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Marius963
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Pe: 2014-04-12T08:43:06+03:00 In: In:

trigonometrie

Sa se rezolve in multimea numerelor reale ecuatia: sin(x+pi/6)+cos(pi/3-x)=1

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  1. PhantomR
    expert (VI)

    Avem \cos \frac{\pi}{3}-x=\sin \frac{\pi}{2}-(\frac{\pi}{3}-x)=\sin \frac{\pi}{2}-\frac{\pi}{3}+x=\sin \frac{\pi}{6}+x=\sin x+\frac{\pi}{6}, deci ecuatia se scrie echivalent

    2\sin (x+\frac{\pi}{6}=1 \Rightarrow \sin (x+\frac{\pi}{6}=\frac{1}{2} \Rightarrow x+\frac{\pi}{6}=(-1)^k\arcsin \frac{1}{2}+k\pi= (-1)^k\frac{\pi}{6}+k\pi \Rightarrow x=[(-1)^k-1]\frac{\pi}{6}+k\pi,k\in\mathbb{Z}.

    NOTA: Am folosit relatia \sin(\frac{\pi}{2}-y)=\cos y (in ordinea inversa). Are loc si \cos \frac{\pi}{2}-y=\sin y si relatii analoage pentru tangenta si cotangenta.

    NOTA: Ideea pentru a realiza acea transformare la inceput se poate vedea mai usor daca scrieti in grade ecuatia: \sin (x+30)+\cos(60-x)=1.

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