trignometric-ratios-and-identities-4

Some nice manipulations:

1. 
   Why ? = cos² sin² 2sin cos
   = 1 2sincos
   = 1 sin²
   
   
   
2. 
   Why ?
   
   =
   
   
3. cos Asin A =
   Why !
   cos A sin A =
   =
   =
   
   Expressing sin in terms of sin A:
   
   =
   =
   The ambiguity of sign is removed from following figure.
   
   
   
   Dumb Question: Where does this diagram come from ?
   Ans:-
   So, if is in I ^st or II^nd quadrent
   then is + ve.
   So, A/2 lie from 2_n for to be + ve
   Similarily for
   
   Illustration 7:
   If cos 25⁰ + sin 25 ⁰ = P , then find value of cos 50⁰ in trems of P ?
   Ans:- cos50⁰ = cos² 25 ⁰ - sin² 25⁰
   = (cos 25⁰ + sin25⁰)(cos 25⁰ - sin25⁰)
   P(cos25⁰ - sin25⁰)
   Also, (cos25⁰ - sin25⁰)² + (cos25⁰ - sin25⁰)² = 1 + 1
   cos25⁰ - sin25⁰ = +
   (+ve sign as cos 25⁰ > sin 25⁰)
   cos50⁰ =
   
   The qreatest and least valume of expression (a sin + b cos)
   
   

• a sin + b cos
  
  
  Why?
  Let a = r cos
  b = r sin so that r =
  So, a sin + b cos = r (sin cos + cossin )
  = r sin ( + )
  Now sin ( + has minimum and maximum value as + 1 and - 1 esppectively.
  So, - r rsin ( + ) r
  So, - a sin + b cos
  
  Illustration 8:
  Find the minimum and maximum value of
  6 sin x cos x + 4cos² 2x ?
  Ans:- 6 sin x cos x + 4cos² 2x = (2sin x cos x ) + 4cos 2x
  = 3 sin 2x + 4cos 2x .
  
• 3sin 2x + 4cos2x
  => Minimum value of 6sin x cos x + 4 cos 2x is - 5
  and maximum value is 5.
  
  Sum of sine and cosine senies when angles are in AP .
  (1) sin + sin =
  Why ?
  
  2sin
  2sin
  2sin
  By adding these n lines we have.
  2sin
  = 2sin
  S =
  
  (2)
  =
  
  Illustration 9:
  Find sum of sin -…………+ 0 n terms. Ans:- Now, sin
  sin
  sin
  ………………………………………………..
  Hence the series is
  sin …………
  =
  =