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F.4 Additional Mathematics Quiz

(Trigonometry I)

|Name: ___________________ |F.4___ ( ) |2009-02-06 |

Full mark: 38

FORMULAE

|[pic] |[pic] |

1. Let 5sin2x ( 12cos2x ( Rsin(2x ( () where R > 0 and ( is an acute angle.

(a) Find the values of R and (. (Give 3 significant figures if necessary)

(b) Find the maximum values of the following expressions.

(i) [pic];

(ii) [pic].

(5 marks)

2. Consider 3y2 ( (2cosx)y + 1 ( sinx = 0 - - - - - - (*)

as a quadratic equation in y.

(a) Show that the discriminant of (*) is 4(cos2x + 3sinx ( 3).

(b) Suppose (*) has double roots, find the value of x for 0 < x < 2(.

(6 marks)

3. (a) Express sin6x + cos6x in terms of cos4x .

(b) Find the maximum and minimum values of (sin6x + cos6x ( [pic])2 .

(8 marks)

4. Show that

(a) tan(A + B) ( [pic].

(b) [pic]

(8 marks)

5. (a) If sin(( + () = ksin(( ( (), where cos(cos( ( 0, show that (k ( 1)tan( = (k + 1)tan( .

(b) Given that 2( + ( = 90(. Using the result of (a), solve the equation

sin(( + () = 2sin(( ( ()

for ( and ( where 0( ( ( ( 180( and 0( ( ( ( 180(.

(10 marks)

Bonus questions (0.5 mark each)

6. Let x = tan1( , evaluate tan1(tan2( + tan2(tan3( + tan3(tan4( + ... + tan44(tan45( in terms of x .

7. Show that sin1((sin2((sin3((…(sin90( = [pic].

END OF PAPER

Marking scheme of F.4 additional mathematics quiz (trigonometry I) 2009-02-06

1.

(a) 5sin2x ( 12cos2x ( 13cos(sin2x ( 13sin(cos2x where ( = tan(1([pic]) = 67.4( 2M+ 1A

( 13sin(2x ( (), thus, R = 13 1A

(b) (i) [pic] ( [pic]

maximum value = [pic] = [pic] 1A

(ii) [pic] ( [pic]

maximum value = [pic] = [pic] 1A

2.

(a) ( = [((2cosx)]2 ( 4(3)(1 ( sinx) = 4(cos2x + 3sinx ( 3) 2M

(b) (*) has double roots, ( = 0

cos2x + 3sinx ( 3 = 0 ( 1 ( sin2x + 3sinx ( 3 = 0 ( sin2x ( 3sinx + 2 = 0 2M

sinx = 1 or sinx = 2 (rejected) ( x = [pic] 2A

3.

(a) sin6x + cos6x ( (sin2x + cos2x)(sin4x ( sin2xcos2x + cos4x) 1M

( sin4x ( sin2xcos2x + cos4x

( (sin2x + cos2x)2 ( 2sin2xcos2x ( sin2xcos2x 1M

( 1 ( 3sin2xcos2x

( 1 ( 3([pic])2 1M

( 1 ( [pic]([pic]) 1M

( [pic]cos4x + [pic] 1A

(b) (sin6x + cos6x ( [pic])2 ( ([pic]cos4x + [pic] ( [pic])2 ( [pic]cos24x 1M

∴ maximum value = [pic] ; minimum value = 0 2A

4.

(a) sin(( + () = ksin(( ( () => sin(cos( + cos(sin( = k(sin(cos( ( cos(sin() 1M

(k + 1)sin(cos( = (k ( 1)sin(cos( => [pic] 1M

( (k ( 1)tan( = (k + 1)tan( 1

(b) ∵ 2( + ( = 90( and 0( ( ( , ( ( 180( ( both ( and ( are acute. 1M

sin(( + () = 2sin(( ( () => (2 ( 1)tan( = (2 + 1)tan( 1M

tan( = 3tan(90( ( 2() = [pic] 1M

= [pic] 1M

( 5tan2( = 3

( tan( = [pic] or ([pic] (rejected ∵ 0( ( ( ( 90() 1M

( ( = 37.8( 1A

( = 90( ( 2(37.8() = 14.5( 1A

5.

(a) [pic]

( [pic] 1M

( [pic] 1M

( [pic] 1M

( tan(A + B) 1

(b) [pic]

( [pic] ( [pic] 1M

( [pic] 1M

( [pic] 1M

( [pic] ( (tan3( 1

6. (a) [pic] 0.5A

(b) [pic] is suggested. 0.5M

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