obtuse angle, find the value of
(a) cosx;
(b) cos2x. [3]


30 (IB/sl/2015/May/paper1tz2/q2)
[Maximum mark: 6]
Let f(x)=asinbx, where b>0. The following diagram shows part of the graph of f.
(a) (i) Find the period of f.
(ii) Write down the amplitude of f.
(b) (i) Write down the value of a.
(ii) Find the value of b. [3]


31 (IB/sl/2015/May/paper2tz1/q8)
[Maximum mark: 13]
The following diagram shows a straight shoreline, with a supply store at S, a town at T, and an island L.

A boat delivers supplies to the island. The boat leaves S, and sails to the island. Its path makes an angle of 204 with the shoreline.
(a) The boat sails at 6 km per hour, and arrives at L after 1.5 hours. Find the distance from S to L

It is decided to change the position of the supply store, so that its distance from L is 5 km. The following diagram shows the two possible locations P and Q for the supply store.
(b) Find the size of S\hat{P} L and of SÓL.
(c) The town wants the new supply store to be as near as possible to the town.
(i) State which of the points P or Q is chosen for the new supply store.
(ii) Hence find the distance between the old supply store and the new one. [6]


32 (IB/s1/2015/May/paper2tz2/q1)
[Maximum mark: 6]
The following diagram shows triangle ABC.
BC=10 cm,ABC=80 and BAC=35
(a) Find AC.
(b) Find the area of triangle ABC.


33 (IB/sl/2015/November/Paper2/q8)
[Maximum mark: 14]
The following diagram shows the quadrilateral ABCD.

AD=6 cm,AB=15 cm,ABC=44,ACB=83 and DAC=θ
(a) Find AC.
(b) Find the area of triangle ABC.

The area of triangle ACD is half the area of triangle ABC.
(c) Find the possible values of θ.
(d) Given that θ is obtuse, find CD. [3]


Answer (Trigo)
1 (a) Show (b)(i) P0=(0,0),P1(2π,2π) (ii) y=x (c) Show 
2 (d) 33
3 (a) (i) t=2 (ii) 9.65 (b) k=5.94
4 (a)(i) 6.25 (ii) 0.9 (b) (i) p=0.45 (ii) q=4π25  (iii) r=1.05 (d) 23:00
5 (a) p=2.2 (b) q=π7 (c) 7.01
6 (a)(i) c=11 (ii) b=π6 (iii) a=6 (b) (i) k=2.5 (ii) g(x)=6sin(π6(x2.5))+11 (c)(i) w=8.5 (ii) π
7 (a)(i) 3 (ii) 4 (b) Graph
8 (a) 0.284 (b) t=0.180 (c) 2 (d) 0.744 (e) 3.28
9 (a) h(0)=2 (b) k=1.48 (c) 5.2
10 (a) x=7π6,11π6 (b) p=712
11 1181
12 (a) A(0.395,13) (b) (i) 13 (ii) 2π (c) f(x)=13cos(x+0.395) 
13 (d) 13 (e) 1.02
14 (a) k=6 (b) a=55.5 (c) t=18.5
15 (a) cosθ=23 (b) cos2θ=19
16 34.6
17 0x<π2,7π2<x<9π2
18 (a) 45 (b) 725 (c) 50
19 (a) 5.50 (b) θ=92.0 or 1.61
20 (a) Show (b) 37.5
21 perimeter =40.6
22 (a) DB=9.57 (b) DC=10.6
23 (a) BC=4.13 (b) Area =3.19
24 PR=1322
25 (a) 235 (b) 146 (c) 193 (d) 27.3
26 x=94
27 (a) AˆBC=135 (b) 12.1 (c) AˆCB=17.0
28 (a) BD=9.42 (b) BˆDC=1.51



29(a) cosx=74 (b) cos2x=18
30(a)(i) π (ii) 3 (b) (i) a=3 (i) b=2
31(a) 9 (b) SˆQL=142,SˆPL=38.0 (c) (i) new store is at Q (ii) 4.52
32(a) AC=17.2 (b) area =77.8
33(a) AC=10.5 (b) area =62.9 (c) θ=86.7,93.3 (d) 12.4

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