1. Additional formulae sin (A + B) = sin A cos B + sin B cos A
cos (A + B) = cos A cos B - sin A sin B
cos (A - B) = cos A cos B + sin A sin B

2. Examples Find the exact value of sin 75
sin (A + B) = sin A cos B + sin B cos A
sin ( ) = sin 30 cos 45 + sin 45 cos 30

3. Examples Express cos (x + /3) in terms of cos x and sin x
cos (A + B) = cos A cos B - sin A sin B
cos (x + /3) = cos x cos /3 - sin /3 sin x

4. Examples
L.H.S.
= R.H.S.

5. Double angle formulae sin (A + B) = sin A cos B + sin B cos A
sin (A + A) = sin A cos A + sin A cos A
sin 2A = 2 sin A cos A
cos (A + B) = cos A cos B - sin A sin B
cos (A + A) = cos A cos A- sin A sin A
cos (A + A) = cos2A - sin2A
cos 2A = cos2A - sin2A
cos 2A = 2cos2A - 1
cos 2A = 1 – 2sin2A

6. Double angle formulae

7. Examples Given that cos A = 2/3, find the exact value of cos 2A.
cos 2A = 2cos2A - 1
Given that sin A = ¼ , find the exact value of sin 2A.
sin 2A = 2 sin A cos A A 4 1
15

8. Solving equations Solve cos 2A + 3 + 4 cos A = 0 for 0  x  2

9. Solving equations Solve sin 2A = sin A for -   x  
=2sin A cos A = sin A
=2 sin A cos A – sin A = 0
= sin A(2 cos A – 1) = 0
 sin A = 0 or cos A = ½
sin A = 0  A = -  or 0 or 
cos A = ½  A = - /3 or /3
Complete solution: A = -  or - /3 or 0 or /3 or 

10. Solving equations Solve tan 2A + 5 tan A = 0 for 0 x  2
tan A = 0  A = 0 or  or 2
7 – 5tan2 A = 0 tan A =  7/5  A = 0.97 , 2.27, 4.01 or 5.41c
Complete solution: A= 0.97 , 2.27, 4.01, 5.41c 0,  or 2

11. Harmonic form If a and b are positive
a sin x + b cos x can be written in the form R sin( x +  )
a sin x - b cos x can be written in the form R sin( x -  )
a cos x + b sin x can be written in the form R cos( x -  )
a cos x - b sin x can be written in the form R cos( x +  )

12. Examples Express 3 cos x + 4 sin x in the form R cos( x -  )
R cos( x -  ) = R cos x cos  + R sin x sin 
3 cos x + 4 sin x = R cos x cos  + R sin x sin 
R cos  = 3 [1] R sin  = [2]
[1]2 + [2]2 : R2 sin2 x + R2 cos2 x =
R2(sin2 x + cos2 x ) =
R2= = 25  R = 5
[2]  [1]: tan  = 4/3   = 53.1
3 cos x + 4 sin x = 5 cos( x  )

13. Examples Express 12 cos x + 5 sin x in the form R sin( x +  )
R sin( x +  ) = R sin x cos  + R cos x sin 
12 cos x + 5 sin x = R sin x cos  + R cos x sin 
R cos  = [1] R sin  = [2]
[1]2 + [2]2 : R2 cos2 x + R2 sin2 x =
R2(cos2 x + sin2 x ) =
R2= = 169  R = 13
[2]  [1]: tan  = 5/12   = 22.6
12 cos x + 5 sin x = 13 sin( x  )

14. Examples Express cos x - 3 sin x in the form R cos( x +  )
R cos( x +  ) = R cos x cos  - R sin x sin 
cos x - 3 sin x = R cos x cos  - R sin x sin 
R cos  = [1] R sin  =  [2]
[1]2 + [2]2 : R2 cos2 x + R2 sin2 x = 12 + (3 ) 2
R2(cos2 x + sin2 x ) =
R2= = 4  R = 2
[2]  [1]: tan  = 3   = 60
cos x + 3 sin x = 2 cos( x + 60 )

15. Solving equations Solve 7 sin x + 3 cos x = 6 for 0 x  2
R sin( x +  ) = R sin x cos  + R cos x sin 
7 sin x + 3 cos x = R sin x cos  + R cos x sin 
R cos  = 7 [1] R sin  = [2]
R2 =  R = 7.62
[2]  [1]: tan  = 3/7   = 0.405c (Radians)
7 sin x + 3 cos x = 7.62 sin( x )
7.62 sin( x ) = 6  x = sin-1(6/7.62)
x = or
x = 0.502c or c