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Consider the formula below:cOS + isin 8This formula is called Euler' $ Formula and one of the most famous formulas in all of Mathematics shows the connection b...

Question

Consider the formula below:cOS + isin 8This formula is called Euler' $ Formula and one of the most famous formulas in all of Mathematics shows the connection between trigonometric functions and the exponential functionProve Euler' $ Formula_ (Hint: Expand both sides using the MacLaurin Series)Use Euler' $ Formula to prove Euler's Identity: eti + 1 = 0 beautiful formulasin all of math, as it relates the most famous Note: This is one of the most constants that exist!prove De Mo

Consider the formula below: cOS + isin 8 This formula is called Euler' $ Formula and one of the most famous formulas in all of Mathematics shows the connection between trigonometric functions and the exponential function Prove Euler' $ Formula_ (Hint: Expand both sides using the MacLaurin Series) Use Euler' $ Formula to prove Euler's Identity: eti + 1 = 0 beautiful formulasin all of math, as it relates the most famous Note: This is one of the most constants that exist! prove De Moivre Formula: Use Euler' $ Formula to (cos 0 + isin 0)" cos(ne) + i sin(n0)



Answers

Use Euler's Fomula to prove that the identity holds. Note the similarity between these relationships and the definitions of the hyperbolic sine and cosine functions.
$$ \sin z=\frac{e^{i z}-e^{-i z}}{2 i} $$

So if we're told that our value of Z is equal to our times e to power of I times, they, uh well, we want to find Z to the one over end. And so if we take this to the one over and we have our e to the i times data to the power of one over end And so if we expand, what we get is the to anyone over an is equal to our to anyone over end times E times I and then we have times data over and and so So now if we use, uh, Oilers formula and and substitute data for as data over end, we have Z to the one over. End is equal to R two D one over and times co sign. And now we have co sign of data over and And if we want to find our next value, if we want to find our next you've got to we have to add two pi, okay, over end and K is dependent on how many how many routes we have. So K is equal to K is equal to end minus one right, And so so now we have. We have this co sign and we can say we have plus plus I sine of data divided by N plus two pi ke over and and so remember, since in this first value this this K is equal to zero. So we have data over end plus zero. And so this is why this this formula works, and we derived this formula using using Mueller's Oilers formula.

So if we're told that our value for Z is equal to E or rather are one rather are one times e times I times data and then we're told we want to find Z to the power. And so if we take the end power, both sides we're left with is our one times e times I time stada to the end power. And so now if we distribute here, we're left with Z to the end is equal to our one to the end Power times e to the i Times data and and so if we use Oilers formula here, well, then Z and is equal to our 12 or rather are just Let's they are This is just our our times co sign and we're going to replace data with data. And so we have data and plus I sine of beta times and and so this is how we obtain our formula

All right, we want to use the Oilers formula to prove this. Um So oilers formula talks about E. To the I. Z. And um it writes E. To the I. Z. As co sign Z plus I. Sine Z. And then we can do the same thing with you to the I. To the negative. I see. So this is we're working with the right hand side. Um and now we're just putting in negative Z for Z into this formula. So this is going to be co sign negative Z plus I sine negative Z. And that's all over two. That's what the right hand side is. And now if we think about the unit circle, co sign of negative Z. And co signs are the same. So co sign of negative Z is just equal to co sign of Z. But then sign of negativity, the Y coordinates of those two points are different. So sign of negative Z is negative signs. So um we get that. But then the eye signs is canceled. And on top we get to cosign Z and on the bottom we get to, so this is equal to cosine Z. Which is definitely equal to the left hand side. So we've done it

So if we have our Z one z one is equal to we're we're told that this C one is equal to are one e times b times I time state a one and rz two is equal to R two times e times I time state or two. So if we multiply these two together we have r one r two times e I ate a one and then e times i data to. So what we're left with is is our one times are two times Mm, I data one plus I data to and so So now we can say since we know that since we know that e to the I data yeah, to the I favor. Or we could just say we could just say we have our one times are two times and then we have e i data. So this gives us a value of this. Gives us co sign of this would be data one plus data to and then now we have Plus, I sign of data one plus data, too. And so we have We have proved we have proved this theory from thrown from oilers identity


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