graphing-functionstrigonometry
Knowing $ y =\ sin\ x$
What does the graph of $ y = \frac{\ sin\ x}{x}$ look like.
I realise that when $x = 0$ there is a vertical asymptote.
I have tried making $x = 90$ and $x = 270$ and get $y = \frac{1}{90}$ and $y = -\frac{1}{270}$ respectively.
How do I progress from here?
The main difference between your first and second graph is the fact $y = \sin x$ uses $\mathbb{R}$ as its domain and $y = \sin n$ uses $\mathbb{N}$ as it's domain. Furthermore, the points on the second graph are not connected which implies that the second graph is not continuous and via inspection, you can notice that the $x$-axis only has natural numbers.
The surface of equation $$z=\sin x+\sin y$$ has the shape of an "egg tray". It has maxima $z=2$ on a periodic grid, and this is why the level curves $z=1$ are regularly spaced approximate circles.
Now if we replace $x$ by $x^2$, we deform space horizontally so that $x^2$ increases faster and faster, giving a "compression effect".
By replacing $y$ with $y^2$, we get the effect on both axis.
Of course, the picture is symmetric by reflection, as the square function is even.
Remains to explain the "spikes" against the axis. If $y$ is small,
$$\sin x^2+\sin y^2=1\to y\approx\sqrt{1-\sin x^2}=\sqrt2\left|\cos\frac{x^2}2\right|$$
shows them.
Best Answer