Prove that for all whole numbers n ≥1, (d^n)/(d x^n) sinx=sin(x+(n π)/(2))

Name: Problem 109, Chapter 3: Calculus: Early Transcendentals
Uploaded: 2021-07-18T18:56:13-08:00
Duration: 5 min 36 s
Channel: Stark Ledbetter
Description: Prove that for all whole numbers n ≥1, (d^n)/(d x^n) sinx=sin(x+(n π)/(2))

Prove that for all whole numbers n ≥1, (d^n)/(d x^n)...

Key Concepts

Differentiation of Trigonometric Functions

Differentiation involves calculating the rate at which a function changes, and trigonometric functions like sine and cosine have well-known derivative formulas. The derivative of sin x is cos x, and the derivative of cos x is -sin x. These formulas are crucial in understanding how repeated differentiation (nth derivatives) can result in a cyclic pattern, which is essential in proving the given statement.

Cyclic Pattern in Derivatives

The repeated differentiation of sine and cosine functions results in a cycle: sin x, cos x, -sin x, -cos x, and then back to sin x. This cyclic behavior means that taking the nth derivative of sin x rotates through these functions, which is reflected by the addition of n?/2 to the argument of the sine function. Recognizing this cyclic pattern is key to establishing the relationship between the nth derivative of sin x and sin(x + n?/2).

Mathematical Induction

Mathematical induction is a fundamental proof technique used to establish statements for all natural numbers. It involves proving a base case, showing that the property holds for the first case, and then proving that if it holds for an arbitrary natural number n, it must also hold for n+1. This method ensures that the property is valid for every whole number starting from the base case.

Transcript

00:01 All right, this is kind of funny worded question because this is prove.

00:03 And we want to prove this fact.

00:05 We want to prove that this equation is true for every whole number n greater than equal to 1.

00:11 This is actually kind of a silly question.

00:13 It asks for the nth derivative of sine x.

00:17 Okay, let's just, but like we know the derivatives of sine x, right? we have sine x, and then the first derivative of sine x is cosine x.

00:25 We know this.

00:27 And then the second derivative of sinex is the derivative of cosine x, which is negative sinex.

00:38 And then the third derivative is negative derivative of sinex is cosinex.

00:46 And then let's do one more.

00:48 The fourth derivative of sinex is negative derivative of cosinex, which is negative sinex.

00:56 So we get positive sinex again.

00:58 And now we're at the beginning.

00:58 If we took the fifth derivative, so we would just keep cycling through those four things.

01:04 Okay, so like what is going on over here? okay, well, sine x, let's just do, let's set n equal to four, since that's where it repeated.

01:15 Okay, so if n equals four, then the right hand side is sine x plus four pi over two, which is sign of x plus two pi.

01:26 Oh, that's good, because sine is periodic with a period of 2 pi.

01:30 So sine of x plus 2 pi is really just sine x.

01:33 Okay, so for sure it's true when n equals 4, and actually it's true when n is any multiple of 4 because we can add 4 pi, 8 pi, and so on.

01:45 Okay, so i believe that one.

01:48 True for n equals 4k.

01:53 If you wanted to, you could write that in.

01:56 N equals 4k.

01:57 Let's just do that.

01:58 That's 2 pi k, which is certainly sine x when k is an integer.

02:04 Okay, that's good.

02:06 Okay, so what about all this other stuff? so like what is what about n equals 4k plus one? right, then we should be in this case, hopefully.

02:18 Okay, well, then the right hand side is sine of x plus, and then the 4k part will be the same.

02:29 Right plus 4k pi over 2 that'll work out and then the plus 1 will introduce an extra pi over 2 okay and then we got to know some stuff about signs and cosines so you can either get this from the unit circle or from the graphs but actually sign so that part's irrelevant that part can go away so that's really equal to sign of x plus pi over 2 and then from trig functions somehow you know that that sine of x plus pi over two is cosine x.

03:01 You can actually even use an addition formula...

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