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Tito Omburo (talk | contribs) →Definition via integration: quarter period behavior |
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<math display="block">\cos\theta = \frac{1-t^2}{1+t^2},\quad \sin\theta = \frac{2t}{1+t^2}</math>
for <math>(t,\theta)</math> a point of the graph of the function <math>\theta=2\arctan(t)</math>. This argument also shows that the cosine and sine, defined in this way, have semiperiod <math>\pi</math>.
Note that the substitution <math>t\to 1/t</math> transforms the integrand defining <math>\theta(t)</math> to itself, and so the quarter period is
<math display="block">\frac{\pi}{2} = \int_0^1\frac{2\,dt}{1+t^2}.</math>
Using this, one can then show that
<math display="block">\theta\left(\frac{1+t}{1-t}) = \theta(t) + \frac{\pi}{2}</math>
by making the rational substitution
<math display="block">\tau = \frac{1+\sigma}{1-\sigma}</math>
in the integral defining <math>\theta(t)</math>. Hence we have
<math display="block">\sin\left(\theta + \frac{\pi}{2}\right) = \frac{2\left(\frac{1+t}{1-t}\right)}{1+\left(\frac{1+t}{1-t}\right)^2} = \frac{1-t^2}{1+t^2} = \cos(\theta).</math>
===Definitions using functional equations===
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