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1) Use Eulerβs formula to derive a formula for cos of four π in terms of cos π.
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2) Use Eulerβs formula to drive a formula for sin of four π in terms of cos π and sin π.
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For part one, weβll use the properties of the exponential function.
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And weβll write π to the four ππ as π to the ππ to the power of four.
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And now, we can use Eulerβs formula.
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And we write the left-hand side as cos four π plus π sin four π.
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And on the right-hand side, we can say that this is equal to cos π plus π sin π all to the power of four.
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Now, weβre going to apply the binomial theorem to distribute cos π plus π sin π to the power of four.
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In our equation, π is equal to cos of π, π is equal to π sin of π, and π is the power; itβs four.
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And this means we can say that cos π plus π sin π to the power of four is the same as cos π to the power of four plus four choose one cos cubed π times π sin π and so on.
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We know that four choose one is four, four choose two is six, and four choose three is also four.
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We also know that π squared is negative one, π cubed is negative π, and π to the power of four is one.
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And we can further rewrite our equation as shown.
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Now, weβre going to equate the real parts of this equation.
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And that will give us a formula for cos of four π in terms of cos π and sin π.
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Letβs clear some space.
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The real part on the left-hand side is cos four π.
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And then on the right-hand side, we have cos π to the power of four.
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Weβve got negative cos squared π sin squared π.
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And weβve got sin π to the power of four.
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So we equate these.
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But weβre not quite finished.
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We were asked to derive a formula for cos four π in terms of cos π only.
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So here, we use the identity cos squared π plus sin squared π is equal to one.
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And we rearrange this.
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And we say that well, that means that sin squared π must be equal to one minus cos squared π.
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And we can rewrite this as cos π to the power of four plus six cos squared π times one minus cos squared π plus one minus cos squared π squared.
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We distribute the parentheses.
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And our final step is to collect like terms.
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And we see that weβve derived the formula for cos of four π in terms of cos π.
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cos four π is equal to eight cos π to the power of four minus eight cos squared π plus one.
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For part two, we can repeat this process equating the imaginary parts.
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They are sin of four π on the left.
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And then on the right, we have four cos cubed π sin π, negative four cos π sin cubed π.
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And we see this sin four π must be equal to four cos cubed π sin π minus four cos π sin cubed π.
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And we could β if we so wish β factor four cos π sin π.
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And weβll be left with four cos π sin π times cos squared π minus sin squared π.
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And last, weβve been asked to derive a formula for sin four π in terms of cos π and sin π.
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You might now see a link between sin four π and the double angle formulae.