online user counteronline user counter

MATHMAA

Only Search Your Site

16th KVJ Solutions6

Probability Introduction :

Here you can find the 16th KVJ Solutions6 in detail explanation. Feel Free to send your opinion  by clicking on  SHARE YOU ENORMOUS EFFORT AND SMART EXAMPLES HERE

6Q) If \( cos \alpha + cos \beta + cos \gamma = sin \alpha + sin \beta + sin \gamma = 0 \), Prove that

 \( cos (2\alpha) + cos (2\beta) + cos (2\gamma)\) =  \( sin (2\alpha) + sin (2\beta) + sin (2\gamma) =0\)

Solution:

 Let x= \( cos ( \alpha ) + i sin (\alpha ) \)

      y = \( cos (\beta ) + i sin (\beta )\)

      z= \( cos (\gamma) + i sin (\gamma )\)

 x+y+z = \( cos( \alpha) + cos (\beta) + cos (\gamma) + i ( sin (\alpha) + sin (\beta) + sin (\gamma) \)

 x+y+z = 0 + i 0 =0

 \( (x+y+z)^{2} = 0\)

 \( x^{2}+y^{2}+z^{2} = -2(xy+yz+zx) \)

 \( x^{2} + y^{2} + z^{2} = -2xyz(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}) \) -----------(1)

\( \frac{1}{x}\) = \(\frac{1}{cos (\alpha) + i sin (\alpha)} \)

 \( \frac{1}{x} = cos( \alpha) - i sin( \alpha )\)

Similarly we get

  \( \frac{1}{y} = cos (\beta) - i sin (\beta )\)

  \( \frac{1}{z} = cos (\gamma) - i sin (\gamma )\)

  \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = cos(\alpha)+cos(\beta)+cos(\gamma) - i(sin(\alpha)+sin(\beta)+sin(\gamma)) \)

    =0 - i (0) =0

Therefore from one we get

\( x^{2}+y^{2}+z^{2} =-2xyz(0) \)

\( x^{2} + y^{2} + z^{2} = 0 \) ----------(2)

 Using De Moivre's Theorem we get

 \( x^{2} = ( cos (\alpha) + i sin(\alpha))^{2} \)

               = \( cos (2 \alpha) + i sin (2 \alpha) \)

Similarly we get

 \( y^{2} = cos ( 2\beta ) + i sin (2 \beta) \)

 \( z^{2} = cos (2 \gamma ) + i sin (2 \beta ) \)

 \( x^{2} + y^{2} + z^{2} =0 \)

 \( x^{2} + y^{2} + z^{2} = cos (2 \alpha ) + cos(2\beta)+cos(2\gamma) + i (sin(2\alpha)+sin(2\beta)+sin(2\gamma)) =0 \)

\( cos(2\alpha) +cos(2 \beta ) + cos (2 \gamma) =0 \)

\( sin (2 \alpha) + sin(2\beta) + sin(2 \gamma) = 0 \)

Hence proved .

Please feel free to send you opinions on  16th KVJ Solutions6


<<BACK<<                                QUESTION PAPER                                         >>NEXT>>



SHARE YOU ENORMOUS EFFORT AND SMART EXAMPLES HERE

!! NEED MORE HELP !!

SBI! Case Studies