47104 三角比相關的不等式的一個小備註

\begin{align} \hbox{試證對所有}\ \theta\in(0, \frac\pi 2),\ \hbox{恆有}\ \theta \lt\frac{\sin \theta +\tan \theta}2\hbox{。}\label{1} \end{align}

\begin{align} \hbox{當}\ \theta \in (0,\frac\pi 2)\ \hbox{時, 恆有} \sin\theta \lt\theta \lt\tan\theta , \label{2} \end{align}

\begin{align} f'(\theta )\gt\sqrt{\cos\theta \sec^2\theta}-1=\sqrt{\sec\theta}-1\gt 0,\label{3} \end{align}

\begin{align*} \sin\theta =\,&\theta -\frac{1}{3!} \theta ^3+\frac{1}{5!} \theta ^5-\frac{1}{7!} \theta ^7+\frac{1}{9!} \theta ^9-\cdots,\\ \tan\theta =\,&\theta +\frac 1 3 \,\theta ^3+\frac 2{15} \,\theta ^5+\frac {17}{315} \,\theta ^7+\frac {62}{2835} \,\theta ^9+\cdots, \end{align*}

\begin{align*} \sin\theta \gt\,&\theta -\frac{1}{3!} \,\theta ^3,\\ \tan\theta \gt\,&\theta +\frac 1 3 \,\theta ^3\hbox{。} \end{align*}

$$\frac{\sin \theta +\tan \theta}2\gt\theta +\frac 1{12} \,\theta ^3\gt\theta \hbox{。}$$

\begin{align} \frac{2 \sin \theta +\tan \theta }3\gt\theta , \label{4} \end{align}

\begin{align} g'(\theta )\gt\root 3\of{\cos\theta\cos\theta\sec^2\theta}-1=1-1=0,\label{5} \end{align}

$$\cos\theta =1-\dfrac{1}{2!} \,\theta ^2+\frac{1}{4!} \,\theta ^4-\frac 1{6!} \,\theta ^6+\frac 1{8!} \,\theta ^8-\cdots,$$

$$\theta \cos\theta =\theta -\frac{1}{2!}\, \theta ^3+\frac 1{4!}\, \theta ^5-\frac 1{6!}\, \theta ^7+\frac 1{8!}\, \theta ^9-\cdots\hbox{。}$$

\begin{align} \frac{2\theta \cos\theta +3 \tan\theta }5\gt\theta \hbox{。}\label{6} \end{align}