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5.6 SPM Questions

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5.6 SPM Questions

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Xk Hein
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5.6 SPM Questions

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Xk Hein

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SPM

Chapter 5 : Trigonometric Functions


Paper 1
1. Given that t = tan ,
0 0
90 0 < < , express, in terms of t :
(a)
, cot
(b)
). 90 sin(
[ 3 mars !
( SPM "003 P1 #o."0)
". So$ve the e%&ation ' sec
"
( ) 13 tan ( * 0,
0 0
3'0 + + 0 A .
[ , mars !
( SPM "003 P1 #o."1)
3. So$ve the e%&ation cos
"
x ) sin
"
x * sin x for
0 0
3'0 + + 0 x . [ , mars !
( SPM "00, P1 #o.1-)
,. So$ve the e%&ation 3 cos "x * - sin x ./ for
0 0
3'0 + + 0 x .
[ , mars !
( SPM "00/ P1 #o.10)
/. So$ve the e%&ation 1/ sin
"
x * sin x 1 , sin 30
o
for
0 0
3'0 + + 0 x .
[ , mars !
( SPM "00' P1 #o.1/)
'. So$ve the e%&ation cot x 1 " cos x * 0 for
o o
x 3'0 + + 0 .
[ , mars !
( SPM "000 P1 #o.1/)
0. Given that
p = sin
, 2here p is a constant an3
0 0
1-0 90 .
4in3 in terms of p :
(a)
ec cos
,
(b)
" sin
. [ 3 mars !
( SPM "00- P1 #o.10)
Paper 2
1. (a) Prove that
" cos " cot tan ec = +
. [ , mars !
(b) (i) Setch the 5raph
x y
"
3
cos " =
for
6 " + x + 0
.
Trigonometric Functions 1
SPM
(ii) 4in3 the e%&ation of a s&itab$e strai5ht $ine for so$vin5 the e%&ation
1 .
,
3
"
3
cos x x

=
.
7ence, &sin5 the same axes, setch the strai5ht $ine an3 state the n&mber of
so$&tions to the e%&ation
1 .
,
3
"
3
cos x x

=
for
6 " + x + 0
.
[ ' mars!
( SPM "003 P" #o. - )
". (a) Setch the 5raph y * cos "x for
0 0
1-0 + + 0 x . [ 3 mars !
(b) 7ence, b8 3ra2in5 a s&itab$e strai5ht $ine on the same axes, fin3 the n&mber of so$&tions
satisf8in5 the e%&ation
1-0
. " sin "
"
x
x =
for
0
1-0 + + 0 x
o
.
[ 3 mars !
( SPM "00, P" #o. 3 )
3. (a) Prove that cosec
"
x ) " sin
"
x ) cot
"
x * cos "x. [ " mars !
(b) (i) Setch that 5raph of 8 * cos "x for
6 " + x + 0
.
(ii) 7ence, &sin5 the same axes, 3ra2 a s&itab$e strai5ht $ine to fin3 the n&mber of
so$&tions to the e%&ation 3( cosec
"

x ) " sin
"
x ) cot
"
x) *
1 .

x
for
" + + 0 x
.
State the n&mber of so$&tions. [ ' mars !
(SPM "00/ P" #o./)
,. (a) Setch the 5raph 8 * ."cos x for
" + + 0 x
. [ , mars !
(b) 7ence, &sin5 the same axis, setch a s&itab$e 5raph to fin3 the n&mber of so$&tions
to the e%&ation
0 cos " = + x
x

for
" + + 0 x
. State the n&mber of so$&tions.
[ 3 mars !
( SPM "00' P" #o. , )
/. (a) Setch the 5raph
x y " cos 3 =
for
" + + 0 x
. [ , mars !
(b) 7ence, &sin5 the same axis, setch a s&itab$e 5raph to fin3 the n&mber of so$&tions
to the e%&ation
"
" cos 3 "
x
x =
for
6 " + x + 0
. State the n&mber of so$&tions.
[ 3 mars !
( SPM "000 P" #o. 3 )
Trigonometric Functions "
SPM
'. (a) Prove that x
x
x
" tan
sec "
tan "
"
=

. [ " mars !
(b) (i) Setch the 5raph
x y " tan =
for
x 0
.
(ii) 7ence, &sin5 the same axes, setch a s&itab$e strai5ht $ine to fin3 the n&mber of
so$&tions for the e%&ation 0
sec "
tan " 3
"
=

+
x
x x

for
x 0
.
State the n&mber of so$&tions.
[ ' mars!
( SPM "00- P" #o. - )
Answers :
Paper 1:
(1). (a)
t
1
(b)
1 1 t
t
"
("). 33.'9
o
, /'.31
o
, "13.'9
o

, "3'.31
o
(3). 30
o
, 1/0
o
, "00
o
(,) ,1.-1
o
, 13-.19
o
(/) "3./-
o
, 1/'.,"
o
, 199.,0
o
, 3,0./3
o
(') 90
o
, "10
o
, "00
o
, 330
o
(0).(a)
p
1
(b)
"
1 " p p

Paper " :
(1). (b)
#&mber of so$&tions * 3
Trigonometric Functions 3
8
x
3
.3
0
" . x
"
3

= y

"
3
cos 3 = y
3
6
3
6 "
6
3
6 ,
3
6 / 6 "
SPM
(") (a) , (b)
#&mber of so$&tions * "
(3) (b) (i)
(ii) #&mber of so$&tions * ,
,. (a), (b)
Trigonometric Functions ,
8
x
1
.1
0
3
1
.
3
x
y =
x " cos * 8
,
6
"
6
6
,
3
6
,
6 / 6 "
,
6 0
"
6 3
8
x
"
."
0
x
6
* 8
x y cos " . =
"
6 6
6 "
"
6 3
8
x
0 ,/
o
90
o
13/
o
1-0
o
1
.1
1 .
1-0
x
y =
x y " cos =
SPM
#&mber of so$&tions * "
/(a), (b)
'. (b)
Trigonometric Functions /
8
x
0

x
y
3
=
x y tan " =
"
6 6
6 "
"
6 3
#&mber of so$&tions * 3
8
x
3
#&mber of so$&tions * -
0
"
1
.
6 "
x
* 8
x y " cos 3 =
,
6
"
6
6
,
3
6
,
6 / 6 "
,
6 0
"
6 3
"
1

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