Question

Consider the function

\(f(x) = \tan(x) := \tan(x) := \frac{sin(x)}{cos(x)}\)

a) Determine the maximal Interval of deinfintion \(I\) and \(0\)

Answer 1

First, as zarkon said, you need to find when \(\cos(x)=0\). The best way to do this is to refer to the unit circle. This is something you should memorize. This is a good resource for that http://everobotics.org/projects/robo-magellan/img/UnitCircle.gif

The x-coordinates are the cosine of the angle, and the y-coordinates are the sine of the angle.

Answer 2

As for the actual zeroes, you'll find that \(\cos(x)=0\) when \[x=\pm\frac{\pi}{2}\]

Answer 3

ok..

Answer 4

Next step, we need to find the zeroes for \(\sin(x)\). Once again, we can refer to the unit circle, and find that \(\sin(x)=0\) when \(x=0,\pi\).

Answer 5

ok..

Answer 6

Now how do we find the interval of definition? Basically, this depends only on \(\cos(x)\). Since \(\cos(x)=0\) every time you add or subtract a multiple of \(\pi\) from \(\frac{\pi}{2}\), you know that the interval must be from \((-\frac{\pi}{2},\;\frac{\pi}{2})\).

Meanwhile, we already found the zero to be at \(x=0\) since \(\sin(x)=0\) at \(x=0\).

Answer 7

ok George, thank you it looks good, and this circle you send i like it, i need all to memorize or is there some important ones ?

Answer 8

If you know some tricks, you just need to memorize the first quadrant of it. However, it would be good to memorize all of it for now.

Answer 9

ok i get it, thank you Geoge

Answer 10

You're welcome.