Leave - Homeworkhomework.m34maths.com/files/documents/Trig-Eqns...Leave blank 14 *H26315RB01424* 6....

8
Leave blank 14 *H26315RB01424* 6. (a) Use the double angle formulae and the identity cos( ) cos cos sin sin A 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 cos sin sin cos sec , ( ) x x x x x x n 1 1 2 2 1 2 + + + + π . (4) (ii) Hence find, for 0 x < < 2 π , all the solutions of cos sin sin cos x x x x 1 1 4 + + + = . (3) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ physicsandmathstutor.com January 2008 a cos 3 x cos 2x x cos 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 lt I sink Cosa cask Hsin c Itsinx cos a coszxtltzsinxts.in It Sinx Cosa

Transcript of Leave - Homeworkhomework.m34maths.com/files/documents/Trig-Eqns...Leave blank 14 *H26315RB01424* 6....

Page 1: Leave - Homeworkhomework.m34maths.com/files/documents/Trig-Eqns...Leave blank 14 *H26315RB01424* 6. (a) Use the double angle formulae and the identity cos( ) cos cos sin sinAB A B

Leave blank

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)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

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

Page 2: Leave - Homeworkhomework.m34maths.com/files/documents/Trig-Eqns...Leave blank 14 *H26315RB01424* 6. (a) Use the double angle formulae and the identity cos( ) cos cos sin sinAB A B

Leave blank

15

Turn over*H26315RB01524*

Question 6 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

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

Page 3: Leave - Homeworkhomework.m34maths.com/files/documents/Trig-Eqns...Leave blank 14 *H26315RB01424* 6. (a) Use the double angle formulae and the identity cos( ) cos cos sin sinAB A B

Leave blank

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)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

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

Page 4: Leave - Homeworkhomework.m34maths.com/files/documents/Trig-Eqns...Leave blank 14 *H26315RB01424* 6. (a) Use the double angle formulae and the identity cos( ) cos cos sin sinAB A B

Leave blank

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)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

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

Page 5: Leave - Homeworkhomework.m34maths.com/files/documents/Trig-Eqns...Leave blank 14 *H26315RB01424* 6. (a) Use the double angle formulae and the identity cos( ) cos cos sin sinAB A B

Leave blank

17

*H31123A01728* Turn over

Question 6 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

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

Page 6: Leave - Homeworkhomework.m34maths.com/files/documents/Trig-Eqns...Leave blank 14 *H26315RB01424* 6. (a) Use the double angle formulae and the identity cos( ) cos cos sin sinAB A B

Leave blank

4

*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)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

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

Page 7: Leave - Homeworkhomework.m34maths.com/files/documents/Trig-Eqns...Leave blank 14 *H26315RB01424* 6. (a) Use the double angle formulae and the identity cos( ) cos cos sin sinAB A B

Leave blank

22

*N35381A02228*

8. Solvecosec2 2x – cot 2x = 1

for 0 - x - 180°.(7)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

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

Page 8: Leave - Homeworkhomework.m34maths.com/files/documents/Trig-Eqns...Leave blank 14 *H26315RB01424* 6. (a) Use the double angle formulae and the identity cos( ) cos cos sin sinAB A B

Leave blank

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)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

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