If , , and , then the roots of , where ,…

MATHEMATICS · Single correct · ISC · Mathematics · Sets and Functions · Sets and Functions

Problem

If xsin⁡a+ysin⁡2a+zsin⁡3a=sin⁡4ax\sin a+y\sin2a+z\sin3a=\sin4a, xsin⁡b+ysin⁡2b+zsin⁡3b=sin⁡4bx\sin b+y\sin2b+z\sin3b=\sin4b, and xsin⁡c+ysin⁡2c+zsin⁡3c=sin⁡4cx\sin c+y\sin2c+z\sin3c=\sin4c, then the roots of t3−z2t2−y+24t+z−x8=0t^3-\frac{z}{2}t^2-\frac{y+2}{4}t+\frac{z-x}{8}=0, where a,b,c≠nπa,b,c\ne n\pi, are (1) sin⁡a\sin a, sin⁡b\sin b, sin⁡c\sin c (2) cos⁡a\cos a, cos⁡b\cos b, cos⁡c\cos c (3) sin⁡2a\sin2a, sin⁡2b\sin2b, sin⁡2c\sin2c (4) cos⁡2a\cos2a, cos⁡2b\cos2b, cos⁡2c\cos2c

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