find-the-number-of-solutions-of-6-cos-x-7sin-2-x-cos-x-0-

Question Number 144329 by gsk2684 last updated on 24/Jun/21

find the number of solutions

of \sqrt{6 - \cos x + {7 \sin}^{2} x} + \cos x = 0

Commented by MJS_new last updated on 25/Jun/21

let c = \cos x \Rightarrow - 1 ⩽ c ⩽ 1

\sqrt{13 - c - 7 c^{2}} + c = 0 \Rightarrow c < 0

\sqrt{13 - c - 7 c^{2}} = - c

13 - c - 7 c^{2} = c^{2}

8 c^{2} + c - 13 = 0 \land c < 0 \Rightarrow c = - \frac{1 + \sqrt{417}}{16} < - 1

\Rightarrow no solution

[- 1 + \sqrt{7} ⩽ \sqrt{6 - \cos x + {7 \sin}^{2} x} + \cos x ⩽ \frac{- 1 + 2 \sqrt{730}}{14}]

Commented by gsk2684 last updated on 25/Jun/21

thank you mr MJS

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