Think of an equilateral triangle. Since it has all three sides the same, of length s, say, it must have all three angles the same. And since the angles in a triangle add to 180 degrees, each must be 180/3= 60. Now draw a line from one vertex to the midpoint of the opposite side. That divides the equilateral triangle into two congruent triangles and it is easy to show that this new line is also perpendicular to opposite side and bisects the angle at the vertex. That means we now have two right triangles with angle 60 degrees and 30 degrees. The hypotenuse of either is a side of the original triangle, of length s. The leg opposite the 30 degree angle is half a side of the original triangle, of length s/2. Calling the length of other leg "x", by the Pythagorean theorem we must have \(\displaystyle s^2= x^2+ s^2/4\) so \(\displaystyle x^2= (3/4)s^2\) and \(\displaystyle x= \frac{\sqrt{3}}{2}s\).
Now we can write down all of the trig functions for both 30 and 60 degree. We have a right triangle with hypotenuse of length s, leg opposite the 30 degree angle of length s/2, and leg adjacent to the 30 degree angle of length \(\displaystyle \frac{\sqrt{3}}{2}s\). sin(30)= "opposite side over hypotenuse"= (s/2)/s= 1/2. cos(30)= "adjacent side over hypotenuse"\(\displaystyle = (\sqrt{3}/2)s/s= \frac{\sqrt{3}}{2}\). tan(30), cot(30), sec(30), and csc(30) can now be calculated from their definitions in terms of cosine and sine.
sin(60)= "opposite side over hypotenuse" but now the side opposite the 60 degree angle has length \(\displaystyle \frac{\sqrt{3}}{2}s\) so \(\displaystyle sin(60)= (\frac{\sqrt{3}}{2})s/s= \frac{\sqrt{3}}{2}\).
You can do the same thing for 45 degrees. An isosceles right triangle has both legs the same length, s, say, so both angles the same, 90/2= 45 degrees. The hypotenuse has length, by the Pythagorean theorem, has length \(\displaystyle \sqrt{2}s\) so sin(45) and cos(45) are both \(\displaystyle \frac{s}{\sqrt{2}s}= \frac{1}{\sqrt{2}}= \frac{\sqrt{2}}{2}\). From that, tan(45)= cot(45)= 1 and \(\displaystyle sec(45)= csc(45)= \sqrt{2}\).