Решите уравнение [tex] | sinx| + |cosx| = \sqrt{2} [/tex]. Created by Migma11. matematika-ru

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July 2022 1 9 Report

Решите уравнение
[tex] | sinx| + |cosx| = \sqrt{2} [/tex]

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nafanya2014

Verified answer

Возводим в квадрат:

sin²x+2|sinx|·|cosx|+cos²x=2, так как sin²x+cos²x=1, то

2|sinx|·|cosx|=1,

$\left \{ {{sinx\cdot cosx \geq 0} \atop {sin2x=1}} \right.$ или $\left \{ {{sinx\cdot cosx$

$\left \{ {sinx\cdot cosx \geq 0} \atop {2x=\frac{\pi }{2} +2\pi k, k \in Z}} \right.$ или $\left \{ {sinx\cdot cosx < 0} \atop {2x=-\frac{\pi }{2} +2\pi n, n \in Z}} \right.$

$\left \{ {sinx\cdot cosx \geq 0} \atop {x=\frac{\pi }{4} +\pi k, k \in Z}} \right.$ или $\left \{ {sinx\cdot cosx < 0} \atop {x=-\frac{\pi }{4} +\pi n, n \in Z}} \right.$

О т в е т. $x=\pm\frac{\pi }{4} +\pi m, m \in Z$

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