If-x-is-a-positive-acute-angle-and-sinx-sin-2-x-sin-3-x-1-then-find-minimum-value-of-cot-2-x-

Question Number 216253 by MATHEMATICSAM last updated on 01/Feb/25

If x is a positive acute angle and

\sin x + \sin^{2} x + \sin^{3} x = 1 then find

minimum value of \cot^{2} x .

Commented by Frix last updated on 03/Feb/25

{There}^{'} s exactly 1 solution to the given equation

thus {there}^{'} s exactly one value of \cot^{2} x

\sin x = \frac{- 1 + {(\frac{17}{27} - \frac{\sqrt{33}}{9})}^{\frac{1}{3}} + {(\frac{17}{27} + \frac{\sqrt{33}}{9})}^{\frac{1}{3}}}{3} \approx . 543689

x \approx 32.9351 °

\cot^{2} x = {(2 - \frac{2 \sqrt{33}}{9})}^{\frac{1}{3}} + {(2 + \frac{2 \sqrt{33}}{9})}^{\frac{1}{3}} \approx 2.38298

Answered by AntonCWX last updated on 03/Feb/25

l e t u = s i n (x)

u + u^{2} + u^{3} = 1

u^{3} + u^{2} + u - 1 = 0

B y {L a g r a n g e}^{'} s R e s o l v e n t,

\Rightarrow z^{2} + (2 {(1)}^{3} - 9 (1) (1) + 27 (- 1)) z + {(1^{2} - 3 (1))}^{3} = 0

\Rightarrow z^{2} - 34 z - 8 = 0

\Rightarrow z = 17 \mp 3 \sqrt{33}

u = \frac{- (1) + \sqrt[3]{17 - 3 \sqrt{33}} + \sqrt[3]{17 + 3 \sqrt{33}}}{3} = 0.543689

s i n (x) = 0.543689 \Rightarrow {s i n}^{2} (x) = 0.295598

x = 32.94 °

{c o t}^{2} (x) = 2.38209

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