Why unit circle has radius 1

Question

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

Please I need a hint.

Answer 1

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.

Answer 2

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

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