Multiple Choice
Express the complex number in the polar form.
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17. Graphing Complex Numbers
Polar Form of Complex Numbers
Multiple Choice
Convert the complex number from polar to rectangular form.
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Verified step by step guidance1
Identify the given complex number in polar form: z = \(\sqrt{2}\) \(\left\)( \(\cos\) \(\frac{7\pi}{4}\) + i \(\sin\) \(\frac{7\pi}{4}\) \(\right\)).
Recall that the polar form of a complex number is given by z = r (\(\cos\) \(\theta\) + i \(\sin\) \(\theta\)), where r is the magnitude and \(\theta\) is the angle.
Convert the angle \(\frac{7\pi}{4}\) to its equivalent in the unit circle. Note that \(\frac{7\pi}{4}\) is in the fourth quadrant, where cosine is positive and sine is negative.
Calculate the rectangular form using the formulas: x = r \(\cos\) \(\theta\) and y = r \(\sin\) \(\theta\). Here, x and y are the real and imaginary parts, respectively.
Substitute r = \(\sqrt{2}\), \(\cos\) \(\frac{7\pi}{4}\) = \(\frac{1}{\sqrt{2}\)}, and \(\sin\) \(\frac{7\pi}{4}\) = -\(\frac{1}{\sqrt{2}\)} into the formulas to find the rectangular form: z = x + yi.
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