Question

(FAAP) Resolver a equação:
$\frac{ \sqrt{3} }{2} \sin(x) + \frac{1}{2} \cos(x) = \frac{ \sqrt{3} }{2}$
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Answer 1

$\frac{\sqrt{3}}{2}sen(x) + \frac{1}{2}cos(x) = \frac{\sqrt{3}}{2} \\ \\ \sqrt{3}sen(x) + cos(x) = \sqrt{3} \\ cos(x) = \sqrt{3}[1 - sen(x)] \\ \\ sen^2(x) + cos^2(x) = 1 \\ sen^2(x) + [\sqrt{3}(1 - sen(x))]^2 = 1 \\ sen^2(x) + [9(1 - 2sen(x) + sen^2(x)] = 1  \\ \\ sen^2(x) + 9 - 18sen(x) + 9sen^2(x) = 1 \\ 10sen^2(x) - 18sen(x) + 9 - 1 = 0 \\ 10sen^2(x) - 18sen(x) + 8 = 0 \\ 5sen^2(x) -9sen(x) + 4 = 0 \\ \\ sen(x)_1 + sen(x)_2 = \frac{9}{5} \\ \\ sen(x)_1.sen(x)_2 = \frac{4}{5}$

$sen(x) = 1 \\ \\ ou \\ \\ sen(x) = \frac{4}{5} \\ \\ logo \\ x = \frac{\pi}{2} \\ ou \\ x = 90^o \\ \\ ou \\ \\ x = arcsen \frac{4}{5}$