If-x-asin2t-bcos2t-prove-that-dx-dt-2-a-2-b-2-x-2-Solution-

Question Number 141785 by 7770 last updated on 23/May/21

If x = asin 2 t + bcos 2 t, prove that

\frac{dx}{dt} = 2 \sqrt{a^{2} + b^{2} - x^{2}}

Solution \dots .

Answered by iloveisrael last updated on 23/May/21

\frac{d x}{d t} = 2 a \cos 2 t - 2 b \sin 2 t

\frac{d x}{d t} = 2 \sqrt{{(a \cos 2 t - b \sin 2 t)}^{2}}

\frac{d x}{d t} = 2 \sqrt{a^{2} \cos^{2} 2 t + b^{2} \sin^{2} 2 t - a b \sin 4 t}

\frac{d x}{d t} = 2 \sqrt{a^{2} (1 - \sin^{2} 2 t) + b^{2} (1 - \cos^{2} 2 t) - a b \sin 4 t}

\frac{d x}{d t} = 2 \sqrt{a^{2} + b^{2} - (a^{2} \sin^{2} 2 t + b^{2} \cos^{2} 2 t + a b \sin 4 t}

\frac{d x}{d t} = 2 \sqrt{a^{2} + b^{2} - x^{2}}

(1) x^{2} = a^{2} \sin^{2} 2 t + b^{2} \cos^{2} 2 t + a b \sin 4 t