Indian Statistical Institute B.Math & B.Stat : Trigonometry

Let $\theta_1 = \frac{2 \pi}{3}, \theta_2 = \frac{4 \pi}{7}, \theta_3 = \frac{7 \pi}{3}$. Then show that $( \sin \theta_1)^{ \sin \theta_1} < ( \sin \theta_3)^{ \sin \theta_3} < ( \sin \theta_2)^{ \sin \theta_2}$. First note that $\pi > \theta_1 > \theta_3 > \theta_2 > 0$ and all of them belong to the $second$ quadrant. $Sine$ function strictly decreases from $1$ to $0$ in the $second$ quadrant. Also $\sin \theta_1 \neq \sin \theta_2 \neq \sin \theta_3 \neq 0$ and each of them are posititve. Using the strictly decreasing property of $Sine$ in the second quadrant we have $\sin \theta_1 < \sin \theta_3 < \sin \theta_2$. Now the result follows the standard inequality $x^c < y^d$ for $x,y,c,d > 0 \quad where \quad x < y, \quad c < d$.