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Chapter 7: Q8E (page 371)

Use de Moivre鈥檚 formula to express cos(3胃) and sin (3胃) in terms of cos 胃 and sin 胃.

Short Answer

Expert verified

cos3胃=cos2胃-3cos胃sin2胃sin3胃=3cos2胃sin胃-sin3胃

Step by step solution

01

Step 1:The de Moivre formula

The de Moivre formula (also known as de Moivre's theorem and de Moivre's ID) states that in any real number x and the whole number n holds that (cosx+isinx)n=cosnx+isinnx

02

Expressed cos (3θ) and sin(3θ) in terms of cos θ and sin θ:

Let 2鈭�C,|z|=1

Now 鈭�胃<0.2蟺]

z=cos胃+isin胃

By de Moivre's formula, we can see that

z=cos胃+isin胃

However, by direct computation, we can see that

z3=cos3胃-3cos胃sin2胃+i3cos2胃sin胃-sin3胃

We conclude that the solution is

cos3胃=cos2胃-3肠辞蝉胃蝉颈苍2胃sin3胃=3cos2胃蝉颈苍胃-sin3胃

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Most popular questions from this chapter

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

[2-2001-100003-4002-3]

7:For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

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If a vector v鈬赌is an eigenvector of both Aand B, isv鈬赌necessarily an eigenvector of A+B?

Consider ann脳n matrix such that the sum of the entries in each row is . Show that the vector

e鈫�=11..1

InRn is an eigenvector of A. What is the corresponding eigenvalue?

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

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