
How do you write the complex number $ - 2i$ in polar form?
Answer
444.9k+ views
Hint: We are given a complex number and we have to write it in the polar form. For that, we must know the process of converting a complex number into polar form. Complex numbers are of the form $a \pm ib$ , where $a$ is the real part and $ib$ is the imaginary part of the complex number. $r\cos \theta + ir\sin \theta $ is the polar form of the complex number $a + ib$ where $r = \sqrt {{a^2} + {b^2}} $ , $\cos \theta = \dfrac{a}{r}$ and $\sin \theta = \dfrac{b}{r}$ . Using these formulas, we can convert the complex number into polar form.
Complete step-by-step solution:
The complex number given to us is $ - 2i$ , comparing it to the standard form of the complex number $a + ib$ , we get –
$a = 0$ and $b = - 2$
We know that the polar form of $a + ib$ is $r\cos \theta + if\sin \theta $ -
$
\Rightarrow r = \sqrt {0 + 4} = 2 \\
\Rightarrow \cos \theta = \dfrac{0}{2} = 0 \\
\Rightarrow \sin \theta = \dfrac{{ - 2}}{2} = - 1 \\
$
We know that
$
\Rightarrow \cos \dfrac{{3\pi }}{2} = 0 \\
\Rightarrow \cos \theta = \cos \dfrac{{3\pi }}{2} \\
$
And
$
\Rightarrow \sin \dfrac{{3\pi }}{2} = - 1 \\
\Rightarrow \sin \theta = \sin \dfrac{{3\pi }}{2} \\
$
Hence, the polar form of $ - 2i$ is $2(\cos \dfrac{{3\pi }}{2} + i\sin \dfrac{{3\pi }}{2})$ .
Note: The complex numbers and real numbers are the two types of numbers. A number that can be shown on the number line is known as a real number, and the numbers that cannot be shown on the number line is known as a complex number. We cannot find the square root of negative numbers, so we have supposed $\sqrt { - 1} $ to be iota $(i)$ , this way $\sqrt { - n} $ is written as $\sqrt n i$ . The complex number $a + ib$ is written as $(r,\theta )$. Thus the polar form of $ - 2i$ is written as $(2,\dfrac{{3\pi }}{2})$ . Converting complex numbers into a polar form is used for their expression on the graph paper, so, with a similar approach, we can convert any complex number into polar form.
Complete step-by-step solution:
The complex number given to us is $ - 2i$ , comparing it to the standard form of the complex number $a + ib$ , we get –
$a = 0$ and $b = - 2$
We know that the polar form of $a + ib$ is $r\cos \theta + if\sin \theta $ -
$
\Rightarrow r = \sqrt {0 + 4} = 2 \\
\Rightarrow \cos \theta = \dfrac{0}{2} = 0 \\
\Rightarrow \sin \theta = \dfrac{{ - 2}}{2} = - 1 \\
$
We know that
$
\Rightarrow \cos \dfrac{{3\pi }}{2} = 0 \\
\Rightarrow \cos \theta = \cos \dfrac{{3\pi }}{2} \\
$
And
$
\Rightarrow \sin \dfrac{{3\pi }}{2} = - 1 \\
\Rightarrow \sin \theta = \sin \dfrac{{3\pi }}{2} \\
$
Hence, the polar form of $ - 2i$ is $2(\cos \dfrac{{3\pi }}{2} + i\sin \dfrac{{3\pi }}{2})$ .
Note: The complex numbers and real numbers are the two types of numbers. A number that can be shown on the number line is known as a real number, and the numbers that cannot be shown on the number line is known as a complex number. We cannot find the square root of negative numbers, so we have supposed $\sqrt { - 1} $ to be iota $(i)$ , this way $\sqrt { - n} $ is written as $\sqrt n i$ . The complex number $a + ib$ is written as $(r,\theta )$. Thus the polar form of $ - 2i$ is written as $(2,\dfrac{{3\pi }}{2})$ . Converting complex numbers into a polar form is used for their expression on the graph paper, so, with a similar approach, we can convert any complex number into polar form.
Recently Updated Pages
Master Class 11 Accountancy: Engaging Questions & Answers for Success

What percentage of the area in India is covered by class 10 social science CBSE

The area of a 6m wide road outside a garden in all class 10 maths CBSE

What is the electric flux through a cube of side 1 class 10 physics CBSE

If one root of x2 x k 0 maybe the square of the other class 10 maths CBSE

The radius and height of a cylinder are in the ratio class 10 maths CBSE

Trending doubts
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Why is there a time difference of about 5 hours between class 10 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What constitutes the central nervous system How are class 10 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Explain the Treaty of Vienna of 1815 class 10 social science CBSE
