What is the period of 𝑓𝑥 is equal to two sin 𝑥 plus 𝜋 over three?
First of all, to help us understand this question, I’m actually gonna quickly make a sketch of the graph of sin 𝑥. So here you go. I’ve actually made the sketch of 𝑦 equals sin 𝑥, as you can see. And we can see that actually it’s a repeating wave and it actually goes through the origin. Okay, so why I’ve drawn this to help us with this question? Well it’s actually this word here that we want to look at first. So we wanna find what is the period.
So what’s the period of this graph? So what’s the period of 𝑦 equals sin 𝑥? 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 I’ve put on with our red arrows. And we see that because it’s 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 𝑦 equals sin 𝑥, would be two 𝜋. Because we if we look, we go from negative two 𝜋 to zero. So that’s a difference of two 𝜋. And again, if we’re going on the positive side, we’ve got zero to two 𝜋. So we can see that definitely yep, the period of 𝑦 equals sin 𝑥 is two 𝜋.
Okay, so this will help you understand what a period is but also really important because we’ll be using that later in the problem. Now, if we actually take a look at our function, which is two sin of 𝑥 plus 𝜋 over three, what we’re gonna want to do is actually gonna rewrite it.
And the way we’re gonna rewrite is we’re actually gonna rewrite in the form 𝑎 sin and then 𝑏𝑥 minus 𝑐 plus 𝑑, which will give us that the function 𝑓𝑥 is equal to two sin. And then, we’ve got a third 𝑥 plus 𝜋 on three. We don’t have a plus 𝑑 because we didn’t 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 𝜋 divided by the absolute value of 𝑏. And that 𝑏 comes from our function when we rewrite it in the form 𝑎 sin of 𝑏𝑥 minus 𝑐 plus 𝑑.
And just kind of a look at the formula we’re using, we’ve got the two 𝜋 on top. And that two 𝜋 actually comes from — I know we mentioned it earlier and I said we’ll come back to it — it comes from the fact that the period of the function 𝑦 equals sin 𝑥 is actually two 𝜋. So that’s where that comes from.
Okay, great! Now, we’ve got this formula. Let’s 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 𝜋 divided by the absolute value of a third. And we get that cause a third is the coefficient of 𝑥. So we’ve seen so that’s our 𝑏.
And what we were saying about the absolute values, we’re 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 𝑥 plus 𝜋 over three, is equal to six 𝜋. And we get that because we’ve got two 𝜋 divided by a third. If we divide it by a third, it’s the same as multiplying by three. So two 𝜋 multiplied by three gives us six 𝜋.