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

Please I need a hint.

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

2 Answers 2

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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.

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  • $\begingroup$ 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) $\endgroup$
    – Bruno Rozendo
    Sep 4, 2016 at 1:23
  • $\begingroup$ @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. $\endgroup$
    – Fly by Night
    Sep 4, 2016 at 1:26
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$$(x,y) = (\cos \theta, \sin \theta)$$

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

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