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Trigonometric Equations

Trigonometric Equations. When the graph of y = sin a is displaced by b units horizontally, its equation becomes y = sin ( a+b ). Angles such as ( a + b ) are called compound angles .

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Trigonometric Equations

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  1. Trigonometric Equations • When the graph of y = sin a is displaced by b units horizontally,its equation becomes y = sin (a+b) Angles such as (a +b) are called compound angles. Here we will learn how to ‘expand’ trigonometric functions like the above in terms of sin  and cos  The name for such expansions is addition formulae

  2. Formula for sin (+ ) sin (+ ) = sin  cos  + cos  sin  Example a) expand sin (60+  )0 sin 600 cos  + cos 60 sin  sin (60+  )0 = But we know from previous exercises that: sin (60+  )0 =

  3. b) Find the exact value of sin 75º sin 75º = sin ( 30 + 45)º = sin 30º cos 45º + cos 30º sin 45º

  4. sin (a - b) = sin a cos b – cos a sin b Find the exact value of sin 15º sin 15º = sin ( 45 - 30)º = sin 45º cos 30º - cos 45º sin 30º

  5. cos (a ±b) = cos a cos bsin a sin b sin (a ±b) = sin a cos b ± cos a sin b Similarly

  6. cos (a ±b) = cos a cos bsin a sin b

  7. Trigonometric Identities We can use the addition formulae to help us prove complex trigonometric identities. Example Prove that cos ( x + 45)º + cos ( x - 45)º LHS = (cos xºcos 45º - sin xºsin 45º )+(cos xºcos 45º + sin xºsin 45º ) = RHS

  8. b) Prove that LHS

  9. c) Prove that LHS = RHS

  10.  3 3 1 L P N Applications of addition formulae M For the diagram opposite show that MN = LM =

  11. Formulae involving 2 Since 2  can be written as (+ ), we can form equations using the addition formula. These results are referred to as double angle formulae because the angle on the left of the equation (2) is double that on the right of the equation .

  12. a) Given that  is acute with , calculate 5 4  3

  13. Trigonometric Equations Double angle formulae often occur in trigonometric equations. These can be solved by substituting in the expressions given previously. Generally, if you have to choose which version of cos 2 to use remember the following: If cos appears in equation then use cos2 If sinappears in equation then use cos2

  14. a) Solve the equation

  15. b) Solve the equation

  16. The double angle formulae cos2 and cos2 can be rearranged as follows. Formulae for cos2 and sin2

  17. a) Express in terms of

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