### Video Transcript

What is the period of ๐๐ฅ is equal to two sin ๐ฅ plus ๐ over three?

First of all, to help us understand this question, Iโm actually gonna quickly make a sketch of the graph of sin ๐ฅ. So here you go. Iโve actually made the sketch of ๐ฆ equals sin ๐ฅ, as you can see. And we can see that actually itโs a repeating wave and it actually goes through the origin. Okay, so why Iโve drawn this to help us with this question? Well itโs actually this word here that we want to look at first. So we wanna find what is the period.

So whatโs the period of this graph? So whatโs the period of ๐ฆ equals sin ๐ฅ? Well, if you think about this good definition for the period, what we can say is that the period is the length of the function cycle. So if we look back at our sketch, we can actually see that the function actually goes through two complete cycles here, which Iโve put on with our red arrows. And we see that because itโs actually repeated. So it starts, it goes up, curves down to the bottom at negative one, and then back up to the same start position.

We can actually see from our graph that actually the period of our function, which is ๐ฆ equals sin ๐ฅ, would be two ๐. Because we if we look, we go from negative two ๐ to zero. So thatโs a difference of two ๐. And again, if weโre going on the positive side, weโve got zero to two ๐. So we can see that definitely yep, the period of ๐ฆ equals sin ๐ฅ is two ๐.

Okay, so this will help you understand what a period is but also really important because weโll be using that later in the problem. Now, if we actually take a look at our function, which is two sin of ๐ฅ plus ๐ over three, what weโre gonna want to do is actually gonna rewrite it.

And the way weโre gonna rewrite is weโre actually gonna rewrite in the form ๐ sin and then ๐๐ฅ minus ๐ plus ๐, which will give us that the function ๐๐ฅ is equal to two sin. And then, weโve got a third ๐ฅ plus ๐ on three. We donโt have a plus ๐ because we didnโt have that in the original function.

Okay, great! Why did we do this? The reason we do this is because of this formula. And this formula here shows us that the period is equal to two ๐ divided by the absolute value of ๐. And that ๐ comes from our function when we rewrite it in the form ๐ sin of ๐๐ฅ minus ๐ plus ๐.

And just kind of a look at the formula weโre using, weโve got the two ๐ on top. And that two ๐ actually comes from โ I know we mentioned it earlier and I said weโll come back to it โ it comes from the fact that the period of the function ๐ฆ equals sin ๐ฅ is actually two ๐. So thatโs where that comes from.

Okay, great! Now, weโve got this formula. Letโs put the values in and work out the period of our function. So therefore, we get that the period of our function is equal to two ๐ divided by the absolute value of a third. And we get that cause a third is the coefficient of ๐ฅ. So weโve seen so thatโs our ๐.

And what we were saying about the absolute values, weโre only interested in positive values. And well, a third is positive anyway. So that would be great. And then, we can calculate this. So therefore, we can say the period of our function, which is two sine of ๐ฅ plus ๐ over three, is equal to six ๐. And we get that because weโve got two ๐ divided by a third. If we divide it by a third, itโs the same as multiplying by three. So two ๐ multiplied by three gives us six ๐.