For which angles between $0^{\circ}$ and $360^{\circ}$ do at least two of the following inequalities hold:

$$
\sin 2 x>\sin x, \quad \cos 2 x>\cos x, \quad \operatorname{tg} 2 x>\operatorname{tg} x, \quad \operatorname{ctg} 2 x>\operatorname{ctg} x
$$

Is there an angle for which none of the given inequalities hold? (As a starting point, we can draw the graphs of the functions in question!)
As a preparatory step, we solve each of the four inequalities separately, reduce them to 0, and express the functions of $2x$ in terms of functions of $x$, factoring the left side.

I. $\sin 2x - \sin x = \sin x (2 \cos x - 1) > 0$

This holds if and only if the two factors have the same sign. The first factor is positive when $0^\circ < x < 180^\circ$. The second factor is positive when $\cos x > 1/2$, that is, when $0^\circ < x < 60^\circ$ or $300^\circ < x < 360^\circ$. Therefore, the inequality holds when

$$
0^\circ < x < 60^\circ \quad \text{or} \quad 300^\circ < x < 360^\circ
$$

II. $\tan 2x > 0$

This is clearly true if and only if

$$
z = \frac{\tan x}{1 - \tan^2 x} > 0
$$

Here, the denominator $\neq 0$, because $\tan x = \pm 1$ when $x = k \cdot 45^\circ$, where $k = 1, 3, 5, 7$, so $2x = k \cdot 90^\circ$, and thus $\tan 2x$ is undefined. For $0^\circ < x < 45^\circ$, $z > 0$. The numerator changes sign at $90^\circ, 180^\circ$, and $270^\circ$, and the denominator changes sign when the absolute value of $\tan x$ exceeds 1, i.e., at $45^\circ, 135^\circ, 225^\circ$, and $315^\circ$. These points are distinct, so $z$ changes sign at every $45^\circ$ multiple, and thus retains its original sign every two transitions. Therefore, $z > 0$ when

$$
90^\circ < x < 135^\circ \quad \text{or} \quad 225^\circ < x < 270^\circ
$$

III. $\cot 2x < 0$

This holds if and only if $\cot x < 0$, i.e., when $90^\circ < x < 180^\circ$ or $270^\circ < x < 360^\circ$.

![](https://cdn.mathpix.com/cropped/2024_05_02_effb34e49617cddb90a4g-1.jpg?height=439&width=978&top_left_y=1193&top_left_x=560)

The found intervals are illustrated in the diagram. From this, it can be read that two or three inequalities hold when

$$
0^\circ < x < 45^\circ, \quad 90^\circ < x < 180^\circ, \quad 180^\circ < x < 240^\circ, \quad 270^\circ < x < 315^\circ
$$

(The 2nd and 3rd intervals are adjacent but not connected, because for $x = 180^\circ$, only the 2nd inequality holds.) No inequality holds if $60^\circ \leq x < 90^\circ$, and also, considering the left endpoint of the given interval as part of it, if $x = 0^\circ$.