logarytmy z trygonometrią, sprawdzenie zadania

[matekleliczek]

\(\displaystyle{ 4log_{16} cos2x+2log_{4} sinx+log_{2} cosx+3 < 0}\)




założeniea

\(\displaystyle{ cos2x>0 \; i \; sinx>0 \; i \; cosx>0}\)

1.

\(\displaystyle{ sinx>0}\)
\(\displaystyle{ x\in (2k\pi, \pi+2k\pi)}\)

2.

\(\displaystyle{ cosx>0}\)
\(\displaystyle{ x\in (- \frac{\pi}{2}+2k\pi, \frac{\pi}{2}+2k\pi)}\)

3.

\(\displaystyle{ cos2x>0}\)
\(\displaystyle{ x\in (-\frac{\pi}{4}, \frac{\pi}{4}+k\pi)}\)

4. (razem 3 punkty)

\(\displaystyle{ x\in (2k\pi,\frac{\pi}{4}+2k\pi)}\)


wracając do zadania

\(\displaystyle{ log_{2}cos2x+log_{2}sinx+log_{2}cosx+3 cosx (\frac{5\pi}{24}+\frac{1}{2}k\pi,\frac{13\pi}{24}+\frac{1}{2}k\pi)}\)

biorac pod uwage załozenie daje nam to odp.

\(\displaystyle{ x\in (\frac{5\pi}{24}+2k\pi,\frac{\pi}{4}+2k\pi)}\)