1. oblicz bez uzycia tablic: \(\displaystyle{ cos24^{o}+cos48^{o}-cos84^{o}-cos12^{o}}\).
2. wykaz, ze: \(\displaystyle{ \frac{sin\frac{\pi}{6}\cdot cos\frac{5\pi}{6}\cdot cos\frac{2\pi}{3}}{sin(-\frac{2\pi}{3})} = -\frac{1}{4}}\)
3. sprawdz tozsamosci:
a) \(\displaystyle{ \frac{sin + sin3 }{cos + cos3 }=tg2 }\)
b) \(\displaystyle{ \frac{tg2 tg }{tg2 - tg }=sin2 }\)
obliczenie bez tablic, tozsamosci trygonometryczne
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Justka
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obliczenie bez tablic, tozsamosci trygonometryczne
3 b)
\(\displaystyle{ P=\frac{tg2\alpha tg\alpha}{tg2\alpha - tg\alpha}=\frac{\frac{sin2\alpha}{cos2\alpha}\cdot \frac{sin\alpha}{cos\alpha}}{\frac{sin2\alpha}{cos2\alpha}-\frac{sin\alpha}{cos\alpha}}=\frac{\frac{2sin^2\alpha}{cos^2\alpha-sin^2\alpha}}{\frac{2sin\alpha cos^2\alpha-sin\alpha(cos^2-sin^2\alpha)}{(cos^2\alpha-sin^2\alpha)cos\alpha}}=\frac{2sin^2\alpha cos\alpha}{sin\alpha(2cos^2\alpha-cos^2\alpha+sin^2\alpha)}=\frac{2sin^2\alpha cos\alpha}{sin\alpha}=2sin\alpha cos\alpha=sin2\alpha=L}\)
\(\displaystyle{ P=\frac{tg2\alpha tg\alpha}{tg2\alpha - tg\alpha}=\frac{\frac{sin2\alpha}{cos2\alpha}\cdot \frac{sin\alpha}{cos\alpha}}{\frac{sin2\alpha}{cos2\alpha}-\frac{sin\alpha}{cos\alpha}}=\frac{\frac{2sin^2\alpha}{cos^2\alpha-sin^2\alpha}}{\frac{2sin\alpha cos^2\alpha-sin\alpha(cos^2-sin^2\alpha)}{(cos^2\alpha-sin^2\alpha)cos\alpha}}=\frac{2sin^2\alpha cos\alpha}{sin\alpha(2cos^2\alpha-cos^2\alpha+sin^2\alpha)}=\frac{2sin^2\alpha cos\alpha}{sin\alpha}=2sin\alpha cos\alpha=sin2\alpha=L}\)
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matshadow
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obliczenie bez tablic, tozsamosci trygonometryczne
2)
\(\displaystyle{ \cos (\pi-\alpha)=-\cos(\alpha)}\)
\(\displaystyle{ \cos (\pi-2\alpha)=-\cos(2\alpha)}\)
\(\displaystyle{ \sin(-\alpha)=-\sin\alpha}\)
\(\displaystyle{ \sin(\pi-2\alpha)=\sin(2\alpha)}\)
\(\displaystyle{ \sin(2\alpha)=2\sin\alpha\cos\alpha}\)
\(\displaystyle{ \alpha=\frac{\pi}{6}}\)
\(\displaystyle{ cos\frac{5\pi}{6}=\cos(\pi-\alpha)=-\cos(\alpha)=-\cos\frac{\pi}{6}}\)
\(\displaystyle{ cos\frac{2\pi}{3}=\cos(\pi-2\alpha)=-\cos(2\alpha)=-\cos\frac{2\pi}{6}}\)
\(\displaystyle{ \sin(-\frac{2\pi}{3})=-\sin\frac{2\pi}{3}}\)
\(\displaystyle{ -\sin\frac{2\pi}{3}=-\sin(\pi-2\alpha)=-\sin(2\alpha)=-\sin\frac{2\pi}{6}}\)
\(\displaystyle{ \frac{sin\frac{\pi}{6}\cdot cos\frac{5\pi}{6}\cdot\cos\frac{2\pi}{3}}{sin(-\frac{2\pi}{3})}=\frac{sin\frac{\pi}{6}\cdot -\cos\frac{\pi}{6}\cdot -\cos\frac{2\pi}{6}}{-\sin\frac{2\pi}{6}}}\)
\(\displaystyle{ \frac{sin\frac{\pi}{6}\cdot -\cos\frac{\pi}{6}\cdot -\cos\frac{2\pi}{6}}{-\sin\frac{2\pi}{6}}=
\frac{sin\frac{\pi}{6}\cdot -\cos\frac{\pi}{6}\cdot -\cos\frac{2\pi}{6}}{-2\sin\frac{\pi}{6}\cos\frac{\pi}{6}}=-\frac{\cos\frac{\pi}{3}}{2}=-\frac{1}{4}}\)
\(\displaystyle{ \cos (\pi-\alpha)=-\cos(\alpha)}\)
\(\displaystyle{ \cos (\pi-2\alpha)=-\cos(2\alpha)}\)
\(\displaystyle{ \sin(-\alpha)=-\sin\alpha}\)
\(\displaystyle{ \sin(\pi-2\alpha)=\sin(2\alpha)}\)
\(\displaystyle{ \sin(2\alpha)=2\sin\alpha\cos\alpha}\)
\(\displaystyle{ \alpha=\frac{\pi}{6}}\)
\(\displaystyle{ cos\frac{5\pi}{6}=\cos(\pi-\alpha)=-\cos(\alpha)=-\cos\frac{\pi}{6}}\)
\(\displaystyle{ cos\frac{2\pi}{3}=\cos(\pi-2\alpha)=-\cos(2\alpha)=-\cos\frac{2\pi}{6}}\)
\(\displaystyle{ \sin(-\frac{2\pi}{3})=-\sin\frac{2\pi}{3}}\)
\(\displaystyle{ -\sin\frac{2\pi}{3}=-\sin(\pi-2\alpha)=-\sin(2\alpha)=-\sin\frac{2\pi}{6}}\)
\(\displaystyle{ \frac{sin\frac{\pi}{6}\cdot cos\frac{5\pi}{6}\cdot\cos\frac{2\pi}{3}}{sin(-\frac{2\pi}{3})}=\frac{sin\frac{\pi}{6}\cdot -\cos\frac{\pi}{6}\cdot -\cos\frac{2\pi}{6}}{-\sin\frac{2\pi}{6}}}\)
\(\displaystyle{ \frac{sin\frac{\pi}{6}\cdot -\cos\frac{\pi}{6}\cdot -\cos\frac{2\pi}{6}}{-\sin\frac{2\pi}{6}}=
\frac{sin\frac{\pi}{6}\cdot -\cos\frac{\pi}{6}\cdot -\cos\frac{2\pi}{6}}{-2\sin\frac{\pi}{6}\cos\frac{\pi}{6}}=-\frac{\cos\frac{\pi}{3}}{2}=-\frac{1}{4}}\)