Trigonometry

**Ostronomos** · Feb 21, 2026 08:43 PM

If we are asked to sketch an angle of radian measure [b]π/6 [/b]we use the principle of proportionality to calculate what fraction of the full circumference it corresponds to. That would be 1/12 of the full circumference.

Therefore, we conclude that the angle that subtends the arc of the unit circle corresponds to 1/12 of the full circumference.

Let P=(x,y) be the coordinates of the terminal point corresponding to the terminal radius of the circle. The following diagram illustrates this.

We say that x=cosθ and y=sinθ. From this the following property is immediate:

sinθ^2 + cosθ^2 = 1.

**confused2**

Right angle triangle .. Pythagoras..
a²+b²=c²
sin(θ)=b/c
cos(θ)=a/c

sin²(θ)+cos²(θ)= a²/c²+b²/c²
a²/c²+b²/c²=(a²+b²)/c²
since (a²+b²)=c²
it follows..
sin²(θ)+cos²(θ)=1

**Ostronomos** · Mar 18, 2026 08:47 PM

The language of mathematics is structured in such a way that is impervious to logical refutation. With the attempt by Pythagoreas to capture the essence of right triangles we have what can be called the eternal forms, hidden ever so beautifully behind the physical world. By the time of Sir Isaac Newton, the clockwork workings of a predetermined universe was firmly established. If we all adopted the idea of a predetermined universe there would be less conflict among us, I believe. As it would stand to reason that human beings are nothing more than biological machines fulfilling a predetermined destiny. But to embrace this to the core would betray our ideals that we hold about humanity so dearly. It requires us to embrace a clockwork universe governed by the principles of mathematics. A difficult feat no doubt. Nevertheless, logic once again pierces the veil of illusion.