So I know that cosecant is 1/sine, secant is 1/cosine, and cotangent is 1/tangent. This is all seems extremely redundant to me and next to useless. I am aware of the geometric reasoning of cosecant and secant. But even that, I don't feel like justifies the need to create 3 new functions for use. So, is there a good reason?
My math teacher showed us a cool pattern to remember the special sine values (the sine values that connect with the 45-45-90 and 30-60-90 triangles). sin(0) = (sqrt0)/2, sin(pi/6) = (sqrt1)/2, sin(pi/4)=(sqrt2)/2, sin(pi/3) = (sqrt3)/2, sin(pi/2) = (sqrt4)/2.
I understand how each of these values independently are found, but why is the pattern of consecutive sqrtn/2 itself true? It doesn't really seem like a coincidence.
What's a proof of the fact: if a quadrilateral's opposite angles add up to 180 degrees, then it's a cyclic quadrilateral? I managed to prove the converse of this statement, but I don't know how to prove this statement itself.
I'm aware of the law of sines for triangles (a/sinA = b/sinB = c/sinC). My question is: is there a similar rule for quadrilaterals, involving their sides and angles, that must hold true? Also, is there a law of cosines-equivalent for quadrilaterals?
I've learned about the Triangle Inequality Theorem, where adding up the two shorter sides will always be greater than the longest side in a triangle. But is there a similar rule/inequality that also controls the lengths of a quadrilateral?