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What is the frequency of the function π of π‘ equals π cos ππ‘ minus π plus πΎ?
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We have a function of π‘.
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We can see π‘ here.
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What does this function π do to the input π‘?
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Well, it multiplies it by π to get ππ‘.
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And then, to this product we subtract π.
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We find the cosine of this difference.
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We multiply that by π.
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And then, finally, we add πΎ.
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π, π, π, and πΎ are just some numbers.
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And to see what role they play in this function, itβs best to just try graphing this function for different values of π, π, π, and πΎ.
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You should find that whatever values of π, π, π, and πΎ you pick, the graph looks a bit like a sine curve.
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We say it is sinusoidal.
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But if you increase the value of π while keeping π, π, and πΎ the same, you should find that the wave gets bigger.
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The oscillations are more pronounced.
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This value π is called the amplitude of the wave.
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This amplitude is the height of the wave when measured from the average value of the wave.
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Itβs half the height of the wave if you measure the height from the highest point of the wave to the lowest point.
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So, thatβs π.
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What about to π?
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Well, letβs see what happens if we increase π while keeping π, π, and πΎ the same.
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As we increase π, we see that the distance between successive peaks, thatβs the highest points on the graph, decreases.
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The peaks get closer together as we increase π.
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And the same is true of the troughs, or lowest points on the wave.
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This function is periodic, and we see that its period is decreasing.
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Remember we have time π‘ on the π₯-axis.
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And so, the peaks looking closer together on the graph actually tells us that the peaks are becoming more frequent.
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So, we can say that as π increases, frequency increases.
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Notice that, unlike before when I said π is the amplitude, I havenβt said that π is the frequency.
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Just that as π increases, frequency increases.
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This is because the frequency is not simply π.
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Weβll see why later.
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Moving on, we can see that increasing π, the graph is translated to the right.
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π is related to something called the phase of the wave.
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And increasing πΎ, translates the graph up.
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πΎ is the average value of the function.
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The equation of the midline of the graph is π¦ equals πΎ.
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Letβs get back to our question, which is what is the frequency of the function π of π‘ equals π cos ππ‘ minus π plus πΎ.
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Well, weβve seen that only the value of π affects the frequency, but Iβve said that the frequency is not π.
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Consider a slightly different function, the function π defined by π of π‘ equals π΄ times cos two πππ‘ plus π plus πΎ.
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And just for a moment, compare this to the function π.
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We can see that the lowercase π in the function π has been replaced by the uppercase, or capital, π΄ in the function π.
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π in the function π has been replaced by two ππ in the function π.
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The minus π has been replaced by a plus π.
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We can also think of this as replacing π by negative π.
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And πΎ has kept its role.
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Now I can tell you the amplitude, frequency, phase, and average value of this function π.
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Capital π΄ is its amplitude.
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And this isnβt surprising, as this capital π΄ plays the same role in π that the lower case π plays in π.
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And we said that the lowercase π was the amplitude of our function π.
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π is the frequency of the function π.
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And note here that π is not simply equal to π.
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π equals two ππ.
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We were right to say that the frequency is not π, although it is related to π.
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If we divide by two π on both sides, we find that the frequency is in fact π over two π.
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This is the frequency of the function π of π‘ equals π΄ cos ππ‘ minus π plus πΎ that weβre looking for.
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So, we found the answer to our question.
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Itβs π divided by two π.
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But for completeness, I should say that π is the phase of the function π.
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And so, negative π is the phase of π.
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And πΎ is, of course, the average of both functions π and π.
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Now, of course, you could ask why two π is involved in the definition of the function π.
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Well, let capital π be the period of the function π, thatβs the time it takes before the function repeats.
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So, π of π‘ plus capital π is π of π‘.
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We use the definition of the function π.
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And we can subtract πΎ from both sides and then divide through on both sides by π΄.
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And so, we see that cos of two ππ times π‘ plus capital π plus π equals cos of two πππ‘ plus π.
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Now, one way for this equation to hold is that the inputs to cos are both the same.
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But we could also add or subtract any multiple of two π and their cosines would be the same.
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And by the definition of the period, exactly one oscillation has occurred.
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And so, we just have one two π to add.
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After one period of the function, weβve moved by two π radians to repeat again.
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Now itβs just some more algebra.
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We cancel the πs, distribute two ππ over the terms in parentheses on left-hand side, which allows us to cancel further.
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We can also divide through by two π now to get ππ‘ on the left-hand side and just one on the right-hand side.
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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.
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And so, by definition, π is the frequency of our function π.
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This has justified our claim that π is indeed the frequency of the function π.
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And we could perform basically the same procedures to show that the frequency of the original function π is π over two π.