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Grade 12Trigonometry

what are the general solutions for x in 1) 1 + sin^3 x + cos^3 x = 3/2(sin2x) 2) root 3 cos x - 3sin x = 4sin 2x cos 3x 3) 3root3 sin^3 x + cos^3 x + 3root3 sin x + cos x = 1 (pg 9 from book trigonometry equation of bansal classes)

Sushmin Saha

12 Years agoGrade 12

1 Answer

Arun Kumar

12 Years ago

Hi

I will solve only first question.

$\\1 + sin^3 x + cos^3 x = 3/2(sin2x) \\1 + (sinx + cosx)(sin^2x - sinx.cosx + cos^2x) = 3sinx.cosx \\1 + (sinx + cosx)(1 - sinx.cosx) = 3sinx.cosx \\s=(sinx + cosx) \\s^2=1+2sinxcosx \\1+s(1-(1-s^2)/2)=3(1-s^2)/2 \\s^3 + 3s^2 - 3s - 5 = 0 \\(s + 1)(s^2 + 2s - 5) = 0 \\but\, range \,of\, s \,is\, [-\sqrt2,\sqrt2] \\s=-1 \\s=\sqrt2(sin(x+\pi/4))=-1 \\=>sin(x+\pi/4)=-1/\sqrt2 \\=>x+\pi/4=-\pi/4+n(-1)^n\pi$

Thanks & Regards, Arun Kumar, Btech,IIT Delhi, Askiitians Faculty

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