radians

The idea of radians is to define degrees in terms of pi. In a previous blog, I showed how Archimedes' proof for the area of a circle can be used to establish pi.

Definition Radian: There are 2π radians in a circle.

In other words 2π radians = 360 degrees.

A straight line has π radians and a right angle has π/2 radians.

This is useful because it enables trigonometric functions to be expressed in terms of taking π as a value.

Here are some examples:

sin (π/2) = 1
sin (π) = 0
sin(2π) = 0

cos (π/2) = 0
cos(π) = -1
cos(2π) = 1

Lemma 1: sin(nπ) = 0 where n is any integer

Proof:

(1) sin(0) = 0 and sin(π)=0 [See Property 1, here]

(2) sin(x + &2pi;) = x [See Property 5, here]

(3) So, we can see that:

if n is even, then n = 2x and sin (nπ) = sin(x*2π) = sin(0 + x*2π) = sin(0) = 0

if n is odd, then n = 2x+1 and sin(nπ) = sin(x*2π + π) = sin(π) = 0

QED

Lemma 2: The area of a sector = (1/2)θr²

Proof:

(1) We know that the area of a circle = πr² (see Corollary 2, here)

(2) An angle of a circle = (θ/2π) of a circle.

(3) So therefore, the area of a sector = (θ/2π)(πr²) = (1/2)θr²

QED