Question

If \(z=\left[(1+7 i) /(2-i)^{2}\right]\) then the polar form of \(z\) is (a) \(\sqrt{2}[\cos (3 \pi / 4)+\mathrm{i} \sin (3 \pi / 4)]\) (b) \(\sqrt{2}[\cos (\pi / 4)+\mathrm{i} \sin (\pi / 4)]\) (c) \(\sqrt{2}[\cos (7 \pi / 4)+\mathrm{i} \sin (7 \pi / 4)]\) (d) \(\sqrt{2}[\cos (5 \pi / 4)+\mathrm{i} \sin (5 \pi / 4)]\)

Short answer

The short answer is: \(z = \sqrt{2}[\cos(3 \pi / 4)+\mathrm{i} \sin (3 \pi / 4)]\).

Step-by-step solution

Simplify the given complex number

To simplify the complex number, we first compute the square of (2-i) and then divide (1+7i) by the result. So, the square of (2-i) is: \((2-i)^2 = (2-i)(2-i) = 2(2-i) - i(2-i) = 4 - 2i - 2i + i^2 = 4 - 4i - 1 = 3 - 4i\) Now, let's divide (1+7i) by (3-4i): \(z = \frac{(1+7i)}{(3-4i)} = \frac{(1+7i)(3+4i)}{(3-4i)(3+4i)}\) Now multiply the numerators and denominators: Numerator: \((1+7i)(3+4i) = 3 + 4i + 21i + 28i^2 = 3 + 25i - 28\) Denominator: \((3-4i)(3+4i) = 3(3+4i) - 4i(3+4i) = 9 +12i -12i -16i^2 = 9 -16(-1) = 9+16\) So the complex number z is: \(z = \frac{-25 + 25i}{25}\) Simplified form of z is: \(z = -1 + i\)

Convert the complex number into polar form

To convert the complex number into polar form, we first determine its magnitude and angle. Magnitude of z: \(|z| = \sqrt{(-1)^2 + 1^2} = \sqrt{2}\) Angle of z (theta): \(\theta = \tan^{-1}(\frac{Im(z)}{Re(z)}) = \tan^{-1}(\frac{1}{-1}) = \tan^{-1}(-1) = \frac{3\pi}{4}\) (as the angle is in the second quadrant) Now, write the polar form of z using the magnitude and angle: \(z = \sqrt{2}[\cos(\frac{3\pi}{4})+i\sin(\frac{3\pi}{4})]\) Now compare this result with the given options, it matches option (a). Thus, the polar form of the given complex number is: \(z = \sqrt{2}[\cos(3 \pi / 4)+\mathrm{i} \sin (3 \pi / 4)]\)

Key concepts

Polar Form

Complex numbers can be expressed in several forms, and the polar form is one of the most useful. It involves using the magnitude and angle (argument) of a complex number to write it as:
\[ z = r \, \text{cis} \, \theta \]
Where \(r\) is the magnitude and \(\theta\) is the argument. The "cis" notation is shorthand for \(\cos(\theta) + i \sin(\theta)\).

• The polar form provides an intuitive understanding of complex numbers' geometric representation.
• In this form, the complex number's magnitude tells us how "large" the number is, while the argument tells us its direction from the positive real axis.

This form is especially advantageous when multiplying or dividing complex numbers, as it simplifies the computation by separating magnitude and angle.

Magnitude of a Complex Number

The magnitude (or modulus) of a complex number, denoted as \(|z|\), measures its distance from the origin in the complex plane. For a complex number of the form \(a+bi\), the magnitude is calculated using the formula:
\[ |z| = \sqrt{a^2 + b^2} \]

• To find the magnitude, square both the real part \(a\) and the imaginary part \(b\), add them together, and then take the square root of the sum.
• It represents the "length" of the vector in the complex plane.

In our original problem, we found the magnitude of \(-1 + i\) to be \(\sqrt{2}\). This tells us the complex number is \(\sqrt{2}\) units away from the origin.

Argument of a Complex Number

The argument of a complex number is the angle that the line representing the number makes with the positive real axis in the complex plane. It is usually expressed in radians. For a complex number \(a+bi\), the argument \(\theta\) can be found with:
\[ \theta = \tan^{-1} \left( \frac{b}{a} \right) \]

• The argument requires considering the signs of \(a\) and \(b\) to determine the correct quadrant.
• Typically, angles are adjusted to fit within the interval \((0, 2\pi)\).

In this example, the complex number \(-1+i\) has an argument of \(\frac{3\pi}{4}\), as it lies in the second quadrant.

Complex Number Division

Dividing complex numbers might seem tricky due to their multiple components, but it becomes much easier when considering them in polar form. For two complex numbers \(z_1 = r_1 \text{cis} \theta_1\) and \(z_2 = r_2 \text{cis} \theta_2\), the division is:
\[ \frac{z_1}{z_2} = \frac{r_1}{r_2} \text{cis} (\theta_1 - \theta_2) \]

• This method of division involves simply dividing the magnitudes \(r_1\) by \(r_2\) and subtracting the arguments \(\theta_1 - \theta_2\).
• This approach leverages the properties of exponential functions, making calculations straightforward.

Through our exercise, the division involved the simplification of complex numbers \((1+7i)\) by \((3-4i)\), ultimately helping find the polar form by using division properties and simplifying magnitude and argument computations.