2.9

## University of Padova

Skip to 0 minutes and 12 secondsHello welcome to the section, "It's Your Turn on Graphs of Trigonometric Functions". We have to compute the domain and the minimal period of the function tangent of 3 times x and sketch its graph. Let us recall some very easy facts about the function tangent of x.

Skip to 0 minutes and 43 secondsOK. The domain of this function, tangent of x, is the set of real numbers minus all the values of the shape pi over 2 plus any integer multiple of pi.

Skip to 1 minute and 14 secondsOK. And this function has minimal period pi.

Skip to 1 minute and 25 secondsPeriod pi. Good. And now let us consider our function. We call it f.

Skip to 1 minute and 35 secondsThe tangent of 3 times x. OK. Which is the domain of this function? We have to understand where is defined this. And this is defined if and only if 3 times x is different from these values.

Skip to 2 minutes and 1 secondIs defined if and only if 3 times x is different from pi over 2 plus an integer multiple of pi. But now dividing by 3, what we get equivalently? That the x is different from pi over 6 plus any integer multiple of pi over 3. Therefore, the domain of our function f, the tangent of 3 times x, is the set of real numbers minus all the values of the shape pi over 6 plus any integer multiple of pi over 3.

Skip to 3 minutes and 4 secondsOK. And now let us compute the minimal period of our function f. What is the minimal period? The minimal period is a value t-- the minimal value t, such that f of x is equal to f of x plus t for each x in the domain of f. OK. Let us try to compute this value. OK. What we have, that the tangent of 3 times x is equal to our function f of x. And it is true that this is equal to f of x plus t, if and only if the tangent of 3 times x is equal to the tangent of 3 times x plus t. That is the tangent of 3 times x plus 3 times t. OK.

Skip to 4 minutes and 28 secondsWhen we have the equality between these two terms for each x in the domain of f, if and only if 3 times t is a multiple of the minimal period of the tangent function, but we are looking for the minimal value of t such that we have this equality. Therefore, which is the minimal multiple is 1 times the period of the tangent. Therefore, we get that 3 times t has to be equal to pi. That is the minimal period is pi over 3. Good. Now we know the domain and the periodicity of our function f. Let us try to sketch its graph. OK. We know which is the graph of the tangent function. OK.

Skip to 5 minutes and 37 secondsThe tangent function has a graph of this shape-- good-- which is the tangent function is not defined on minus pi over 2. It's not defined on pi over 2 in the integer multiple. And then you complete the graph, just repeating this shape along the x-axis. OK. And now we are looking for the tangent of 3 times x. What happens? Now the period-- the period is not anymore pi, which is the distance between these two values. But now the period is pi over 3. And which is the graph? No surprise. You see, you just have the same shape as the tangent, but now in a stricter interval.

Skip to 6 minutes and 38 secondsAnd then you repeat the same graph, because of the periodicity, on all the x-axis. Again, the tangent of 3 times x of course is not defined in minus pi over 6. It's not defined in pi over 6. It's not defined on all the integer multiples of these values. OK. Thank you very much for your attention.

# It's your turn on graphs of trigonometric functions

Do your best in trying to solve the following problems. In any case some of them are solved in the video and all of them are solved in the pdf file below.

### Exercise 1.

Compute the domain and the minimal period of the function $$x\mapsto \tan(3x)$$ and sketch its graph.

### Exercise 2.

Compare $$\tan x$$ with $$\tan(\pi-x)$$, $$\tan(\pi+x)$$, $$\tan(\pi/2-x)$$, and $$\tan(\pi/2+x)$$ whenever they are defined.