Question Video: Finding the Frequency of a Trigonometric Function | Nagwa Question Video: Finding the Frequency of a Trigonometric Function | Nagwa

# Question Video: Finding the Frequency of a Trigonometric Function

What is the frequency of the function π(π‘) = π cos(ππ‘ β π) + πΎ?

06:41

### Video Transcript

What is the frequency of the function π of π‘ equals π cos ππ‘ minus π plus πΎ?

We have a function of π‘. We can see π‘ here. What does this function π do to the input π‘? Well, it multiplies it by π to get ππ‘. And then, to this product we subtract π. We find the cosine of this difference. We multiply that by π. And then, finally, we add πΎ. π, π, π, and πΎ are just some numbers. And to see what role they play in this function, itβs best to just try graphing this function for different values of π, π, π, and πΎ.

You should find that whatever values of π, π, π, and πΎ you pick, the graph looks a bit like a sine curve. We say it is sinusoidal. But if you increase the value of π while keeping π, π, and πΎ the same, you should find that the wave gets bigger. The oscillations are more pronounced. This value π is called the amplitude of the wave. This amplitude is the height of the wave when measured from the average value of the wave. Itβs half the height of the wave if you measure the height from the highest point of the wave to the lowest point. So, thatβs π. What about to π?

Well, letβs see what happens if we increase π while keeping π, π, and πΎ the same. As we increase π, we see that the distance between successive peaks, thatβs the highest points on the graph, decreases. The peaks get closer together as we increase π. And the same is true of the troughs, or lowest points on the wave. This function is periodic, and we see that its period is decreasing.

Remember we have time π‘ on the π₯-axis. And so, the peaks looking closer together on the graph actually tells us that the peaks are becoming more frequent. So, we can say that as π increases, frequency increases. Notice that, unlike before when I said π is the amplitude, I havenβt said that π is the frequency. Just that as π increases, frequency increases. This is because the frequency is not simply π. Weβll see why later.

Moving on, we can see that increasing π, the graph is translated to the right. π is related to something called the phase of the wave. And increasing πΎ, translates the graph up. πΎ is the average value of the function. The equation of the midline of the graph is π¦ equals πΎ. Letβs get back to our question, which is what is the frequency of the function π of π‘ equals π cos ππ‘ minus π plus πΎ. Well, weβve seen that only the value of π affects the frequency, but Iβve said that the frequency is not π.

Consider a slightly different function, the function π defined by π of π‘ equals π΄ times cos two πππ‘ plus π plus πΎ. And just for a moment, compare this to the function π. We can see that the lowercase π in the function π has been replaced by the uppercase, or capital, π΄ in the function π. π in the function π has been replaced by two ππ in the function π. The minus π has been replaced by a plus π. We can also think of this as replacing π by negative π. And πΎ has kept its role.

Now I can tell you the amplitude, frequency, phase, and average value of this function π. Capital π΄ is its amplitude. And this isnβt surprising, as this capital π΄ plays the same role in π that the lower case π plays in π. And we said that the lowercase π was the amplitude of our function π. π is the frequency of the function π.

And note here that π is not simply equal to π. π equals two ππ. We were right to say that the frequency is not π, although it is related to π. If we divide by two π on both sides, we find that the frequency is in fact π over two π. This is the frequency of the function π of π‘ equals π΄ cos ππ‘ minus π plus πΎ that weβre looking for. So, we found the answer to our question. Itβs π divided by two π.

But for completeness, I should say that π is the phase of the function π. And so, negative π is the phase of π. And πΎ is, of course, the average of both functions π and π. Now, of course, you could ask why two π is involved in the definition of the function π. Well, let capital π be the period of the function π, thatβs the time it takes before the function repeats. So, π of π‘ plus capital π is π of π‘.

We use the definition of the function π. And we can subtract πΎ from both sides and then divide through on both sides by π΄. And so, we see that cos of two ππ times π‘ plus capital π plus π equals cos of two πππ‘ plus π.

Now, one way for this equation to hold is that the inputs to cos are both the same. But we could also add or subtract any multiple of two π and their cosines would be the same. And by the definition of the period, exactly one oscillation has occurred. And so, we just have one two π to add. After one period of the function, weβve moved by two π radians to repeat again. Now itβs just some more algebra.

We cancel the πs, distribute two ππ over the terms in parentheses on left-hand side, which allows us to cancel further. We can also divide through by two π now to get ππ‘ on the left-hand side and just one on the right-hand side. And if we divide through by the period capital π of our function π, we find that π is one over this period capital π.

And so, by definition, π is the frequency of our function π. This has justified our claim that π is indeed the frequency of the function π. And we could perform basically the same procedures to show that the frequency of the original function π is π over two π.

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