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Additional formulae sin (A + B) = sin A cos B + sin B cos A 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)

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Presentation on theme: "Additional formulae sin (A + B) = sin A cos B + sin B cos A 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)"— Presentation transcript:

1 Additional formulae sin (A + B) = sin A cos B + sin B cos A 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) = cos 2 A - sin 2 A cos 2A = cos 2 A - sin 2 A cos 2A = 2cos 2 A - 1 cos 2A = 1 – 2sin 2 A

6 Double angle formulae

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

8 Solving equations Solve cos 2A cos A = 0 for 0 x 2 =2 cos 2 A cos A = 0 =2 cos 2 A + 4 cos A + 2= 0 = cos 2 A + 2 cos A + 1 = 0 = (cos A + 1) 2 = 0 = cos A = - 1 A =

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 Complete solution: A= 0.97, 2.27, 4.01, 5.41 c 0, or 2 tan A = 0 A = 0 or or 2 7 – 5tan 2 A = 0 tan A = 7/5 A = 0.97, 2.27, 4.01 or 5.41 c

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 cos x + b sin x can be written in the form R cos( 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 + )

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 = 4 [2] [1] 2 + [2] 2 : R 2 sin 2 x + R 2 cos 2 x = R 2 (sin 2 x + cos 2 x ) = R 2 = = 25 R = 5 [2] [1]: tan = 4/3 = 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 = 12 [1] R sin = 5 [2] [1] 2 + [2] 2 : R 2 cos 2 x + R 2 sin 2 x = R 2 (cos 2 x + sin 2 x ) = R 2 = = 169 R = 13 [2] [1]: tan = 5/12 = 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 [1] R sin = 3 [2] [1] 2 + [2] 2 : R 2 cos 2 x + R 2 sin 2 x = ( 3 ) 2 R 2 (cos 2 x + sin 2 x ) = R 2 = = 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 = 3 [2] R 2 = R = 7.62 [2] [1]: tan = 3/7 = c (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 = c or c


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