# Similarity between graphs of sin and tan inverse

Why is it that the graphs of tan inverse and sin in the interval $$\left[-\frac \pi 2 , \frac \pi 2\right]$$ are so similar.

Is it just some coincidence or something deeper?

• What are your thoughts on this problem? They are both functions used in trigonometry: of course they are similar. – clathratus Mar 20 at 20:01

Both $$f(x) = \arctan x$$ and $$g(x) = \sin x$$ are increasing on $$(-\pi/2,\pi/2)$$ and both have an inflection point at $$x = 0$$. Their somewhat similar appearance is due to those properties and the fact that $$\arctan \dfrac \pi 2$$ is remarkably close to $$1$$.
The first few terms of their Taylor series around $$0$$ are $$x-x^3/6+x^5/120$$ and $$x-x^3/3+x^5/5$$, respectively, so it’s not terribly surprising that they look similar in a relatively small interval around the origin. I doubt that there’s any deep connection beyond that, though.