Deriving Special Relativity Formulae Steps

[Clever Penguin]

I was bored, so I decided to derive the special relativity formulae.
I drew the following diagram of a light clock:

In order to find t, I did sinθ=d/ct
Which gives tsinθ=d/c
Which gives t=d/csinθ

If v = 0, vt = 0, and θ = 90
sin90 = 1
t = d/csinθ = d/c
We call this t₀If v is greater than 0, vt is greater than zero, and θ is less than 90
sin90 is less than1
t = d/csinθ is greater than d/c

We use Pythagoras to get t₀=t(1- (v²/c²))^-1/2

What are the steps involved in getting to this answer?

 

[Dale]

In order to find t, I did sinθ=d/ct

Usually both t and θ are considered unknowns. So writing one equation in two unknowns doesn't help. You should use the Pythagorean theorem instead to get one equation in one unknown.

 

[Clever Penguin]

Usually both t and θ are considered unknowns. So writing one equation in two unknowns doesn't help. You should use the Pythagorean theorem instead to get one equation in one unknown.

Valid point

so (ct)² = d² + (vt)²

 

[Dale]

Valid point

so (ct)² = d² + (vt)²

Yes. Then rearranging and using $d/c=t_0$ gives you the desired formula

 

[Clever Penguin]

Yes. Then rearranging and using $d/c=t_0$ gives you the desired formula

thanks