Chapter 9: Q. 39 (page 592)

Findzwand $\frac{z}{w}$ . Leave the answers in polar form.
role="math" localid="1646661399572" $z = 2 + 2 i w = \sqrt{3} - i$

Short Answer

Expert verified

zwin polar form is $4 \sqrt{2} (\cos 15 ° + i \sin 15 °)$ .

$\frac{z}{w}$ in polar form is $\sqrt{2} (\cos 75 ° + i \sin 75 °)$ .

Step by step solution

Step 1. Given information.

Consider the given question,

$z = 2 + 2 i w = \sqrt{3} - i$

The multiplication of the polar form of complex numbers is given by,

$z_{1} z_{2} = r_{1} r_{2} [\cos (θ_{1} + θ_{2}) + i \sin (θ_{1} + θ_{2})] . . . . . . (i) \frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}} [\cos (θ_{1} - θ_{2}) + i \sin (θ_{1} - θ_{2})] . . . . . . (i i)$

We need to write the given numbers in polar form.

$r = \sqrt{x^{2} + y^{2}}, θ = \tan^{- 1} (\frac{y}{x})$

Step 2. Find the polar form of w.

Determine the polar form of z,

$r_{z} = \sqrt{2^{2} + 2^{2}} r_{z} = 2 \sqrt{2} θ_{z} = \tan^{- 1} (\frac{2}{2}) θ_{z} = 45 °$

Therefore, $z = 2 \sqrt{2} (\cos 45 ° + i \sin 45 °)$ .

Determine the polar form of w,

$r_{w} = \sqrt{{(\sqrt{3})}^{2} + {(- 1)}^{2}} r_{w} = 2 θ_{w} = \tan^{- 1} (\frac{- 1}{\sqrt{3}}) θ_{w} = - 30 °$

Therefore, $w = 2 (\cos (- 30 °) + i \sin (- 30 °))$

Step 3. Find zw and zw.

Substitute the values of z,w in equation (i),

$z \cdot w = 2 \sqrt{2} [\cos (45 ° + i \sin 45 °)] \cdot 2 [\cos (- 30 °) + i \sin (- 30 °)] z \cdot w = (2) 2 \sqrt{2} [\cos (45 ° + (- 30 °)) + i \sin (45 ° + (- 30 °))] z \cdot w = 4 \sqrt{2} (\cos 15 ° + i \sin 15 °)$

Substitute the values of z,w in equation (ii),

$\frac{z}{w} = \frac{2 \sqrt{2} (\cos 45 ° + i \sin 45 °)}{2 (\cos (- 30 °) + i \sin (- 30 °))} \frac{z}{w} = \frac{2 \sqrt{2}}{2} [\cos (45 ° - (- 30 °)) + i \sin (45 ° - (- 30 °))] \frac{z}{w} = \sqrt{2} (\cos 75 ° + i \sin 75 °)$

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Findzwand $\frac{z}{w}$ . Leave the answers in polar form.
role="math" localid="1646661399572" $z = 2 + 2 i w = \sqrt{3} - i$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the polar form of w.

Step 3. Find zw and zw.

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Findzwand zw. Leave the answers in polar form.role="math" localid="1646661399572" z=2+2iw=3-i

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the polar form of w.

Step 3. Find zw and zw.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Probability and Statistics

Mechanics Maths

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Findzwand $\frac{z}{w}$ . Leave the answers in polar form.
role="math" localid="1646661399572" $z = 2 + 2 i w = \sqrt{3} - i$