Question

if cosx+cosy+cosz=0=sinx+siny+sinz then the possible value of cos(x-y/2)=k then |2k|=?
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Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

GIVEN

$\sf{ \cos x + \cos y + \cos z = 0 \: \: \: \: and\: }$

$\sf{ \sin x + \sin y + \sin z = 0 \: \: \: \: and\: }$

$\displaystyle \sf{\cos \frac{x - y}{2} \: = k }$

TO DETERMINE

$\sf{ |2k| }$

CALCULATION

$\sf{ \cos x + \cos y + \cos z = 0 \: \: \: \: \: }$

$\implies \: \sf{ \cos z = - ( \cos x + \cos y) \: \: \: \: \: }$

Squaring both sides

$\sf{ { \cos}^{2} z = {\cos }^{2} x +{ \cos}^{2} y\: + 2 \cos x \cos y \: \: \: \: }....(1)$

Again

$\sf{ \sin x + \sin y + \sin z = 0 \: \: \: \:\: }$

$\implies \: \sf{ \sin z = - ( \sin x + \sin y ) \: \: \: \:\: }$

Squaring both sides

$\sf{ { \sin}^{2} z = {\sin }^{2} x +{ \sin}^{2} y\: + 2 \sin x \sin y }$

On addition (1) & (2)

$\sf{ { \cos}^{2} z + \sf{ { \sin}^{2} z }= {\cos }^{2} x +{ \cos}^{2} y\: + {\sin }^{2} x +{ \sin}^{2} y\: + 2 \sin x \sin y + 2 \cos x \cos y }$

$\implies \: \sf{1 = 1 + 1 + 2 \sin x \sin y + 2 \cos x \cos y }$

$\implies \: \sf{ 2 \sin x \sin y + 2 \cos x \cos y = - 1 }$

$\displaystyle \: \implies \: \sf{ \cos x \cos y + \: \sin x \sin y = - \frac{1}{2} }$

$\displaystyle \: \implies \: \sf{ \cos (x - y ) = - \frac{1}{2} }$

$\displaystyle \: \implies \: \sf{ 2{\cos}^{2} \frac{ (x - y ) }{2} - 1= - \frac{1}{2} }$

$\displaystyle \: \implies \: \sf{ 2{\cos}^{2} \frac{ (x - y ) }{2} = 1- \frac{1}{2} }$

$\displaystyle \: \implies \: \sf{ 2{\cos}^{2} \frac{ (x - y ) }{2} = \frac{1}{2} }$

$\displaystyle \: \implies \: \sf{ {\cos}^{2} \frac{ (x - y ) }{2} = \frac{1}{4} }$

$\displaystyle \: \implies \: \sf{ {\cos} \frac{ (x - y ) }{2} = \pm \: \: \frac{1}{2} }$

Comparing with

$\displaystyle \sf{\cos \frac{x - y}{2} \: = k } \: \: \: \: \: \: \: we \: get$

$\displaystyle \sf{k = \pm \: \frac{1}{2} }$

Hence

$\displaystyle \sf{ | \: 2k\: | }$

$= \displaystyle \sf{ | \pm \: 1\: | } \: \: \: ( \because \: \: k \: = \pm \: \frac{1}{2}$

$= \displaystyle \sf{ 1 }$

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LEARN MORE FROM BRAINLY

tan⁻¹(x/y)-tan⁻¹(x-y/x+y)=.......(x/y≥0),Select Proper option from the given options.

(a) π/4

(b) π/3

(c) π/2

(d) π

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