# Why unit circle has radius 1 [closed]

I'm looking for how demonstrate why unit circle has radius 1, but I don't know how start.

• In truth, it doesn't matter. Just a radius of $1$ happens to be most simple for the equations. Sep 4, 2016 at 1:10
• It is not clear what you are asking. Are you asking why $\cos^2 x+ \sin^2 x = 1$? Sep 4, 2016 at 1:11
• Because that's what "unit" means. "Unit" means $1,$ "unit circle" means circle of radius $1.$
– bof
Sep 4, 2016 at 1:38
• A unit circle by definition has radius 1. That's what unit circle means. This the first thing you know about a unit circle. Sep 4, 2016 at 4:37
• "Unit" means 1 unit. If the radius were 2 or 3 we wouldn't call it a "unit". Are you asking why we like the radius 1 instead of 2 or 3? Sep 4, 2016 at 4:39

Sine, Cosine and Tangent are ratios of lengths.

If you chose a circle of radius 2 then you would have made all lengths and all heights twice as big. When you divide these new lengths and heights, the common factor of 2 would cancel.

$$\sin \theta = \frac{\text{opp}}{\text{hyp}} = \frac{(2\times\text{opp})}{(2 \times\text{hyp})} = \frac{(3 \times \text{opp})}{(3 \times \text{hyp})} = \cdots$$

Picking the radius (hypotenuse) to have length one is the most convenient choice.

• Just it? So, I can choose a another number if I want. 1 it's an ordinary number (but I can pikeup 2 or 100) Sep 4, 2016 at 1:23
• @BrunoRozendo You can pick any non-zero radius you want. The only difference is the intermediate steps would involve bigger numbers. But these bigger numbers will cancel down to give the usual answers. Sep 4, 2016 at 1:26

$$(x,y) = (\cos \theta, \sin \theta)$$

If you choose a different radius (and thereby, a different hypotenuse length), you lose the simplicity.