Where does the sine rule come from?

For the mathematical puritans out there the sine rule derives from the following:

(you really don’t need to know this)

Sine and Cosine are used to find the angle of a shape

Find “a” KNOWING the angle A and length b and c.

Redraw the diagram

Finding the length of A if B and C are known

Find Y in the triangle I

First find the length of Y, the height of the triangle

Use trigonometry which tells us that $sin\ theta=(O)/H$

$sinA=Y/b$

Therefore: $Y=bsinA$

Find “a” in triangle II

You can now find A in conjunction with sine

Use trigonometry which tells us that $sin\ theta=(O)/H$

$sinB=(bsinA)/a$

Therefore $a=(bsinA)/sinB$

Or $a/sinA=b/sinB$ (the sine rule)

But again we emphasise that you really don’t need this.