What is the intersecting point of $y=\sin3x$ and y=$\frac{3x}{5\pi}$?

Question

This is one of many question from Japanese University Examination.

Answer 1

We know that $ -1 \leq \sin \Delta \leq 1 $. Then, $$ -1 \leq \sin 3x \leq 1 $$ $$ -1 \leq \frac{3x}{5 \pi} \leq 1 $$ $$ -\frac{5 \pi}{3} \leq x \leq \frac{5 \pi}{3} $$.
Also, the period of $ \sin 3x = \sin 2 \pi \frac{3}{2 \pi} x $ is $ \frac{2 \pi}{3} $.

Notice from the figure that for any line (with any slope) there are two intersectiona per each period in the non-negative domain.

Postive part: $[0,\infty)$
1) From the equations above, we have that $ x \leq \frac{5 \pi}{3} $.
2) A relveant question to answer is: how many periods $ T $ are there contained in $ 0 \leq x \leq \frac{5 \pi}{3} $? Why? Because if we find $ T $ we can also find the number of intersections (two intersections per period).
3) The number of periods where intersections occur is $ T = \lceil \frac{\frac{5 \pi}{3}}{\frac{2 \pi}{3}} \rceil = 3 $. Thus, in the range $ 0 \leq x \leq \frac{5 \pi}{3} $ there are 3 periods where intersections occur, and in each happen 2 intersections. Thus, a total of 6 intersections can be seen in the positve part.

Negative part: $(- \infty, 0]$
In the negative part, we will find another 6 intersections. However, since the intersection at zero was counted twice, we need to substract 1, which gives a total of 11 intersections.

Answer 2

$\sin3x$ is a periodic function. Very much familiar.

$y=\frac{3x}{5\pi}$ is equation of straight line of form $y=mx$ where y-intercept is $0$.

Clearly, $x=0$ is one of point of the intersection. In fact, there are a total of $11$ point of intersections.