1
Teaching and learning module Additional mathematics form 5
CHAPTER 5
NAME:\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026
FORM :\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026
Date received : \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 Date completed \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 . Marks of the Topical Test : \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 Prepared by : Additional Mathematics Department Sek Men Sains Muzaffar Syah Melaka For Internal Circulations Only Formulae
f) sin (A \ u 0 0 b 1 B) = sinAcosB \ u 0 0 b 1 cosAsinB
a) sin 2A + cos 2A = 1 b)
sek2A = 1 + tan2A
c)
kosek2 A = 1 + kot2 A
m g) cos (A \ u 0 0 b 1 B) = cos AcosB
tanA \u00b1 tanB
h) tan (A \ u 0 0 b 1 B) =
d)
sin2A = 2 sinAcosA
e)
cos 2A = cos2A \u2013 sin2 A = 2 cos2A-1 = 1- 2 sin2A
1 m tanA tanB
sinAsinB
2 Students will be able to: 1. Understand the concept of positive and negative angles measured in degrees and radians. 1.1 Represent in a Cartesian plane, angles greater than 360 or 2 \u03c0 radians for: a) positive angles b) negative angles.
1.1
a) Positive angles are angle measured in the anticlockwise direction from the positive x \u2013axis. b) Negative angle are angle measured in the clockwise direction from the positive x \u2013 axis
\u03b8
\u2212 \u03b8
C) The Position of an angle \ u 0 3 b 8 that is greater than 360 or 2 \u03c0 radians can be obtained using o
o
= n(2\u03c0 = n(360 ) + \u03b1 the relation\u03b8 or \u03b8 ) + \u03b1 o o c) One full rotation = 360 or 2 \u03c0 , so two full rotation = 720 or 4 \u03c0 d) A Cartesian plane can be divided into four quadrant
Quadrant 1
Quadrant 1I
o
180
\u03c0 < \u03b8 < \u2264 \u22 690 4 \u03b8 or \u03c0 o
2
Quadrant III o
1270
o
< \u03b8 < 180
or
3\u03c0 2
< \u03b8 < \u03c0
o
< \u03b8 < 0o
90
or
\u03c0 < \u03b8 < 0o
2
Quadrant 1V o
270
o
\u2264 \u22 6360 4 \u03b8 or
3\u03c0 2
< \u03b8 < 2\u03c0
Sketch the angle for each of the following angle in separate Cartesian planes Hence which quadrant the angle is in . Example 1 o a) 520
Exercise 1 o g) - 135
b) 1050
h) -45
o
o
c)780
o
i)
\u2212 430o
d)
j)
7 2
rad \u03c0
\u03c0 \u2212 rad
4
e)
k)
10 3
rad \u03c0
7
\u03c0 \u2212
2
Homework Text book Page 111 Exercise 5.1 No 1 \u2013 2 Students will be able to: 2.0 Understand and use the six trigonometric functions of any angle 2.1 Define sine, cosine and tangent of any angle in a Cartesian plane. 2.2 Define cotangent, secant and cosecant of any angle in a Cartesian plane. 2.3 Find values of the six trigonometric functions of any angle. 2.4 Solve trigonometric equations.
f)
19\u03c0
l)
\u03c0 \u2212
6
8
3
3 2.1 Define sine, cosine and tangent of any angle in a Cartesian plane. Refer to the following diagram , When \ u 0 3 b 8 lies on the first quadrant as shown in the diagram PQ = y and r =x2
below , OQ = x, \u2206 OPQ,
y
x
r
r
+
2
y . Refer to
sin
= = Then sin\u03b8 , cos\u03b8 ,
x = tan\u03b8 y
\ u 0 3 b 8
cos \
u 0 3 b 8
tan
\ u 0 3 b 8
y
= sin
\u03b1
=
= cos
\u03b1
=
= tan
\u03b1
=
r x r y
=
x
= =
Quadrant
Graphs
Conclusion :
2.2 Define cotangent, secant and cosecant of any angle in a Cartesian plane. Definition : cotangent \
1
u cot 0 3\u03b8 b = 8 =
\u03b8 tan
= secant \u03b8sek =\u03b8
Example 2: o o 1. Given sin 45 = 0.707 and cos45 = 0,707 , Find o o o o the value of tan 45 ,cot45 ,sec45 and cosek45 Solution : o tan 45 = = = cot45 =
=
=
sek45 =
=
=
o
o
cosek45 = o
=
1
\u03b8 cos
2 Given sin
2 3
value of tan Solution : tan
cot
= sec
2
= \u03c0
3 2 3 2 3
= cosecant \u03b8 =ek\u03b8 cos
= \u03c0
2 3
= 0.866 \u03c0 2
, cot \u03c0
3
and cos
,sec \u03c0
2 3
=\u03c0
2
\u03c0 and
3
=
=
=
=
= \u03c0
=
=
2
=
=
cosec
3
= \u03c0
1
\u03b8 sin
0.5 .Find the
cosec
2 3
\u03c0
4
Exercise 2 o o 1 Given sin 15 = 0.259 and cos15 = 0.966 , Find 4 4 o o o o 2 Given sin π =- 0.866 and cos π = - 0.5 Find the the value of tan 15 ,cot15 ,sec15 and cosek15 3 3 Solution : 4 4 4 4 o tan 15 = value of tan π,cot π ,sec π and cosec π Solution o
cot15 =
tan
4 3
3
3
3
3
4
π
=
sec
π
=
cosec
3
=
π
o
sec15 = cot
o
cosek15 =
4 3
4 3
π
=
Complementary angles o Two angle are called complementary angles if the sum of these angles is equal to 90 . Foe example the O o angle 65 is said to be the complement of angle 25 o a) sinθ = cos(90 − θ )
o c) tanθ = cot(90 − θ )
o e) secθ = cosec(90 − θ )
o b) cosθ = sin(90 − θ )
o d) cotθ = tan(90 − θ )
o f) cosecθ = sec(90 − θ )
Example 3 : Given that sin 52 = p and cos52 = q find the value of each of the following trigonometric functions in terms of p and / or q o o o a) sin 38 b) sec 38 c) cot 38 o
o
Exercise 3 : Given that tan 47 = r and cos47 = s find the value of each of the following trigonometric functions in terms of r and / or s o o o a) cot 43 [ r ] b) sin 43 [ s ] c) sec43 [ 1/(rs) ] o
o
Homework Text book Page 122 Exercise 5.2 No 1 – 10
2.3 Find values of the six trigonometric functions of any angle The value of any trigonometric function of an angle is obtained by following the steps below a) Find the reference angle, α ,which is the acute angle form by the rotating ray and the x-axis in the respective quadrant b) Find the value of the trigonometric function of the reference angle, α . c) Determine the correct sign of the value of the trigonometric function of angle θ according to the respective quadrant.
Example 4.: For each of the following trigonometric functions determine the reference angle . Hence ,find the value of trigonometric function . o o 0 o o o a) sin 135 b) cos(-150 ) tan 143 13' cot325 sek340 cosec(-230 12')
Exercise 4.: For each of the following trigonometric functions determine the reference angle . Hence ,find the value of trigonometric function o o o o o o a) sin 290 b)cosec350 c)cot 300 d)sec(-330 ) e)cos(-300 ) f) tan (-200 )
[0.9397]
[-5.760]
Example 5 : Given that sin
θ
without using a calculator a) cos θ
5
= - , 90o 13
[-0.5773] <θ <
b)cosec θ
Exercise 5 : Given that cos
θ
[1.1547]
12
13
[-0.3640]
o
270 Find the value of each the following trigonometric function
c)sec θ
= - , 90o
[0.5]
o < θ < 180
d) tan θ
e) cot θ
. Find the value of each the following trigonometric
function without using a calculator a) cos θ
cosec θ
sec θ
tan θ ,
cot θ
Homework Text book Page 122 Exercise 5.2 No 11 – 13
2.4 :
Solving trigonometric equations. When solving trigonometric equation, we follow the step below S1 : Obtain the reference angle for the angle using calculator S2 : Determine the relevant quadrants in which angle lie S3 : Determine all the possible solutions in the given range of the angles
Example 6 : Solve each of the following trigonometric equation for a) sin θ = 0.6428 b) sin θ = - 0.9421
o
0
≤θ ≤
o
o
360
b) cos θ = 0.4392
o
Exercise e 6 : Solve each of the following trigonometric equation 0 for ≤ θ ≤ 360 : a) cos θ = -0.6428 b) tan θ = 0.5 c) sin θ =-0.7382
[130 ,230 ] o
o
[26 34', 206 34'] o
o
[227 34',312 26'] o
o
o
2
o
Example 7 Example 6 : Solve each of the following trigonometric equation 0 for ≤ θ ≤ 360 : o (a) cos ( θ -25 ) = 0.9848 tan 2 θ = 1 .732 2tan θ = 3 θ Cos = -0.8192 2
[30 ,120 ,210 ,300 ] o
o
o
o
o
o
Exercise 7 : Solve each of the following trigonometric equation 0 for ≤ θ ≤ 360 o b) tan ( θ + 60 ) = -1 1 1 o a) Cos 2 θ = c) tan ( θ -15 ) = 0.8687
[ 75 ,, 240 ] o
[ 30 ,150 ,210 ,330 ] o
o
d) 2 tan3 θ = -1
2
2
o
[ 56 19', 236 19' ]
o
o
o
o o o o o o [ 111o 58'] [51 7',111 7',171 7',231 9',291 9',513 9']
o
o o Example 8 : Solve each of the following trigonometric equation 0 for ≤ θ ≤ 360 o a) sin θ = - cos48 b) 5 cos θ sin θ = cos θ c) 2sin θ = cos θ
[ 222 , 318 ] o
o
[11 32', 168 ,28',90 ,270 ] o
o
o
[ 26 34', 206 34']
o
o
o o ≤ 360 Exercise 8 : Solve each of the following trigonometric equation for ≤θ 0 o o a) cos θ = - tan 42 b) tan 3 θ = cot 15 c) cos θ = sin
o
θ
[ 154 ,318 ]
[ 25 ,85 ,87 ,205 ,159 ,195 ]
[45 ,225 ]
[ 270 ,225 …]
[19 28', 160 32' , 30 , 150 ]
[ 153 26', 333 ,26',45 ]
o
o
o
o
o
o
o
o
o
o o Example 9 : Solve each of the following trigonometric equation 0 for ≤ θ ≤ 360 : 2 a) (1 + sin x)(cos x) = 0 b) 6 sin x + cosec x = 5 c) 2 tanx-1 = cot x
o
o
o
o
o
o
o
o
o
o
o
3
o
Exercise 9 : Solve each of the following trigonometric equation 0 for ≤ θ ≤ 360 2 2 a) 2kos x + 5 cos x - 3 = 0 b) 3sinx = tan x c) 2 tan x + tan x - 3 = 0
o o [ 0 ,70 40',289 20' ] [ 45 ,12' 303 …. [60 ,300 ] Homework Text book Page 123 Exercise 5.2 No 14– 20 Further Practice Text Book Page 124 No 21 - 30 o
o
o
SPM Question 2 1) Given sin x = p/3 where x is a acute 2.Solve the equation 4 tan x = 1 o o angle. Express cot x in terms of p for 90 < x < 360 [ Answer o o o o 26 34’,153 26’,206 43’ , 333 26’ ] 2 9−p [ Answer ] p
o
1.
o
]
Solve the equation 2
6kos (θ
for 0 < o
[ Ans
θ
θ
−
π
3
) − kos(θ
−
π
3
)= 2
< 360o
= 11.8o,108.2o 180o , 300o ]
Students will be able to: 3.0 Understand and use graphs of sine, cosine and tangent functions. 3.1 Draw and sketch graphs of trigonometric functions: a) y = c + a sin bx, b) y = c + a cos bx, c) y = c + a tan bx, where a, b and c are constants and b>0. 3.2 Determine the number of solutions to a trigonometric equation using sketched graphs. 3.3 Solve trigonometric equations using drawn graphs.
Refer to the text book page 124 – 125 to understand and recognise the characteristics of the graph of trigonometric functions Example 10 Using a scale of 2 cm to 0.5 unit on the x-axis and 2 cm to 1 unit on the y – axis, draw the graph
of y = 4 sin Solution x y
π
2
x
0 0
for 0 ≤ x ≤ 4 . Hence find the solution of equation 4 sin 0.5 2.83
1.0 4
1.5 2.83
2.0 0
2.5 -2.83
π
x +
2
3.0 -4
3 2
x −3 = 0 3.5 -2.8
4.0 0
4
Exercise 10 Using a scale of 2 cm to of y = cos 2x + 1 for π
2
(cos
2x
+
π
8
unit
on
the
3 sin 2x +
2
and
2
cm
to
1
unit
on
the
y
–
axis,
0 ≤ x ≤ π . Hence determine the values of x that satisfy the equation
− x for 0 ≤ x ≤ π . 1) π =
Example 11 1. Sketch the graph of y = 3 sin 2x for 0º ≤ x ≤ 360º. Determine the number of solution to the equation
1
x-axis
2. Sketch the graph of y = ׀tan x ׀for 0 ≤ x ≤ 2 π . Determine the number of solution to the equation ׀
x −2 = 0
Exercise 11 1. Sketch the graph of y = 3 cos2x for 0º ≤ x ≤ 360º. Determine the number of solutions to the equation 3 π cos2x – 2x = 0 ( Answer 3 solution)
tan x = ׀
1
3
x +3 = 0
2. Sketch the graph of y = 1-2sin x for 0 ≤ x ≤ 2 π . Hence , draw a suitable straight line on the same axis to find the number of solutions to the equation π − 2π sin x = 3x , for 0 ≤ x ≤ 2 π .State the number of solutions.
Homework Text book Page 130 Exercise 5.3 N0 1 – 10 Students will be able to: 4.0 Understand and use basic identities. . 4.1 Prove basic identities: a) sin2 A + cos2 A = 1 b) 1 + tan2 A = sec2 A c) 1 + cot2 A = cosec2 A
4.2 Prove trigonometric identities using basic identities. 4.3 Solve trigonometric equations using basic identities.
draw
the
5
Basic Identities 1. 2. 3.
sin x + cos x≡ 1 2 2 sec x ≡ 1 + tan x 2 2 cosec x = 1 + cot x 2
4. tan x 5. sec x 6
Guide to proving trigonometric identities
2
≡
cos x 1
≡
cosec x
7. cot x
1. Pecahkan menggunakan gantian rumus no 4 hingga no 7 1. Samakan penyebut 2. faktorkan atau cari identiti iaitu no 1 hingga 3
sin x
≡
cos x ≡
1 sin x
1 tanx
=
kosx
sin x
Example 12 : Prove each of the following trigonometric identities a) kos x - sin x ≡ 1-2sin x 2
2
2
b) cot x cos x
≡
c) sin y + cos y cosec y 2
cosec x –sin x
Exercise 12 : Prove each of the following trigonometric identities 2 2 2 2 a) tan x-sin x≡ tan x sin x b) tan x + cot x ≡ cosec x secx
1
b)
1 + sin y
+
1 1 − sin y
cosec y
≡
2
= 2sek
Homework Text book Page 134 Exercise 5.4 N0 1 – 2 Example 13 : Solve each of the following trigonometric equation 0 < x <360 a) 6 cos x =1 + 2 sec x c) 6 cosec x = 11 - 4 sin x 3 2 b) 2 cosek x = 7 + o
o
tanx
[ 48 11' , 120 ,240 311 49' ] o
o
o
o
[ 21 48',135 ,201 48' 315 ] o
o
o
o
[48 35',131 25' ] o
o
y
Exercise 13 : Solve each of the following trigonometric equation 0 < x <360 o
a) 5 sin x – 2 = 2 cos x
b) 4 cos x – 3 cot x = 0
2
[53.13 180 , 306.87 ] o
o
o
c)tan x +8 = 7 sekx 2
[48.59 90 131.41 ,270 ] o,
o,
6
o
o
[ 48 11' , 60 ,300 , 311 49' ]
o
o
o
o
0
Homework Text book Page 134 Exercise 5.4 N0 3 – 7
SPM Question
a) Solve the equation 6 cos x = 1 + 2 sec x for 3 2 for b) Solve the equation 2cosec x = 7 + o o o o o 0 ≤ x ≤ 360 [ Answer 48 11’,120 ,240 , 311 49’ ] tanx o
0 ≤ x ≤ 360 [ Answer 21 48’ , 135 ,201 48’ ,315 ] o
o
o
Students will be able to: 5. Understand and use addition formulae and double-angle formulae. 5.1 Prove trigonometric identities using addition formulae for sin (A ± B), cos (A ± B) and tan (A ± B). 5.2 Derive double-angle formulae for sin 2A, cos 2A and tan 2A. 5.3 Prove trigonometric identities using addition formulae and/or double-angle formulae. 5.4 Solve trigonometric equations. Addition Formulae and double Angle Formulae Addition Formulae
sin(A ±
B)
sin A cosB cos(A ± B) = cos A cosB m sin A sin B tan(A ± B) =
=
tan A ± tanB 1m tan A tanB
± cos A sin B
Double angle Formulae sin2A = 2sinAcosA 2 2 Cos2A =Cos A-sin A 2 = 1 - 2sin A 2 = 2Cos A -1 2 tanA tan2A = 2 1− tan A
Half-angle formulae. A
A
2
2
sin A = 2sin kos cos A = kos2 =
A
2
2
− sin
= 2 tan
A
2 tan A = 2 A 1− tan 2
A
2
o
Example I4 o Find the value of sin 15 and o tan 165 without using Calculator
7 12
Given sin A =
, cos B = −
13
4
5
where A and B are obtuse angle Without using calculator find the value of b) tan (A + B) a) cos (A - B)
Exercise 14 : a) Find the value each of the following without using Calculator o o a) sin 15 b) tan75 o o c)tan105 d) cos 165
o
o
IF h = cos10 and k = sin40 , Express each of the following in terms h and / or k o o o a) sin 50 b) sin 20 c) cos 5
c) Find the value each of the following without using Calculator 3 12 o o a) 2 cos30 sin30 where A and B are acute angle . 2 a) 1-2sin 22.5 Find the value each of the following without using Calculator 5
b) If tan A =
4
and cot B
a) sin (A -B)
c) tan 2B
b) cos (A+B)
d) tan
A
2
Homework Text book Page 134 Exercise 5.5 N0 1- 7 Example 15 : Prove each of the following trigonometric Identities 2 a) cos 3x = 4 cos x – 3 cos x 1 − cos2x b) sin 2x =
tanx
c) 2 cot 2x + tan x = cot x
8 Exercise 15 : Prove each of the following trigonometric Identities 2 a) sin 3x = 3sin x - 4 sin x b) cot x– cosec 2x = cot 2x
Homework Text book Page 134 Exercise 5.5 N0 8 - 10 o < x < 360o Example 16 : Solve the following equation0for a) 3 sin2x = 2sinx b) 3 kos2x - 7kosx = -5
[ 70 32',180 ,289 28' ] o
o
o
o
tan2x
=
2
2 − sec x
[48 11' , 60 ,300 ,311 49' ] [41 49',138 11',180 ,221 49', 318 11' ]
o
o
o
o
o
o
o
o
o
o
o
o
o
c)3 kos2x + 4 cosx = 1
[ 90 ,194 29',270 ,345 31' ] [63 26',116 34',180 ,243 26' ,296 34' ] Homework Text book Page 134 Exercise 5.5 N0 11 – 15 o
2 tan x
c)tan2x = 10 tan x
o < x < 360o Example 16 : Solve the following equation0for a) cos x + 2 sin2x = 0 b) 3 tan2x + 2 tan x = 0
o
c)
o
[70 32',180 ,289 28' ] o
o
o
o
9
SPM Questions a) Find the value of A and B that satisfy 1 and b) Given sin (x – y) = the equation 2 sin (A -3B)=0.33 and sin (A+B) = 0.91 for 3 o o cos x sin y = . Find the value 0 ≤ ( A − 3B) ≤ 90 ,and 4 o o o each of the following =42 23’ 0 ≤ ( A + 3B) ≤ 90 [ Answer A o a) sin x cos y b) sin (x + y) B= 7 42’ ]
c) Prove that sin 2x ≡ cot x (1 – cos 2x)
d)Prove that 2 cot 2x + tan x
o o 90 ≤ x ≤ 180 . f) Given 3 tan2x = 4 for 2 Find the value of sin x [ Ans 4/ /5]
≡
cot x
e)Show th a t tan x ≡
sin x
− sin 2x
kosx− 1 − kos2x
o 0 ≤
o
o 0 ≤
o
10
x ≤ 360 x ≤ 90 find g) Solve 16kos( x − π ) sin(x − π ) = 5 for h) Givensin x =m for 2 o o o o i) cos2x in term of m [ Ans 1-2m ] [ Answer 19 21’,70 40’,199 21’250 40’]
ii)
the positive value for m if sin 2x =
2m 3
[ Answer 2/2/3 ]
a)Solve the equation 2 2 sek x = 3 – tan x for 0o (b) Given tanθ =
1
3
c) Solve each of the following for b) Prove cos 2 θ = 2cos θ -1 Given θ is a acute angle and 90o ≤ β ≤ 270o [ SPM95 ] sinθ = p express each of the 2 (a) 2 tan β = 1 following in term of p [ SPM 94] (b) 2 - 3 sin β − cos2β = 0 (i) tan θ (ii) cos(- θ ) (iii) cos 2 θ 2
≤
x < 360o
without using
calculator find the value of (i) tan2 θ o (ii) tan (135 - θ ) [SPM 93]
Solve 4 sin (x- π )cos(x- π ) = 1 for 0 ≤ x ≤ 2π Given tan2y = cos y 2
5
for 90
o
12
Given sin θ = k where θ is a ac ute angle fin d i)sin 2θ in term of k ii)the positive value of k if kos2 θ =k [ SPM 98 ]
[SPM 97 ]
Prove that (cos 2 θ + 1)tan θ = sin 2 θ [ SPM 95 ]
Prove that Cosek2A + Cot2A = Cot A [ SPM 93]
11
2
2
Prove that tanθ - cotθ = sek θ 2
2
- cosecθ
[ SPM 98 ]
Show
sin 2θ
+
1 + kosθ
+
sinθ kos2θ
=
tanθ [ SPM 97]
