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If a.cosK+b.sinK=c then prove that a.sinK-b.cosK=+- root a^2+b^2-c^2

If a.cosK+b.sinK=c then prove that a.sinK-b.cosK=+- root a^2+b^2-c^2

Grade:12

1 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
7 years ago
Hello student,
Please find the answer to your question below
Givena.cosK+b.sinK=c....................(1)
=(a.cosK+b.sinK)2+(a.sinK-b.cosK)2
=a2cos2k+b2sin2k+2absinkcosk+a2sin2k+b2cos2k-2absinkcosk
=a2(cos2k+sin2k)+b2(sin2K+cos2k)
=a2+b2
So(a.cosK+b.sinK)2+(a.sinK-b.cosK)2=a2+b2
From (1) c2+(asink-bcosk)2=a2+b2
So asink-bcosk=\pm \sqrt{a^{2}+b^{2}-c^{2}}

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