Dominik55325viz
Rozwiązane

Kąt α jest ostry oraz cosα=[tex]\frac{\sqrt{3}}{3}[/tex]. Oblicz wartość wyrażenia: [tex]\frac{sin\alpha }{cos\alpha } +\frac{cos\alpha }{1+sin\alpha }[/tex].

Odpowiedź :

Konrad509viz

[tex]\dfrac{\sin \alpha}{\cos \alpha}+\dfrac{\cos \alpha}{1+\sin \alpha}=\\\\\dfrac{\sin \alpha(1+\sin \alpha)}{\cos \alpha(1+\sin \alpha)}+\dfrac{\cos^2 \alpha}{\cos \alpha(1+\sin \alpha)}=\\\\\dfrac{\sin \alpha+\sin^2 \alpha+\cos^2 \alpha}{\cos \alpha(1+\sin \alpha)}=\\\\\dfrac{\sin \alpha+1}{\cos \alpha(1+\sin \alpha)}=\\\\\dfrac{1}{\cos \alpha}\\\\\\\dfrac{1}{\cos \alpha}=\dfrac{1}{\dfrac{\sqrt3}{3}}=\dfrac{3}{\sqrt3}=\dfrac{3\sqrt3}{3}=\sqrt3[/tex]

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