Download trigonometry: compound angle identities 07

TRIGONOMETRY: COMPOUND ANGLE IDENTITIES
07 APRIL 2014
Lesson Description
In this lesson, we:

Revise the trigonometry from Grade 11

Introduce compound angle identities

Introduce double angle identities
Summary
After some revision on grade 11 work the compound angle identities will be introduced
Compound Angle Formulae
Double Angle Formulae
Test Yourself
Question 1
Simplify without the use of a calculator:
2
sin (360
A.
o
- x)
_
-1
cos( x  180 0 ). tan x . sin(90 o  x). cos(360 o  x)
sin(180 o  x)
B. 1
C. -2
D. 2
E. none of the above
Question 2
If
   90 0 ; 270 0 , determine 
0
cos (  + 80 ) =
A.
3
2
without the use of a calculator:
sin(300 0 ) cos 45 0
cos 405 0
B.
3
C. 
3
2
D. 1
E. none of the above
For question 3, 4 and 5 WITHOUT the use of a calculator, determine the value of the following:
Question 3
sin 49 0
cos 410
A.
3
2
B.
C. 
3
1
2
D. 1
E. none of the above
D. 1
E. none of the above
D. 1
E. none of the above
Question 4
0
0
0
0
sin 85 cos 65 + cos 85 sin 65
A.
3
2
B.
1
2
3
C.
3
C. 
Question 5
1
(cos 15 0  3 sin 15 0 )
2
A.
2
2
B.
3
2
Question 6
If
13 sin x  5  0 and x  [0;270] , determine without the use of a calculator, the value of sin2x.
A. 
10
26
B.
120
169
C. 
120
169
D. 1
E. none of the above
D. 1
E. none of the above
D. 1
E. none of the above
D. p
p 2  1 E. none of the above
Q7, 8 and 9 is based on the following statement:
If sin 39° = p, determine the following in terms of p:
Question 7
sin 129°
A.
p2 1
B.
1  p2
C.
B.
1  p2
C.
B.
1  p2
C. 2 p. 1  p
p2
Question 8
tan 321°
A.
 p2 1
p
p
1  p2
Question 9
sin 78°
 p2 1
A.
p
2
Improve your Skills
Question 1
Simplify without using a calculator, showing your working details:
tan 60 cos 240 sin 310
sin 270 cos 220 sin 300
(6)
Question 2
Consider the identity:
sin A
1  cos A
2


1  cos A
sin A
sin A
a.) Prove it!
b.) Give all values of
A 0;360 for which it is undefined.
(5)
(2)
Question 3
Given that sin 23  g , express the following in terms of g:
sin 203
b.) cos 23
c.) cos113
d.) tan 203
a.)
(1)
(2)
(2)
(2)
Question 4
If
cos10  t then determine cos340 in terms of t:
(3)
Question 5
Express
cos 2x in terms of sin x
.
(2)
Question 6
Solve the following equation for
  0;180 : cos 4 cos 20  sin 4 sin 20  0.5
Question 7
Consider the following identity:
a.) Prove it!
sin 3x  sin 7 x
 tan 5 x
cos 3x  cos 7 x
 Hint : 3x  5x  2x  and 7 x  5x  2x 
b.) Hence or otherwise solve:
sin 3x  sin 7 x
 1 for x  0;90
cos3x  cos 7 x
(6)
(3)
Question 8
Prove the following identities:
1
sin(45  x)sin(45  x)  cos 2 x
2
a.)
2
1  sin 2 x  cos x 
 1 

sin 2 x
 sin x 
b.)
(4)
(3)