MATHEMATICS Grade 12
Western Cape Education Department
Telematics Learning Resource 2019
MATHEMATICS Grade 12
Telematics Mathematics Grade 12 Resources
2
February to October 2019
Dear Grade 12 Learner
In 2019 there will be 8 Telematics sessions on grade 12 content and 6 Telematics sessions on grade 11 content. In grade 12 in the June, September and end of year examination the grade 11 content will be assessed. It is thus important that you compile a study timetable which will consider the revision of the grade 11 content. The program in this book reflects the dates and times for all grade 12 and grade 11 sessions. It is highly recommended that you attend both the grade 12 and 11 Telematics sessions, this will support you with the revision of grade 11 work. This workbook however will only have the material for the grade 12 Telematics sessions. The grade 11 material you will be able to download from the Telematics website. Please make sure that you bring this workbook along to each and every Telematics session.
In the grade 12 examination Trigonometry will be + 50 marks and the Geometry + 40 marks of the 150 marks of Paper 2.
Your teacher should indicate to you exactly which theorems you have to study for examination purposes. There are altogether 6 proofs of theorems you must know because it could be examined. These theorems are also marked with (**) in this Telematics workbook, 4 are grade 11 theorems and 2 are grade 12 theorems. At school you should receive a book called "Grade 12 Tips for Success". In it you will have a breakdown of the weighting of the various Topics in Mathematics. Ensure that you download a QR reader, this will enable you the scan the various QR codes.
At the start of each lesson, the presenters will provide you with a summary of the important concepts and together with you will work though the activities. You are encouraged to come prepared, have a pen and enough paper (ideally a hard cover exercise book) and your scientific calculator with you.
You are also encouraged to participate fully in each lesson by asking questions and working out the exercises, and where you are asked to do so, sms or e-mail your answers to the studio.
Remember:" Success is not an event, it is the result of regular and consistent hard work".
GOODLUCK, Wishing you all the success you deserve!
Telematics Mathematics Grade 12 Resources
3
February to October 2019
2019 Mathematics Telematics Program
Day
Date
Term 1: 9 Jan ? 15 March
Tuesday
12 February
Wednesday 13 February
Time
15:00 ? 16:00 15:00 ? 16:00
Grade
12 12
TERM 2: 2 April to 14 June
Monday
8 April
15:00 ? 16:00 12
Tuesday
9 April
15:00 ? 16:00 12
Wednesday 15 May
15:00 ? 16:00 11
Thursday
16 May
15:00 ? 16:00 11
Wednesday 22 May
15:00 ? 16:00 12
Thursday
23 May
15:00 ? 16:00 12
Term 3: 9 July ? 20 September
Monday
29 July
15:00 ? 16:00 12
Tuesday
30 July
Wednesday 07 August
15:00 ? 16:00 12 15:00 ? 16:00 11
Monday
12 August
15:00 ? 16:00 11
Term 4: 1 October ? 4 December
Tuesday
15 October
15:00 ? 16:00 11
Wednesday 16 October
15:00 ? 16:00 11
Subject
Mathematics Wiskunde
Mathematics Wiskunde Mathematics Wiskunde Mathematics Wiskunde
Mathematics Wiskunde Mathematics Wiskunde
Mathematics Wiskunde
Topic
Trigonometry Revision Trigonometrie Hersiening
Trigonometry Trigonometrie Geometry Meetkunde Geometry Meetkunde
Differential Calculus Differentiaalrekening Functions Funksies
Paper 1 Revision Paper 2 Revision
Telematics Mathematics Grade 12 Resources
4
February to October 2019
Session 1: Trigonometry
x Definitions of trigonometric ratios:
o In a right-angled '
SinT opposite hypotenuse
CosT adjacent hypotenuse
TanT opposite adjacent
hypotenuse
opposit
T
adjacent
o On a Cartesian Plane
SinT y
y
r
CosT x r
TanT y x
ry
T
x
x
x Special Angles
o
0?, 90?, 180?, 270?, 360? can be
obtained from the following unit circle
.
T 90q
y
(0 ; 1)
T 180q (-1 ; 0)
T 0q
(1 ; 0) x T 360q
r, the radius is 1 since it is a unit circle
30?, 45? and 60? can be obtained from the following
30q
2
3
60q 1
45q
2
1
45q 1
(0 ; -1)
T 270q
x The "CAST" rule enables you to obtain the
sign of the trigonometric ratios in any of the
four quadrants.
S Sine is +ve in the 2nd quadrant
y
180q- T
T =30q
sin 30q
=
1 2
cos 30q =
3 2
tan 30q = 1
3
T
T Tan is +ve in the3rd quadrant
SinSe
All
x
Tan
Cos
+
180q+T
360q-T
T = 45q sin 45q = 1
2
cos 45q = 1
2
tan 45q = 1
T = 60q
sin 60q =
3 2
cos 60q =
1 2
tan 60q = 3
A - ALL trig ratios are +ve in the first quadrant
C Cos is +ve in the 4nd quadrant
The trigonometric function of angles (180q?T) or (360q?T) or (-T)
becomes
?
Trigonometric function of T
The sign is determined by the "CAST" rule.
(180q T ) sin(180q T) sinT cos(180q T ) cosT tan(180q T) tanT
(180q T ) sin(180q T ) sinT cos(180q T ) cosT tan(180q T) tanT
(360q T ) sin(360q T ) sinT cos(360q T ) cosT tan(360q T ) tanT
(360q T ) sin(360q T) sinT cos(360q T ) cosT tan(360q T ) tanT
(T ) sin(T ) sinT cos(T ) cosT tan(T ) tanT
Telematics Mathematics Grade 12 Resources
5
February to October 2019
x TRIGONOMETRIC IDENTITIES
tan T
sin T cosT
(cosT z 0)
sin 2 T cos2 T 1 ,
sin2 T 1 cos2 T ,
cos2 T 1 sin2 T
x Co-functions or Co-ratios
sin(90q T ) cosT cos(90q T ) sin T
r 90q-T y
sin(90q T) cosT cos(90q T) sinT
x Trigonometric Equations
1. Determine the Reference angle
2. Establish in which two quadrants is.
3. Calculate in the interval
[0q; 360q]
4. Write down the general solution
sinT 0,707 Reference = sin1(0,707) = 45q
? = 45q or = 180q - 45q
? = 45q or = 135q
? = 45q+ k360? or = 135q + k360? where k =
T x
cosT 0,866 Reference = cos1(0,866) = 30q
tanT 1 Reference = tan 1(1) = 45q
? = 180q- 30q or = 180q + 30q ? = 180q - 45q
? = 150q or = 210q ? = r150q ? = r150+ k360? where k =
? = 135q ? = 135q+ k180? & k =
TRIGONOMETRIC GRAPHS
Equation
Shape a > 0
Sine Function
Cosine Function
Tangent Function
a < 0
Amplitude
a
a
Period
SOLUTIONS OF TRIANGLES
x Area Rule
Area of 'ABC = 1 absin C = 1 acsin B = 1 bcsin A
2
2
2
x Sine Rule
sin A = sin B = sin C Or a = b = c
a
b
c
sin A sin B sin C
x Cosine Rule
a2 b2 c2 2bc cos A
or
cos A b 2 c 2 a 2
2bc
Note: "c" refers to the side of the triangle opposite to angle C that is the side
A
c
b
B
a
C
Telematics Mathematics Grade 12 Resources
6
February to October 2019
TRIGONOMETRY SUMMARY
Question Summary of
type
procedure
Example question
1. Calculate the value of a trig expression without using a calculator.
Establish whether you need a rough sketch or special triangles, ASTC rules or compound angles.
1.1 If 13cosD
5 and tan E
3 , 4
D [0q;
270q]
and
E [0q;
180q] .
It is given that sin(D E) sinD.cos E sin E.cosD
Determine, without using a calculator,
a) sinD
b) sin(D E ) .
1.2
Calculate: a)
cos(210$ ).sin2 405$.cos14$ tan 120 $. sin104 $
b) sin 70q cos 40q cos 70qsin 40q
2. If a trig ratio is given as a variable express another trig ratio in terms of the same variable. 3. Simplify a trigonometric al expression.
4. Prove a given identity.
5. Solve a trig equation.
Draw a rough sketch with given angle and label 2 of the sides. The 3rd side can then be determined using Pythagoras. Express each of the angles in question in terms of the angle in the rough sketch. Use the ASTC rule to simplify the given expression if possible. See if any of the identities can be used to simplify it, if not see if it can be factorized. Check again if any identity can be used. This includes using the compound and double angle identities.
Simplify the one side of the equation using reduction formulae and identities until .
Find the reference angle by ignoring the "-"sign and finding sin 1 (0,435)
Write down the two solutions in the interval
2. If sin 27q q , express each of the following in terms of q. a) sin117q b) cos(27q)
3. Simplify: cos ( 720q x) . sin ( 360q x) . tan ( x 180q)
a) sin ( x) . cos (90q x)
b)
sin ( 90q x) . tan ( 360q x)
sin (180q x) . cos (90q x) cos(540q x).cos(x)
sin 2 x cos x cos3 x c)
cos x
sin2 x cos x d)
1 cos2 x
Prove that
a)
tan x . cos3 x
1 sin x
1 sin2 x cos2 x 2
b)
Solve for x [180q; 360q] a) sin x 0,435 b) cos 2x 0,435 c) tan 1 x 1 0,435 2
x [0q; 360q] . Then
write down the general solution of this eq. From the general solution you can determine the solution for the specified interval by using various values of k.
Telematics Mathematics Grade 12 Resources
7
February to October 2019
Question type Summary of procedure
Example question
6. Sketch a trig graph.
1st sketch the trig graph without the vertical or horizontal transformation. Then shift the graph in this case 1 unit up.
7.Find the area of a triangle.
8. Finding an unknown side or angle in a triangle.
If it is a right-angled triangle then
area 1 baseu height , otherwise use 2
the area rule Area of 'ABC = 1 ab sin C
2 Draw a rough sketch with the given information. If it is not a right-angled triangle you will use either the sine or cosine rule.
Sketch b) y 2 cos 3x 1 for x [90q; 120q] c) y sin(x 60) for x [240q; 120q]
'ABC, with B 104,5q , AB 6cm and BC 9cm . Calculate, correct to one decimal place area 'ABC
a) 'ABC, with B 104,5q , AB 6cm and BC 9cm . Calculate the length of AC.
b) 'ABC, with C 43,2q , AB 4,5cm and BC 5,7cm . Calculate the size of A .
Calculate the period
Write down the amplitude if it is a sine or cosine graph.
SKETCHING TRIG GRAPH
Identify the shape of the graph and draw Now do the vertical or horizontal
a sine, cosine or tan graph with
transformation if required.
determined period and amplitude. Label
the other x-intercepts. Repeat this pattern
over the specified domain.
SKETCH y 2 cos 3x 1 for x [90q; 120q]
Period = Amplitude =
360q
2 120q
3
y 2
1
-90 -60 -30 -1
x 30 60 90
-2
y 3
2
1
-60
-30
-1
x
30
60
90
Telematics Mathematics Grade 12 Resources
8
February to October 2019
QUESTION
.1
In the figure below, the point P(?5 ; b) is plotted on the Cartesian plane.
OP = 13 units and RO^ P D .
y
P(?5x; b)
13
O
xx R
Without using a calculator, determine the value of the following:
.1.1
cosD
(1)
.1.2
tan(180q D )
(3)
.2
Consider:
sin(T 360q) sin(90q T ) tan(T ) cos(90q T )
.2.1
Simplify
sin(T
360q) sin(90q T ) tan(T ) cos(90q T )
to
a
single
trigonometric
ratio.
(5)
.2.2
Hence, or otherwise, without using a calculator, solve for T if 0q d T d 360q :
sin(T 360q) sin(90q T ) tan(T ) 0,5
(3)
cos(90q T )
.3
.3.1
Prove that 8 4
4.
sin 2 A 1 cos A 1 cos A
(5)
.3.2
For which value(s) of A in the interval 0q d A d 360q is the identity in
QUESTION 5.3.1 undefined?
(3)
.4
Determine the general solution of 8cos2 x 2 cos x 1 0 .
(6)
[26]
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