Convert the following complex number into
Polar form complex number.
(i)
(ii)

Ask by Lee Daniel. in India

Dec 21,2024

Question

Convert the following complex number into Polar form complex number. (i) $$ z=1-\sqrt{3} i $$ (ii) $$ z+2+2 i $$

UpStudy AI · Accepted Answer

To convert the given complex numbers into polar form, we'll express each in the form:
$$
z = r \left( \cos \theta + i \sin \theta \right)
$$
where:
- $$ r $$ is the modulus (or magnitude) of the complex number.
- $$ \theta $$ is the argument (or angle) of the complex number.

Let's convert each complex number step by step.

---

### (i) $$ z = 1 - \sqrt{3}i $$

**1. Calculate the modulus $$ r $$:**
$$
r = \sqrt{x^2 + y^2} = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2
$$

**2. Determine the argument $$ \theta $$:**
$$
\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{-\sqrt{3}}{1}\right) = \tan^{-1}(-\sqrt{3})
$$
$$
\theta = -60^\circ \quad \text{or} \quad 300^\circ \quad \text{(since the complex number is in the fourth quadrant)}
$$
$$
\theta = -\frac{\pi}{3} \quad \text{or} \quad \frac{5\pi}{3} \quad \text{radians}
$$

**3. Express in polar form:**
$$
z = 2 \left( \cos 300^\circ + i \sin 300^\circ \right) \quad \text{or} \quad 2 \left( \cos \frac{5\pi}{3} + i \sin \frac{5\pi}{3} \right)
$$

---

### (ii) $$ z = 2 + 2i $$

**1. Calculate the modulus $$ r $$:**
$$
r = \sqrt{x^2 + y^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
$$

**2. Determine the argument $$ \theta $$:**
$$
\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{2}{2}\right) = \tan^{-1}(1)
$$
$$
\theta = 45^\circ \quad \text{or} \quad \frac{\pi}{4} \quad \text{radians}
$$

**3. Express in polar form:**
$$
z = 2\sqrt{2} \left( \cos 45^\circ + i \sin 45^\circ \right) \quad \text{or} \quad 2\sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)
$$

---

**Final Summary:**

1. **For $$ z = 1 - \sqrt{3}i $$:**
$$
z = 2 \left( \cos 300^\circ + i \sin 300^\circ \right) \quad \text{or} \quad 2 \left( \cos \frac{5\pi}{3} + i \sin \frac{5\pi}{3} \right)
$$

2. **For $$ z = 2 + 2i $$:**
$$
z = 2\sqrt{2} \left( \cos 45^\circ + i \sin 45^\circ \right) \quad \text{or} \quad 2\sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)
$$

These are the polar forms of the given complex numbers.
$$ z = 1 - \sqrt{3}i $$ in polar form is $$ 2 \left( \cos 300^\circ + i \sin 300^\circ \right) $$ or $$ 2 \left( \cos \frac{5\pi}{3} + i \sin \frac{5\pi}{3} \right) $$. $$ z = 2 + 2i $$ in polar form is $$ 2\sqrt{2} \left( \cos 45^\circ + i \sin 45^\circ \right) $$ or $$ 2\sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right) $$.

Convert the following complex number into
Polar form complex number.
(i)
(ii)

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Convert the following complex number into Polar form complex number. (i) (ii)