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?
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1$\begingroup$ 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 $x$. $\endgroup$– Mark S.Mar 26, 2017 at 19:36
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$\begingroup$ @MarkS. so we can say that the triangle definition is the limiting form of definition got from power series $\endgroup$– Bivas DasMar 26, 2017 at 19:41
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1$\begingroup$ 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 $\sin(\frac{\pi}{2})$ $\endgroup$– PeterMar 26, 2017 at 19:44
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$\begingroup$ You can also extend the power series to complex values unlike the triangle definition. $\endgroup$– Simply Beautiful ArtMar 26, 2017 at 19:44
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1$\begingroup$ This answer may be helpful. $\endgroup$– BlueMar 27, 2017 at 23:56
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