Download cos 375 cos ( 75 ) sin( 50 )sin230 sin40 cos310 °- - ° - ° °

GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME
MATHEMATICS
GRADE 12
SESSION 4
(LEARNER NOTES)
TOPIC 1: TRIGONOMETRY (A)
Learner Note: Trigonometry is an extremely important and large part of Paper 2. You must
ensure that you master all the basic rules and definitions and be able to apply these rules in
many different types of questions. In this session, you will be concentrating on Grade 12
Trigonometry which involves compound and double angles. These Grade 12 concepts will
be integrated with the trigonometry you studied in Grade 11. Before attempting the typical
exam questions, familiarise yourself with the basics in Section B.
SECTION A: TYPICAL EXAM QUESTIONS
QUESTION 1
Simplify the following without using a calculator:
(a)
tan(60)cos(156)cos 294
sin 492
(7)
(b)
cos2 375  cos2 (75)
sin(50)sin 230  sin 40 cos310
(7)
[14]
QUESTION 2
(a)
Show that cos  60     cos  60      3 sin 
(5)
(b)
Hence, evaluate cos105  cos15 without using a calculator.
(5)
[10]
QUESTION 3
Rewrite cos3 in terms of cos .
[6]
Page 1 of 6
GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME
MATHEMATICS
GRADE 12
SESSION 4
(LEARNER NOTES)
SECTION B: ADDITIONAL CONTENT NOTES
Summary of all Trigonometric Theory
sin  
sin  
y
r
cos  
x
r
tan  
y
x
sin  
90   90  
cos  
cos  
tan  
tan  
( x ; y)
180  

360  
180  
sin  
sin  
cos  
cos  
tan  
tan  
Reduction rules
sin(180  )  sin 
sin(180  )   sin 
sin(360  )   sin 
cos(180  )   cos 
tan(180  )   tan 
sin(90  )  cos 
cos(180  )   cos 
tan(180  )  tan 
sin(90  )  cos 
cos(360  )  cos 
tan(360  )   tan 
cos(90  )  sin 
sin()   sin 
cos(90  )   sin 
cos()  cos 
tan()   tan 
Whenever the angle is greater than 360 , keep subtracting 360 from the angle until you
get an angle in the interval  0 ;360 .
Identities
cos 2   sin 2   1
tan  
sin 
cos 
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GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME
MATHEMATICS
GRADE 12
SESSION 4
(LEARNER NOTES)
Special angles
Triangle A
Triangle B
45
2
60
2
1
1
30
45
1
3
From Triangle A we have:
sin 45 
1
cos 45 
1
2
2
From Triangle B we have:

2
2
sin 30 

2
2
cos30 
3
2
tan 30 
1
1
tan 45   1
1
1
2
and
sin 60 
3
2
and
cos 60 
1
2
and
tan 60 
3
3
 3
1
For the angles  0; 90;180;270;360  the diagram below can be used.
90
y
A(0 ;2)
B(1 ; 3)
C( 2 ; 2)
2
60
2
r2
G(  2 ; 0)
45
D( 3 ; 1)
30
180
E(2 ; 0)
x
0
360
F(0 ;  2)
270
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GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME
MATHEMATICS
GRADE 12
SESSION 4
(LEARNER NOTES)
The following identities are important for tackling Grade 12 Trigonometry:
Compound angle identities
sin(A  B)  sin A cos B  cos A sin B
sin(A  B)  sin A cos B  cos A sin B
cos(A  B)  cos A cos B  sin A sin B
cos(A  B)  cos A cos B  sin A sin B
Double angle identities
cos 2   sin 2   cos 2
sin 2  2sin  cos 
cos 2   sin 2 

cos 2  2 cos 2   1

2
1  2sin 
SECTION C: HOMEWORK
QUESTION 1
Determine the value of the following without using a calculator:
(a)
sin 34 cos10  cos34 sin10
sin12 cos12
(3)
(b)
sin(285)
(5)
(c)
cos 2 15  sin15 cos 75
cos 2 15  sin15 cos15 tan15
(6)
[14]
QUESTION 2
1
cos 2
2
(a)
Prove that sin(45  ).sin(45  ) 
(b)
Hence determine the value of sin 75.sin15
(5)
(3)
[8]
QUESTION 3
Prove that:
sin 4  4sin .cos   8sin 3 .cos 
[4]
Page 4 of 6
GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME
MATHEMATICS
GRADE 12
SESSION 4
(LEARNER NOTES)
SECTION D: SOLUTIONS AND HINTS TO SECTION A
1(a)
tan( 60) cos( 156) cos 294
sin 492
( tan 60)(cos156)(  cos 66)

(sin132)

( 3)( cos 24)(  sin 24)
(sin 48)
( 3)( cos 24)(  sin 24)
2sin 24 cos 24
3

2
cos 2 375  cos 2 (75)
sin(50)sin 230  sin 40 cos 310

1(b)

cos 2 15  cos 2 75
( sin 50)( sin 50)  (sin 40)(cos 50)
cos 2 15  sin 2 15
 2
sin 50  (cos 50)(cos 50)
cos 2 15  sin 2 15
sin 2 50  cos 2 50
cos 2(15)

1
 cos 30








( tan 60)(cos156)
 cos 66
sin 48
 3
 sin 24
2sin 24 cos 24
3

2
(7)
cos2 15
cos 2 75
sin 2 50
cos 2 50
cos30
1
3

2






(7)
3
2
[14]
Page 5 of 6
GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME
MATHEMATICS
2(a)
GRADE 12
SESSION 4
cos  60     cos  60   
(LEARNER NOTES)

cos60.cos   sin 60.sin  
cos60.cos   sin 60.sin  
 cos 60.cos   sin 60.sin    cos 60.cos   sin 60.sin  
 3
 3
1
 1
   .cos   
 .sin     .cos   
 .sin 
2
 2
 2 
 2 
  3 sin 
2(b)
cos105  cos15
 cos  60  45   cos  60  45 
(5)
 cos  60  45 
 cos  60  45 
  3 sin 45
  3 sin 45
 2
  3 

 2 


3
3
2
1

2
  3 sin 
2
2
 6

2
 6
2
(5)
[10]
cos 3
 cos(2  )
 cos 2.cos   sin 2.sin 
 cos(2  )
 cos 2.cos   sin 2.sin 
 (2 cos 2   1).cos   (2sin  cos ).sin 
 2 cos3   cos   2sin 2 .cos 




 2 cos3   cos   2(1  cos 2 ) cos 
2cos 2   1
2sin  cos 
1  cos2 
4cos3   3cos 
(6)
 2 cos3   cos   2(cos   cos3 )
 2 cos3   cos   2 cos   2 cos3 
 4 cos3   3cos 
[6]
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