Question

Fie x ∈ ( π/2 , π) și cos 2x = -1/2.
Se cer cos x și sin x.

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Answer 1

Explicație pas cu pas:

$x\in ( \frac{\pi}{2} ; \pi)$

cadranul II, semnul: sinx +, cosx -

$\cos(2x) = 2 \cos^{2} (x) - 1$

$\iff 2 \cos^{2} (x) - 1 = - \frac{1}{2} \\ 2 \cos^{2} (x) = \frac{1}{2} \iff \cos^{2} (x) = \frac{1}{4} \\ \cos(x) = - \frac{1}{2};\cos(x) = \frac{1}{2}$

x in cadran II

$\implies \cos(x) = - \frac{1}{2}$

$\cos(2x) = \cos^{2} (x) - \sin^{2} (x)$

$- \frac{1}{2} = \frac{1}{4} - \sin^{2} (x) \iff \sin^{2} (x) = \frac{1}{2} + \frac{1}{4} = \frac{3}{4}$

$\sin(x) = - \frac{ \sqrt{3} }{2}, \sin(x) = \frac{ \sqrt{3} }{2}$

x in cadran II

[tax]\implies \sin(x) =  \frac{ \sqrt{3} }{2}[tex] \\