I'd like to talk about the comparison of the trigonometric functions of angles.
For example, both angles in degrees, different from one another, are known and guaranteed to be in the same quarter of the unit circle. The question is how to compare their trigonometric functions without using tables or calculators.
How does one need to think when comparing two angles?
For example, we have two angles, one is 11 degrees, and another one is 13 degrees.
Comparing sines is easy since we know that for angles from 0 to 90 degrees, the greater the angle the greater the sine value, so $$\sin11^{\circ}<\sin13^{\circ}$$We also know that for angles from 0 to 90 degrees, the greater the angle the less the cosine value, so $$\cos11^{\circ}>\cos13^{\circ}$$Similar thing goes for tangents and cotangents: $$\tan11^{\circ}<\tan13^{\circ}\\\cot11^{\circ}>\cot13^{\circ}$$
How to compare their sines/cosines with their tangents/cotangents without using tables or calculators? Thank you!