- Thread starter apple2357
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If you use the unit-circle definition (that the sine and cosine of θ are, respectively, the

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One reason could be that beginning trig students have more experience working with right triangles (eg: finding lengths using Pythagorean formula) than they do finding coordinates of points on a circle. That situation seems similar to measuring angles in degrees first, then introducing radian measure later. That is, we start with simpler concepts and work up to more complicated ones.… So why is [trig] often introduced by [right-triangle definitions] rather than the [unit circle]?

\(\displaystyle sin(z) = \frac{e^{iz} - e^{-iz}}{2i}\)

\(\displaystyle cos(z) = \frac{e^{iz} - e^{-iz}}{2}\)

\(\displaystyle sin^2(z) = \left( \frac{e^{iz} - e^{-iz}}{2i} \right)^2 = -\frac{e^{-2iz} + e^{2iz} - 2}{4}\)

\(\displaystyle cos^2(z) = \left( \frac{e^{iz} + e^{-iz}}{2} \right)^2 = \frac{e^{-2iz} + e^{2iz} + 2}{4}\)

\(\displaystyle sin^2(z) + cos^2(z) = \frac{(e^{-2iz} + e^{2iz} + 2) - (e^{-2iz} + e^{2iz} - 2)}{4} = \frac{4}{4} = 1\)