Question

Daca tg x=2 calculati : E=[tex] \frac{sinx}{ sin^{3} cos^{3} }

Answer 1

   
[tex]\displaystyle \text{Voi rezolva problema considerand ca x Prtine cadranului 1} \\ x \in (0; ~ 90^o) ~~sau~~ x \in (0, ~ \frac{\pi}{2} ) \\ \text{Vom folosi urmatoarele formule: } \\ sin\;x = \frac{tg\, x}{ \sqrt{1+tg^2x}}~~~~~si~~~~~ cos\;x = \frac{1}{ \sqrt{1+tg^2x}}\\ \\ Rezolvare: \\ sin\;x = \frac{tg\, x}{ \sqrt{1+tg^2x}}=\frac{2}{ \sqrt{1+2^2}} =\frac{2}{ \sqrt{5}}=\frac{2\sqrt{5}}{5} [/tex]

[tex]\displaystyle cos\;x = \frac{1}{ \sqrt{1+tg^2x}}=\frac{1}{ \sqrt{1+2^2}} =\frac{1}{ \sqrt{5}}=\frac{\sqrt{5}}{5} \\ \\ \frac{sin\,x}{ sin^{3}x~ cos^{3}x}= \frac{~~~~\frac{2\sqrt{5}}{5} }{ \left(\frac{2\sqrt{5}}{5} \right)^{3} \left(\frac{\sqrt{5}}{5} \right)^{3}}=\frac{\frac{2\sqrt{5}}{5} }{ \left(\frac{40\sqrt{5}}{125} \right) \left(\frac{5\sqrt{5}}{125} \right)}= [/tex]


[tex]\displaystyle =\frac{\frac{2\sqrt{5}}{5} }{ \frac{40\sqrt{5} \times 5\sqrt{5}}{125^2}}= \frac{\frac{2\sqrt{5}}{5} }{ \frac{8 \times 5 \times 5 \times 5}{125 \times 125}}= \frac{\frac{2\sqrt{5}}{5} }{ \frac{8 \times 5 \times 5 \times 5}{125 \times 125}}= \frac{\frac{2\sqrt{5}}{5} }{ \frac{8}{125 }}= \\ \\ = \frac{2\sqrt{5}}{5} \times \frac{125}{8 } = \frac{2\sqrt{5} \times 125}{5 \times 8} = \boxed{\frac{25\sqrt{5}}{4} } [/tex]