Sketch y : sin3x for 0 < r < 360'.

One cycle. We have 'compressed'3 cycles ofthe curve into the range 0-360o.

L Draw a sketch graph of the following for 0 < x < 360'.

a y: cos(r - 90o)

d y:3sinxb y:sinx-12 c y: tan(r*90")e y: cos2x

2 Match each equation with one of the sketch graphs below.

B: Y:cos(x-90') C:A, y: sin2xD, y: sin.x * 1 E: y:cosx-1

3 Draw a sketch graph of the following for 0 < r < 360".

F:

y: sin(x - 90")

y :2sinx

a y :2sinxdy=sin(x*90')

b y: tan2xe Y:2sinx*2

.tc y: smztr

L198

IIi

Draw a sketch graph of y : 2 sin(l + 180") for 0 < r < 360o.

Use your graph to write down the solution to the equation

2sin(r * 180.):2 (for0<r<360')

Draw a sketch graph of y : tanlxfor 0 < r < 350o.

How many solutions are there to the equation

tanlx : lz (for 0 < I < 360")

5* Draw a sketch graPh of the following for 0 < r < 360o.

4ab

5ab

7a

b

a Y: -costrd y: sin(-2r)

by:cos(-x) c y:1-cosxe Y: -2 sinx

On the same axes, sketch the curves y : cos4x and y : sin 2x f.or

0<x<90o.How many solutions are there to the equation cos 4x : sin 2r in this range?

8 a Sketchthecurve A :2tanZxfior0 (x{ z.

b In the range 0 < x < z how many solutions are there to the equation2tan2x:11?

1 Calculate the length of the side marked with a letter' All lengths are in cm.

In triangle ABC, a : 4.3, b : 7.2, c : 9. Find C.

nnIn triangle DEF, D : 58o, EF :7.2, DE = 5.4. Find F.

In triangle PQR, p : 8, 4 : 14, r: 7. Find 0.

199

6a

i In triangleh{Z,Y : 97.3",X2: 22, XY : 14. Find Z'

A point T is 11 km due north of a point S. A point_R,-which is east of the line

;oining T and S, is 8 km from T and 7 km from S. Calculate the bearing of R

from S.

A fourth point Q is on a bearing of 320" from s and is 10 km from T.

Calculate angle TQS and hence the bearing of T from Q'

7 Find the area of each triangle' All lengths are in cm'

8 An equilateral triangle has an area of 300 cm2. Calculate the length of the sidesof the triangle.

9 If sin 27" :0.454, give another angle whose sine is 0.454.

t1

Sort the following into pairs of equal value.

f-'i" 3o"l fcos 45"-l ltanrzs.l

li*55.] E,3oFlI tan 45" I l-ri" 45"-l icos6|

10

using surds where necessary.c sin 120.

f cos 30o

12 solve the fo'owing trigonometric equations in the given intervals(to 1 decimal place whEr".,"."rrrry ),asinx:-afor0<r<350o bcosr:_3. ^3 ., pcosr:_itor0<x<360.

c tanx:-0.7for0<x<360. dsinr:_lfor0<x<360o7-

the exact value of the following,b cos.45

e tan 135o

Write downa tan 60o

d cos 120o

For each diagram find:a the length of minor arc ABb the area of the minor sector AOB

Find all values of gbetween 0o and 360'for which

2sin0+8cos20:5,

giving your answers correct to the nearest 0.1o where necessary.

Sketch the following curves for 0 < x < 360"

solve the following trigonometric equations in the given intervals:

" ,ir,(*-:) : -€ 4, 0 -x12n b sin/x-'\ :1fo. 0lx'-2rr\ 2) z """'\^ 3) 2-"'"

. "o,("

. f) : -| rc,0,-x .-2n d tan(x - I):t for -z ( x I rr

a y:3sinrd y:1 * cosr

b y: cos(r - 90')e y :2cos4x

-l

16

1.8

c y: sinZx

Solve the following trigonometric equations in the given intervals:a cos2r * 2sin x - 2:0 for0'<.r<360'(1 exactvalue)b 2sin2 x - cos x - 1.:0 for 0 <.r < 2n (3 exactvalues)c 6cos2r * sin x - 5 :0 for -180o ( ff < 180" (2 exactvalues and 2 values

to 1 decimal place)

d 6sin2x * cos x - 4:0 for -180o (.tr< 180'(2 exactvalues and 2 valuesto 1 decimal place)

e 4cos2 x - 4sinx - 5 - 0 for -zr< x4 rr (2 exact values).

Find the exact solutions to the following trigonometric equations in the givenintervals:

a sin'(x* 3) :1 ro, -rr< x 1r\ 6) 2

b cosr(x- t\ :1 ro, o1x12n\ 3J +

c sin(x+a\:1 ro, o4x,-2,-\ 21 4

201

In triangle X:tz,} : 97'3",X2 : 22'XY : 14' Flr:'.dz'

aApointTis1LkmduenorthofapointS'ApointR'.*T"hiseastofthelinejoining T and s, Js k^ from T u"a /m trom s. Calculate the bearing of R

from S.

b A fourth point Q is on a bearing of 320" from S and is 10 km from T'

Calculate angle f-QS a"d hencJthe bearing of T from Q'

Find

a

the area of each triangle' All lengths are in cm'

Ft" 3o;l

tr" 45;l

|.*45"-l

Fi" 45f

It-'tFlt*'3oFl

F t 60;-l

11 Write downa tan 60"

d cos 120'b cos45

e tan 135o

c sin 120"

f cos 30o

the exact value of the following, using surds where necessary.

L2 solve the following trigonometric equations in the given intervals(to 1 decimal place where,","."rruryi e

ta sin *: -lforo<x<360o-J

c tan x: -0.7 for 0 < x < 360o

b cos *: -?for0<r<360o4

d sin r:-lforo<x{360o2

19 Solve the following trigonometric equations in the given intervals

a sin2(r - 40') :9 for 0o < x < 360" (exact values)4

1

b cos 2x : I for 0 < x 12rr (exact values)2

c 8cos2 x *2sinx-5:0 for0'<tr<360" (to L decimalplace)d sinr : 3 cos r for 0 <.r < 2n (to2 decimal places)

e 2sin2l : sin.r costr for 0 < r < 360o

20 In the diagram shown,r : radius of circlel: arc length0: angle at centre in radiansA : area of sector

a If r : 8cm and 0 : l.S,find l.b If r: 6 cm and 0 : 0.8, find A.c If I : 12crr. and 0 :1,, find r.d If A : 20 cm2 and r :2crr., find 9.

e If / : 50cm and r : 20cm, find 0.

f If A : 3cm2 and I : 3cm, find r and 0.

g If A : 8 cm2 and I : 4cm, findr.

2T Achord PQ of a circle of radius 10 cm subtends an

angle of A radians at the centre O. Calculate theU2exact areas of the two parts into which the chordPQ divides the sector POQ.

22 In the diagram, the circle has centre O and radius r.Angle AOB is g radians and C is the mid-pointof OB. The length of AC is l.

a Express 12 in terms of r and 0.

b Given that I : !r, calanlate the value of g,

correct to 3 decimal places.

c Given thatr :2cn, calculate the area of theshaded region, giving your answer correctto 2 decimal places.

202

29 i sketchthegraph oIy: sinrforvalues of rsuchthat0o <x<720".ii Sketclr, on the same diagram, the graph of y : sin 11, for values of r such

that0o <x<720".

iii State the number of solutions of the equation

sinr: sinlrfor values of r such that 0o < x <720".

iv specify the transformation which transforms the complete graph ofu : sin x (i,e. the graph drawn for all values of x) to ttre complete graph ofi:snir.

v I - IocRI

30 f(r):5sin3ro, 0<x<180.a sketch the graph of. f.(x),indicating the value of x ateach point where the

graph intersects the x-axis.

c

31 a

write down the coordinates of all the maximum and minimum points off(x).

Calculate the values of r for which f.(x) :2.5. IEDEXCELI

Find all the solutions of the equation

sin (3r * 45"):9.7in the interval -90o < r ( 90o, giving your answer to the nearest 0.1o.

No credit will be giaen for simply reading aalues from a graph.

Describe a sequence of geometrical transformations that maps the curvey : sin r onto the curve y : sin (3r + 45"). tAeAI

207

7 Solve the equationS% - 8(3') :0.

8 lf 7v : 3", show lhaly : kx for some constant k which is to be determined.

9 If 8v : 5', show thaty : kx f.or some constant k and find the value of k.

10 a If log13 + log xy - log3x : O,find an equation involving logx and logy.

b Hence find y in terms of x.

11 a If log.l.2 - log xy + Loglx2 : 0, find an equation involving log r and (ogy.

b Hence findy in terms of r.

12 a Write down, in its simplest form, the common difference of the arithmetic

log23 * log29 *log227 + ...b Show that the sum of the first ten terms of A is 55 log2 3.

1-3 Find the value of x for which 23x +"1 - 3'+ 2, giving3 significant figures in youranswer.

1 Express as a single logarithm.

5 If log2 x : \2 then find the following:a log2r3 b log216x

5 Solve the following equations:

d logls (3x + 11 :2B log,15 : 3

b 42'-1:65e logT(x-3):8h log3 9 : logle r

a log2 + 1og5

d log 3 + log 100

e 21og4 + Iog2

Evaluate the following.a log5 25

d tog, (*)

Write aasalogarithm.a 1,V:4

c log2li

c logls (5x) :6t Log,S:4i log5 x: Logd2

c 2Log3

f log48 - log 8i log5+log6-1o910

lo961

logr(|)

b log 20 - log4e log9+log9h 3log2*Log4

b log21,6

e lo91610000

b 6o:17

c

f

[Hint Remember '102 :100 <+ log1e L00 : 2'l

c 7o:3

213

4 Express the following in terms of log a,Iogb andlogc.a logab2

d Los&c

b logc2a

" tog4a

logi6lblog-a

c

f

5 Find.r, correct to 3 significant figures.

a 5':17 b 3':8 c 11,': 100 d 2* - t :5

Solve the equations

alog2(x+5)-Iog2x:3 blog3(x+11)-logr(x-t):4c log5 Gx + 47) - logr (x + 1) :2 d Log,7 :2

_z t,7x+107 a Simpltf1?.x'I 5x

b Find the value of r for which \og2(x2 * 7x * t0) - logz @2 + 5x) :3.

8 Given thatb : lo& 27, express in terms of b:

a togoS b Log,(9a)

9 Given thatb:Iogo2, express in terms of b:

a logo8 b Iog,(1,6a)

10 Given that 5 * 3 log2 x : log2y,show that y : 32x3.

11 a Given that 1 + 2logsx:logt!, show thaty:3x2.b Hence, or otherwise, solve the equation

7 + 2Logsr: logs Vx - 2)

12a

b

c Iog(4a2)

Show that log" m,Iogomn andlogmn2 are three successive terms of anarithmetic sequence whose common difference islogon.Given that mnz : a, show that the sum of the first 5 terms of the arithmeticsequence with first term logo m and common difference log,on is 5.

Given that

2log2x : k andlogr(Zx) : k + 4

find the value of r.

214

1 Express as a single logarithm in its simplest form

log2+2log18 -ltog36

a Show that log2 8 : 3.

b Find the value of

i 1og2 8a

11rii tog2 (,m,)

3 Given that p : logq 16, express in terms of p,

a logr2,

b logr(8q).

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4 a Write down the exact value of r given that 4' : 8.

b use logarithms to find y, correctto 3 decimal places, when 5v: 10. tocRl

5 It is given that ln x : p + 2 andlny :3p.Express each of the following in terms of p:

[nr: log"x]

a ln(xy),

b ln(x3),

. '" (i)

7 The variable r satisfies the equation3,.Ab + 1 - 6)c + 2.

By taking logatithms of both sides, show that r : 1ot'.log 8

y2+4x+g5 a Simolifv ""

- f ---J x.*xb Find the value of x for which logr(r.., * 4x * 3) - logz (x2 + tg1 :4.

IEDEXCELI

tAQAl

215

8i

9ab

L0ab

L1 a

111

Solve the equation

f+4x2-9x-6:0,giving each root in an exact form.

Given that

2ln(x + 2) + lnff : ln (13x + 6),

prove that

x3+4x2-9x-6:0.Hence solve the equation

Zln(x + 2) + lnr : ln (13x + 6). IOCR]

Given that 3 * 2log2x:logz!, show thaty :8*.Hence, or otherwise, find the roots a and B, where o1 F, of the equation

3 + 2Iog2x : logzf. x - 3).

Show that log2 q.: -2.Calculate logr$, giving your answer to 3 significant figures. IEDEXCEL]

Find the smallest integer satisfying the inequality 2 > 100643.

Find the exact solution of the equation

In(2A + 1) - In4: 1.,

giving the answer in terms of e. lReminder lny: Log.yl IOCR]

Given that

Iogx:logo5 * 2log"3,

where a is a positive constant, show that r : 45.

i Write down the value of Log22.

ii Given that

logry : loga2,

find the value of.y. IAQA]

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216

Find the constants A and B in the following.a (x*Z)(Ax +B): x2+7x+']-,0 b (x+1)(Ax+ B):2*+5x+3c (r* 2)(Ax+B): 3x2+8x+ 4 d (x-2)(Ax+B): 3*-5x-2e (2x + 1)(Ar + B): 6f +7x+2 f (2x + 3XAx+ B) = 6x2 +7x-3g @- 1)(Ax+B): x3+2x2-x-2 h (f + 3)(Ax+B):3x3-f *9x-3

Find the constants A, B and C.

a (x * 1,)(Ax2 * Br * C) :2x3 + 13x2 +']-,6x -15

b (r + 2)(Ax2* Br * C) : x3 + 7f + 1,6x + 12

c (tr - 1)(Atr2 * Br* C) : x3 * f - x -'1,d (r+3)(Ax2 *Bx* C):2x3+9*+llx+5e (x - 2)(Ax2 * Bx * C):4x3 - 9x2 + 5x - 6

Use the factor theorem to factorise the following cubics as far as possible:

a f(x'):'x3-7x-6c f(r):x3-3x*2e f(r):f +x2-2gf(x):i3-5f+5x+3i f(r) :2f -t 2x2 - \lx * 3

bf(r):'r3-7x*6d f(r) :.f + x2 - 6x

t f(x):r9+ x2-2x-8hf(x):f+4x2-x-4i f@):4f *1012 +x-6

b 12x3 _ 37x2 _t 2gx _ 6:0d,f-x:0

Find a given that:

a (r - 2) is a factor of f(r) : x3 + af + 3x + 2b (r + 1) is a factor of f(r) : i3 - 2x2 * ax -17

c (x - 1) is a factor of f(x) :2x3 + Sxz - 3x * a

d (r + 3) is a factor of f(r) : flx3 - * + 4x - 6

Find a andb given thata (x - 1) and (x - 2) are factors of f@) :2x3 + af + bx - 6

b (r + 1) and (r - 5) are factors of f(r) : ax3 -'1,4x2 + bx + 10

c (r * 3) and (x - 4) are factors af f@) :3f - 7f + a* * bx + 24

d (x + 2) and (3r + 1) are factors of f(x) :3f + axz + bx - 2

e (x - 1) and (2x + 3) are factors off(x): ax3 + 3x2 + bx - 3

Enda giventhat (r + 4) is a factorof f(x) : f +8x2 + ax* 12 andhencefactorise f(x) completely.

I! Kx) : f + 3f + x - 1, find f(1) and (- 1). Hence factorise f(r).

Sftrow that (r - 2) is a factor of f(x) : x3 - 4x2 + x* 6 and hence solve f(r) : 0.

Fnd the three solutions to each of the following equations (giving youranswers as fractions where necessary):

e 10ri-29x2*25x-5:0c 4f-72x2*5r:0

1,19