William Boggs

Answered

2022-01-16

a) Given:

$z}_{1}=6\mathrm{\angle}{30}^{\circ$ and ${z}_{2}=4+5i$

What is the resultant of the given complex numbers? In rectangular form.

b) Convert i into polar form

What is the resultant of the given complex numbers? In rectangular form.

b) Convert i into polar form

Answer & Explanation

kalfswors0m

Expert

2022-01-17Added 24 answers

Step 1

a) Given:$z}_{1}=6\mathrm{\angle}{30}^{\circ$ and ${z}_{2}=4+5i$

First express$z}_{1$ in rectangular form.

$z}_{1}=6\mathrm{\angle}{30}^{\circ$ implies magnitude of ${z}_{1}=6$ and argument of $z}_{1}={30}^{\circ$

Then,

$z}_{1}=6\mathrm{\angle}{30}^{\circ$

$=6\left({\mathrm{cos}30}^{\circ}+i{\mathrm{sin}30}^{\circ}\right)$

$=6(\frac{\sqrt{3}}{2}+\frac{i}{2})$

$=3\sqrt{3}+3i$

Step 2

Now find the resultant of the given complex numbers as follows.

${z}_{1}+{z}_{2}=3\sqrt{3}+3i+4+5i$

$=(4+3\sqrt{3})+8i$

Therefore, the resultant of the given complex numbers is$(4+3\sqrt{3})+8i$

a) Given:

First express

Then,

Step 2

Now find the resultant of the given complex numbers as follows.

Therefore, the resultant of the given complex numbers is

Philip Williams

Expert

2022-01-18Added 39 answers

Step 1

b) Convert to i into polar form as shown below.

Compare the complex number$z=i$ with $z=x+iy$ and obtain $x=0$ and $y=1$ .

$r=\sqrt{{x}^{2}+{y}^{2}}$

$=\sqrt{0+1}$

$=1$

and,

$\theta ={\mathrm{tan}}^{-1}\left(\frac{y}{x}\right)$

$={\mathrm{tan}}^{-1}\left(\frac{1}{0}\right)$

$=\frac{\pi}{2}$

Step 2

Then the polar form of i is,

$r(\mathrm{cos}\theta +i\mathrm{sin}\theta )=1\{\mathrm{cos}\left(\frac{\pi}{2}\right)+i\mathrm{sin}\left(\frac{\pi}{2}\right)\}$

$=1\{\mathrm{cos}(\frac{\pi}{2}-2\pi )+i\mathrm{sin}(\frac{\pi}{2}-2\pi )\}$

(Since$\mathrm{cos}(x-2\pi )=\mathrm{cos}x$ and $\mathrm{sin}(x-2\pi )=\mathrm{sin}x)$

$=1\{\mathrm{cos}(-\frac{3\pi}{2})+i\mathrm{sin}(-\frac{3\pi}{2})\}$

Therefore, the polar form of i is$\{\{\mathrm{cos}(-\frac{3\pi}{2})+i\mathrm{sin}(-\frac{3\pi}{2})\}$

b) Convert to i into polar form as shown below.

Compare the complex number

and,

Step 2

Then the polar form of i is,

(Since

Therefore, the polar form of i is

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