Leave - Homeworkhomework.m34maths.com/files/documents/Trig-Eqns...Leave blank 14 *H26315RB01424* 6....
Transcript of Leave - Homeworkhomework.m34maths.com/files/documents/Trig-Eqns...Leave blank 14 *H26315RB01424* 6....
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14
*H26315RB01424*
6. (a) Use the double angle formulae and the identity
cos( ) cos cos sin sinA B A B A B+ ≡ −
to obtain an expression for cos 3x in terms of powers of cos x only.(4)
(b) (i) Prove that
cossin
sincos
sec , ( )xx
xx
x x n1
1 2 2 12+
+ + ≡ ≠ + π .
(4)
(ii) Hence find, for 0 x< < 2π , all the solutions of
cossin
sincos
xx
xx1
1 4+
++
= .
(3)
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physicsandmathstutor.com January 2008
a cos 3x cos 2x xcos 2x cos c Sin 2x sink
2co5x 1 cos c 2sinxcosxsinx
2cos3c Cosa 2cosse sink
2cos3x Cosa Zcosa l cosh
2cossoe Cosa 2 cosset 2cos3x
4 cos32 3cosx
b ltI sink Cosa
cask Hsin cItsinx cos a
coszxtltzsinxts.inIt Sinx Cosa
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Turn over*H26315RB01524*
Question 6 continued
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physicsandmathstutor.com January 2008
2t2sHsiao cosx
2 1tsHsiao Cosa
2 2secx as requiredCosx
c 2 Seca 4Seca 2
Lose L2
x cos E S A
Fx I 51T
3
3 J T C
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16
*N30745A01624*
5. (a) Given that sin2 θ + cos2 θ ≡ 1, show that 1 + cot2 θ ≡ cosec2 θ .(2)
(b) Solve, for 0 θ < 180°, the equation
2 cot2 θ – 9 cosec θ = 3,
giving your answers to 1 decimal place.(6)
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physicsandmathstutor.com June 2008
a since cos 0 1
gin cosI LSince Sin O
I t cotta cosec a
b 2cotte 9coseco 3
2 cosec20 l 9 coseco 3
2cosec'd 9coseco 5 0
2 coseco 1 coseco 5 to
coseco or cosec 0 5
s 2 Sino fG sin tS A0 11.50 168.50 1.5T C
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16
*H31123A01628*
6. (a) (i) By writing 3θ = (2θ + θ), show thatsin 3θ = 3 sin θ – 4 sin3θ.
(4)
(ii) Hence, or otherwise, for solve
8 sin3θ – 6 sin θ + 1 = 0.
Give your answers in terms of π.(5)
(b) Using or otherwise, show that
(4)
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0 < <θ π3
,
sin( ) sin cos cos sin ,θ α θ α θ α− = −
sin15 14
° = (√6 −√ 2).
physicsandmathstutor.com January 2009
a i sin 30 sin 20 0
sin 20 cos O t cos 20since
2sinocosocoso l 2sino since
2since I since since 2since
2since 2sink since 2since
3Since 4 sin'd
ii 8since 6since t 1 0
O I Gsince 8since
O I t 2 3since 4since
O it 2 sin 30
I sin302
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*H31123A01728* Turn over
Question 6 continued
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physicsandmathstutor.com January 2009
30 sin E IoG IT
Tp 5I18
b sin O L sincecost cososinx
sin 150 sin 45 30 sin456530 cos45sin3o
Iz 2 I
252
JJ la ETE
1 6524TG E
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*H34264A0428*
2. (a) Use the identity cos2 θ + sin2 θ = 1 to prove that tan2 θ = sec2 θ – 1.(2)
(b) Solve, for 0 - θ < 360°, the equation
2 tan2 θ + 4 sec θ + sec2 θ = 2 (6)
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physicsandmathstutor.com June 2009
a Costa t since I
cost since Itoo Costa
I t Eau'd see'd
taio seok 1 as required
b 2 tank 4 Seco seed 3
2Seco 2 4 Seco Sefo 2 0
3Sec'd 4 seed 4 0
3 Seco 2 Seco 12 o
Seco Seco 2
CosQCosa I
Z2
cos E 600
S A G 120 240
F601T C
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22
*N35381A02228*
8. Solvecosec2 2x – cot 2x = 1
for 0 - x - 180°.(7)
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physicsandmathstutor.com January 2010
I t CoE22a cosec2x
I cotta coC 2x
cot x CoE2x 0
CoE 2x CoE 2x 1 to
cot 2x to coC 2x I
tan 2x D C an 2x I
2x 900,2700 2x 450 2250
2 450 1350 x 22.50 112.50
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2
*H35385A0228*
1. (a) Show that
sincos
tan21 2
θθ
θ+
=
(2)
(b) Hence find, for –180° - θ < 180°, all the solutions of
sincos2
1 21θ
θ+=2
Give your answers to 1 decimal place.(3)
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physicsandmathstutor.com June 2010
a sin 20 2sinacosaI 1 cos20 I 2050 I
2sinocosoZ Coste
sinceCosa
tano as required
b 2siI cosza
2 C an O I
tano Iz
Ean E 26.60
S A 0 26.60 153.40
26.6
T C