nierówności trygonometryczne

[krzysiek]

1. \(\displaystyle{ \frac{1}{2}< 2 + sin x - cos x < \frac{7}{2}}\)
2. \(\displaystyle{ sin^3 xcos x - cos^3 xsin x q \frac{1}{4}}\)
3. \(\displaystyle{ cos x ctg2x q 0}\)
4. \(\displaystyle{ sin^3 x - 4sin^2 x - sin x + 4 q 0}\)

[Skrzypu]

1. \(\displaystyle{ \frac{1}{2}< 2 + \sin x - \cos x < \frac{7}{2}}\)

\(\displaystyle{ \frac{1}{2}< 2 + \sin x - \cos x}\)
\(\displaystyle{ \cos x-\sin x < {3 \over 2}}\)
\(\displaystyle{ \sqrt2 \cos({\Pi \over 4} + x} < {3 \over 2}}\)

\(\displaystyle{ 2 + \sin x - \cos x < \frac{7}{2}}\)
\(\displaystyle{ \cos x - \sin x > -{3 \over 2}}\)
\(\displaystyle{ \sqrt2 \cos({\Pi \over 4} + x}> -{3 \over 2}}\)


2. \(\displaystyle{ \sin^3 x \cos x - \cos^3 x \sin x q \frac{1}{4}}\)
\(\displaystyle{ \sin x \cos x (\sin^2 x - \cos^2 x) q \frac{1}{4}}\)
\(\displaystyle{ {\sin 2x \over 2}(-\cos 2x) q \frac{1}{4}}\)
\(\displaystyle{ \sin 2x \cos 2x q -\frac{1}{2}}\)
\(\displaystyle{ {\sin 4x \over 2} q -\frac{1}{2}}\)
\(\displaystyle{ \sin 4x q -1}\)


3. \(\displaystyle{ \cos x \cot 2x q 0}\)
\(\displaystyle{ \cos x \frac{\frac{\cos x}{\sin x}-\frac{\sin x}{\cos x}}{2} q 0}\)
\(\displaystyle{ \frac{\cos^2 x}{\sin x}-\sin x q 0}\)
\(\displaystyle{ \frac{\cos^2 x-\sin^2 x}{\sin x} q 0}\)
\(\displaystyle{ \frac{\cos 2x}{\sin x} q 0}\)
\(\displaystyle{ \cos 2x \sin x q 0}\)


4. \(\displaystyle{ \sin^3 x - 4\sin^2 x - \sin x + 4 q 0}\)
\(\displaystyle{ \sin^2 x (\sin x - 4)-(\sin x - 4) q 0}\)
\(\displaystyle{ (\sin^2 x-1)(\sin x - 4) q 0}\)
\(\displaystyle{ (\sin^2 x-\sin^2 x -\cos^2 x)(\sin x - 4) q 0}\)
\(\displaystyle{ \cos^2 x (\sin x - 4) q 0}\)