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

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

Step by step solution

01

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: \(z = -1 + i\)
02

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

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