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Question

If $x sin space a plus y sin space 2 a plus z sin space 3 a equals sin space 4 a x sin space b plus y sin space 2 b plus z sin space 3 b equals sin space 4 b comma x sin space c plus y sin space 2 c plus z sin space 3 c equals sin space 4 c$ Then the roots of the equation $t cubed minus open parentheses z over 2 close parentheses t squared minus open parentheses fraction numerator y plus z over denominator 4 end fraction close parentheses t plus open parentheses fraction numerator z minus x over denominator 8 end fraction close parentheses equals 0 comma a comma b comma c not equal to n pi$ are

$\sin a, \sin b, \sin c$
$\cos a, \cos b, \cos c$
$\sin 2 a, \sin 2 b, \sin 2 c$
$\cos 2 a, \cos 2 b, \cos 2 c$

hint Hint:

In this question, we have to find the roots of the given equation. we will assume a, b, c to be the roots $x sin space theta plus y sin space 2 theta plus z sin space 3 theta equals sin space 4 theta$ . Later, we will solve the equation using some trigonometric identity and find the required roots.

The correct answer is: $cos space a comma cos space b comma cos space c$

$x sin space a plus y sin space 2 a plus z sin space 3 a equals sin space 4 a x sin space b plus y sin space 2 b plus z sin space 3 b equals sin space 4 b x sin space c plus y sin space 2 c plus z sin space 3 c equals sin space 4 c L e t space a comma space b comma space c space a r e space t h e space r o o t s space o f x sin space theta plus y sin space 2 theta plus z sin space 3 theta equals sin space 4 theta rightwards double arrow x space sin space theta plus y left parenthesis 2 sin space theta space cos space theta right parenthesis plus z left parenthesis 3 sin space theta plus 4 space sin cubed theta space right parenthesis equals open parentheses 4 space sin space theta. cos space theta space cos space 2 theta close parentheses rightwards double arrow sin space theta open parentheses x plus 2 y. cos space theta plus z left parenthesis 3 minus sin squared theta right parenthesis close parentheses equals sin space theta left parenthesis 4 cos space theta left parenthesis 2 cos squared minus 1 right parenthesis right parenthesis rightwards double arrow sin space theta open parentheses x plus 2 y. cos space theta plus z left parenthesis 3 minus sin squared theta right parenthesis minus 8 cos cubed theta plus 4 space cos space theta close parentheses equals 0 rightwards double arrow sin space theta open parentheses negative 8 sin cubed theta plus 4 z space cos squared theta plus left parenthesis 2 y plus 4 right parenthesis cos space theta minus 4 z plus x plus 3 z close parentheses equals 0 rightwards double arrow sin space theta open parentheses cos cubed theta minus z over 2 cos squared theta minus fraction numerator left parenthesis y plus 2 right parenthesis over denominator 4 end fraction cos theta minus fraction numerator left parenthesis z minus x right parenthesis over denominator 8 end fraction close parentheses equals 0 rightwards double arrow e i t h e r comma space sin space theta equals 0 space o r space cos cubed theta minus z over 2 cos squared theta minus fraction numerator left parenthesis y plus 2 right parenthesis over denominator 4 end fraction cos theta minus fraction numerator left parenthesis z minus x right parenthesis over denominator 8 end fraction equals 0 space t h e n comma space cos space a comma space cos space b comma space cos space c space a r e space t h e space s o l u t i o n space o f space t cubed minus z over 2 t squared minus fraction numerator left parenthesis y plus 2 right parenthesis over denominator 4 end fraction t minus fraction numerator left parenthesis z minus x right parenthesis over denominator 8 end fraction comma space left parenthesis a comma space b comma space c not equal to n straight pi right parenthesis$

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