# Complex Variables : Complex Variables : De Moivre's Theorem and Rectangular Coordinates Theorem and Rectangular Coordinates

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5. Use de Moivre's formula to derive the following trigonometric identities.

(a) cos 3Î¸ = cos3 Î¸ - 3cos Î¸Ö¼sin2 Î¸

(b) sin 3Î¸ = 3cos2 Î¸Ö¼sin Î¸ - sin3 Î¸

6. By writing the individual factors on the left in exponential form, performing the needed operations, and finally changing back to rectangular coordinates, show that

(a) i (1 - sqrt (3)Ö¼i) (sqrt (3) + i) = 2 (1+ sqrt (3)Ö¼i)

(b) 5i / (2 + i) = 1 + 2i

(c) (-1 + i)7 = -8 (1 + i)

(d) (1 + sqrt (3)Ö¼i)-10 = 2-11(-1 + sqrt (3)Ö¼i)

*sqrt means square root.

See the attached file.

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#### Solution Summary

De Moivre's Theorem and rectangular coordinates are investigated. The solution is detailed and well presented.

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