Identities: (Sum and Difference for cos)
1. cos (A – B) = cos A cos B + sin A sin B
cos (A + B) = cos A cos B - sin A sin B
Examples:
cos (A + 45o) = cos A cos 45o – sin A sin 45o
cos 30o cos p + sin 30o sin p = cos (30o – p)
Identities: (Sum and Difference for sin)
2. sin (A – B) = sin A cos B – cos A sin B
sin (A + B) = sin A cos B + cos A sin B
Examples:
sin (A + 45o) = sin A cos 45o + cos A sin 45o
sin 30o cos p - cos 30o sin p = sin (30o – p)
Examples - Simplify the following WITHOUT using a calculator:
1. cos (45o + 30o) = cos 45o cos 30o – sin 45o sin 30o
 2  3   2  1 
6 2


  
= 

 

4
 2  2   2  2 
2. sin 15o = sin (45o – 30o) / sin (60o – 45o)
= sin 45o cos 30o – cos 45o sin 30o / sin 60o cos 45o – cos 60o sin 45o
 2  3   2  1 
 2  3   2  1 
6 2
6 2





  
  
= 
/











4
4
 2  2   2  2 
 2  2   2  2 
3. cos 165o =
=
=
=
cos (180o – 15o)
- cos 15o
- [cos (45o – 30o)]
- (cos 45o cos 30o + sin 45o sin 30o)
 2  3   2  1   6  2

 

=  
 2    2  2  
2
4


 

4. sin 100o cos 20o – cos 280o sin 160o
= sin (180o – 80o) cos 20 - cos (360o - 80o) sin (180o – 20o)
= sin 80o cos 20o - cos 80o sin 20o
= sin (80o – 20o)
= sin 60o
3
=
2
5. sin (α – 30o) + sin (α + 30o)
= (sin α cos 30o – cos α sin 30o) + (sin α cos 30o + cos α sin 30o)
= sin α cos 30o – cos α sin 30o + sin α cos 30o + cos α sin 30o
= 2 sin α cos 30o
3
= 2 sin α
2
=
3 sin α
6. sin (-135o) = - sin 45o
2
= 2
7. Given : 5 cos Ө + 3 = 0, where 180o ≤ Ө ≤ 360o and cos β =
12
where β ε [180o ; 360o].
13
Use sketches to determine the value of sin (Ө + β).
cos Ө =
3
5
x
  ;  y = -4
r
cos β =
12
13
x
  ;  y = -5
r
x = 12
x = -3
y = -4
r=5
r = 13
Ө: x = -3 ; y = -4 ; r = 5
β: x = 12 ; y = -5 ; r = 13
 sin (Ɵ + β) = sin Ɵ cos β + cos Ɵ sin β
x
y x
y
=     +    
=
=
=
y = -5
rr
rr
  4   12 
 3  5

  +  

 5   13 
 5   13 
 48 
 15 
  +  
 65 
 65 
 33 
 
 65 
Double – angle Identities:
1. sin 2A = 2 sin A cos A
Examples:
sin 70o = sin 2(35) = 2 sin 35o cos 35o
2 sin 25o cos 25o = sin 2(25o) = sin 50o
Double – angle Identities:
2. cos 2A = cos2 A - sin2 A
= 2 cos2 A – 1
= 1 – 2 sin2 A
Examples:
cos 70o = cos 2(35o) = cos2 35o – sin2 35o
= 2 cos2 35o - 1
= 1 – 2 sin2 35o
Examples - Simplify the following WITHOUT using a calculator:
1. cos2 22,5o – sin2 22,5o = cos 2(22,5o)
= cos 45o
2
=
2
2. If sin 320 = t , determine the value of each, in terms of t:
(2.1) cos 58o = cos (90o – 32o)
= sin 32o
= t
(2.2) sin (-32o) = - sin 32o
= -t
(2.3) cos 64o = cos 2(32o)
= 1 – 2 sin2 32o
= 1 – 2t2
(2.4) sin 64o = sin 2(32o)
= 2 sin32o cos32o
2
 t   1  t 
= 2  
 1   1 
= 2t 1 t 2
t  y
 
1 r
x2 = r2 – y2
= (1)2 – (t)2
= 1 – t2
sin 32o =
1
x = 1 t 2
 x = 1 t 2 ; y = t ; r = 1
t
Exercise
1. Use the sum and difference identities to expand the following:
(1.1) sin (Ө + 30o)
(1.2) cos (50o + 2x)
(1.3) cos (60o – 10a)
(1.4) sin (2β – 15o)
o
o
(1.5) cos (65 – 30 )
(1.6) sin (60o – β)
(1.7) cos (Ө - 23o)
(1.8) sin (17o – 4x)
2. Determine the following, without using a calculator:
(2.1) cos 285o
(2.2) sin (-105o)
cos110o. cos15o
(2.3) cos (-195o)
(2.4)
sin 200
(2.5) cos 15o
(2.6) sin 105o
3. Simplify, without using a calculator:
(3.1) sin 83o cos 52o + cos 83o sin 52o
(3.3) cos 280o cos 20o – sin 80o sin 200o
(3.5) sin 73o cos 13o – cos 73o sin 13o
(3.2) cos 23o cos 37o – sin 23o sin 37o
(3.4) cos 20o cos 320o + sin 140osin 200o
(3.6) cos 20o cos 25o - cos 250o cos 115o
4. Simplify, without using a calculator:
(4.1) cos (β + 60o) – cos (β - 60o)
(4.2) sin (135o – α) – sin (225o – α)
1
1
and tan α =
with both Ɵ and α being acute angles, use sketches to:
7
10
(5.1) determine the value of cos (Ɵ - α)
5
(5.2) prove that sin (Ɵ + α) =
5
5. If sin Ɵ =
6. Use the double-angle identities to expand the following:
(6.1) sin 2β
(6.2) cos 2Ɵ
7. Determine the value of the following using the double-angle identities:
(7.1) 1 – 2 sin2 15o
(7.2) 2 cos2 22,5o – 1
o
o
(7.3) 2 sin 30 cos 30
(7.4) cos2 15o – sin2 15o
x
(7.5) 2 cos2 15o – 1
(7.6) 1 – 2 sin2
2
x
x
(7.7) 2 sin 22,5o cos 22,5o
(7.8) cos2   - sin2  
2
2
8. If sin x = k, determine the value of each in terms of k:
(8.1) cos2 x
(8.2) cos 2x
(8.3) sin 2x
(8.4) sin 3x
9. Evaluate, without using a calculator:
0
o 2
(9.1) (cos 15 – sin 15 )
cos210o. sin 320o
(9.2)
sin  160o tan225o sin 650o

