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

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

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

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

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

Double angle formulae

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

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

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 

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

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 +  )

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  = 4 [2] [1]2 + [2]2 : R2 sin2 x + R2 cos2 x = 32 + 42 R2(sin2 x + cos2 x ) = 32 + 42 R2= 32 + 42 = 25  R = 5 [2]  [1]: tan  = 4/3   = 53.1 3 cos x + 4 sin x = 5 cos( x + 53.1 )

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  = 12 [1] R sin  = 5 [2] [1]2 + [2]2 : R2 cos2 x + R2 sin2 x = 122 + 52 R2(cos2 x + sin2 x ) = 122 + 52 R2= 122 + 52 = 169  R = 13 [2]  [1]: tan  = 5/12   = 22.6 12 cos x + 5 sin x = 13 sin( x + 22.6 )

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 [1] R sin  = 3 [2] [1]2 + [2]2 : R2 cos2 x + R2 sin2 x = 12 + (3 ) 2 R2(cos2 x + sin2 x ) = 12 + 3 R2= 1 + 3 = 4  R = 2 [2]  [1]: tan  = 3   = 60 cos x + 3 sin x = 2 cos( x + 60 )

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  = 3 [2] R2 = 72 + 32  R = 7.62 [2]  [1]: tan  = 3/7   = 0.405c (Radians) 7 sin x + 3 cos x = 7.62 sin( x + 0.405) 7.62 sin( x + 0.405 ) = 6  x + 0.405 = sin-1(6/7.62) x + 0.405 = 0.907 or 2.235 x = 0.502c or 1.830c