Question

In Exercises 69–76, find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex fifth roots of 32 (cos 5π/3 + i sin 5π/3)

Accepted Answer

Recognize that the problem asks for the fifth roots of a complex number given in polar (trigonometric) form: $$32 \left( \cos \frac{5\pi}{3} + i \sin \frac{5\pi}{3} ight). $$The general form for the $$nth $$roots of a complex number $$r (\cos 	heta + i \sin 	heta) is $$given by De Moivre's Theorem. Identify the magnitude $$r$$ and the argument $$	heta of $$the complex number. Here, $$r = 32$$ and $$	heta = \frac{5\pi}{3}. $$The fifth roots will have magnitude $$r^{1/5}$$ and arguments given by $$\frac{	heta + 2k\pi}{5}$$ for $$k = 0, 1, 2, 3, 4. $$Calculate the magnitude of each root as $$\sqrt[5]{32}$$, which simplifies to $$2$$ because $$2^5 = 32$. Then, write the argument for each root as $$	heta_k = \frac{5\pi/3 + 2k\pi}{5}$$ for k = 0$$, 1, 2, 3, 4$$. Express each root in trigonometric form as 2$$ \left( \cos 	heta_k + i \sin 	heta_k ight)$$, substituting each $$	heta_k$$ value. This gives the five distinct roots evenly spaced around the circle in the complex plane. Convert each root from trigonometric form to rectangular form using the formulas x = r$$ \cos 	heta_k$$ and y = r$$ \sin 	heta_k$$, where x$ is the real part and y$ is the imaginary part. Round each value to the nearest tenth if necessary.

In Exercises 69–76, find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex fifth roots of 32 (cos 5π/3 + i sin 5π/3)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Complex Numbers in Polar Form

De Moivre's Theorem

Conversion Between Polar and Rectangular Forms