Question

polar formof the complex number-2-1​

Answer 1

Step-by-step explanation:

The polar form of a complex number is another way to represent a complex number. The form z=a+bi is called the rectangular coordinate form of a complex number.

The horizontal axis is the real axis and the vertical axis is the imaginary axis. We find the real and complex components in terms of r and θ where r is the length of the vector and θ is the angle made with the real axis.

From Pythagorean Theorem :

r2=a2+b2

By using the basic trigonometric ratios :

cosθ=ar and sinθ=br .

Multiplying each side by r :

rcosθ=a and rsinθ=b

The rectangular form of a complex number is given by

z=a+bi .

Substitute the values of a and b .

z=a+bi    =rcosθ+(rsinθ)i    =r(cosθ+isinθ)

In the case of a complex number, r represents the absolute value or modulus and the angle θ is called the argument of the complex number.

This can be summarized as follows:

The polar form of a complex number z=a+bi is z=r(cosθ+isinθ) , where r=|z|=a2+b2−−−−−−√ , a=rcosθ and b=rsinθ , and θ=tan−1(ba) for a>0 and θ=tan−1(ba)+π or θ=tan−1(ba)+180° for a<0 .

Example:

Express the complex number in polar form.

5+2i

The polar form of a complex number z=a+bi is z=r(cosθ+isinθ) .

So, first find the absolute value of r .

r=|z|=a2+b2−−−−−−√    =52+22−−−−−−√    =25+4−−−−−√    =29−−√     ≈5.39

Now find the argument θ .

Since a>0 , use the formula θ=tan−1(ba) .

θ=tan−1(25)    ≈0.38

Note that here θ is measured in radians.

Therefore, the polar form of 5+2i is about 5.39(cos(0.38)+isin(0.38)) .

Answer 2

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