Use what you learned in calculus to show that Sin(alpha)≤alpha and sin(alpha)/cos(alpha)≥alpha holds for all 0≤alpha≤pi/2
You can use it using geometry.
Look at the picture.
The angle between the x-axis and the red line is $\displaystyle \alpha$.
While the circle is a unit circle.
Thus, the length of the arc made by $\displaystyle \alpha$ is $\displaystyle \alpha$.
While the length of the blue line is $\displaystyle \sin \alpha$.
Now reflect that image through the x-axis (second picture).
We see that the length of the blue line is $\displaystyle 2\sin \alpha$.
And the length of arc made by the blue line is $\displaystyle 2\alpha$.
Thus, $\displaystyle 2\sin \alpha \leq 2\alpha \implies \sin \alpha \leq \alpha$.
This is because the shortest distance between two points is a straight line.