tg\alpha=\frac{\sin\alpha}{cos\alpha}
\frac{8}{15}=\frac{sin\alpha}{cos\alpha}
8cos\alpha=15sin\alpha \ |:8
cos\alpha=\frac{15}{8}sin\alpha
Z jedynki trygonometrycznej:
sin^2\alpha+cos^2\alpha=1
sin^2\alpha+(\frac{15}{8}sin\alpha)^2=1
sin^2\alpha+\frac{225sin^2\alpha}{64}=1\ |*64
64sin^2\alpha+225sin^2\alpha=64
289sin^2\alpha=64
(17sin\alpha)^2=8^2
17sin\alpha=8
sin\alpha=\frac{8}{17}
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cos\alpha=\frac{15}{8}\cdot \frac{8}{17}=\frac{15}{17}
\frac{sin\alpha\cdot cos\alpha}{3}=\frac{8}{17}\cdot \frac{15}{17}\cdot \frac{1}{3}=\frac{120}{867}=\frac{40}{289}
Odpowiedź:
\frac{40}{289}
Sprawdzenie:
tg\alpha=\frac{sin\alpha}{cos\alpha}=\frac{\frac{8}{17}}{\frac{15}{17}}=\frac{8}{17}\cdot \frac{17}{15}=\frac{8}{15}