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For right angled triangle $ABC$, $\cos x = \frac{adjacent}{hypotenuse} = \frac{AB}{BC}$

Again from power series expansion of $\cos x$ we get $$\cos x = 1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+......$$ Are these two definitions same?

Archisman Panigrahi
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Bivas Das
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    Usually we don't allow triangles to have angles outside the range of $(0 ,\mathrm rad,\pi,\mathrm{rad})$, or to have a negative base/hypotenuse. In that sense they're not the same since the power series is fine with any real $x$. – Mark S. Mar 26 '17 at 19:36
  • @MarkS. so we can say that the triangle definition is the limiting form of definition got from power series – Bivas Das Mar 26 '17 at 19:41
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    There is another difference : In the first definition, $x$ is an angle (most written as $\alpha,\beta,etc.$, whereas in the second definition we have a function acting on real numbers (which only correspond to an angle). So you have to convert the angle into a number before you can use the second definition. For example $\sin(90°)$ has the same value as $\sin(\frac{\pi}{2})$ – Peter Mar 26 '17 at 19:44
  • You can also extend the power series to complex values unlike the triangle definition. – Simply Beautiful Art Mar 26 '17 at 19:44
  • The real puzzle is, "What is $x$ in the geometric sketch?" If we want to put that on a firm footing, we really do need to go down the power series route, as that will let us define $\pi$, lengths of circular arcs, and hence define angles. But the geometric sketches are good for our intuitions. – ancient mathematician Mar 27 '17 at 08:04
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    This answer may be helpful. – Blue Mar 27 '17 at 23:56

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