Question

Calculati sin2x,stiind ca cosx=4/5 ,x apartine {3pi/2 ,2pi}

Answer 1

    
[tex]\displaystyle \text{Intervalul: } ~~ ( \frac{3 \pi}{2}, ~2 \pi) ~ \text{ este cadranul IV al cercului trigonometric.} \\ \\ \text{In cadranul IV, cosinusul este pozitiv iar sinusul este negativ.} \\ \\ x\in ( \frac{3 \pi}{2}, ~2 \pi) \\ \\ cos\,x= \frac{4}{5} \\ \Longrightarrow~sin\,x= -\sqrt{1-cos^2x}= -\sqrt{1- \left(\frac{4}{5}\right)^2 }= \\ \\ -\sqrt{1- \frac{4^2}{5^2}}=-\sqrt{1- \frac{16}{25}}=-\sqrt{ \frac{25-16}{25}}=-\sqrt{ \frac{9}{25}}=\boxed{-\frac{3}{5}} [/tex]


$\displaystyle sin\,2x = 2sin\,x \;cos\,x = 2 \cdot \left(- \frac{3}{5} \right) \cdot \frac{4}{5} = -\frac{2 \cdot 3 \cdot 4}{5 \cdot 5} =\boxed{- \frac{24}{25}}$