The polar form of the complex number \(7 - 7i\) is \(\boxed{z = 7\sqrt 2 \left( {\cos \left({\frac{{7\pi }}{4}} \right) - i\sin \left( {\frac{{7\pi }}{4}} \right)} \right)}.\)
Further explanation:
The complex number is \(z = a + ib.\)
The polar form of the complex number can be expressed as follows,
\(\boxed{z = r\left( {\cos {{\theta }} + i\sin {{\theta }}} \right)}\)
Here, r is the modulus of the complex number and \({{\theta }}\) is the argument or angle of complex number.
Explanation:
The given complex number is \(z = 7 - 7i.\)
The value of a is 7 and the value of b is \(-7.\)
The value of r can be obtained as follows,
\(\begin{aligned}{r^2}&= {a^2} + {b^2}\\{r^2}&= {7^2} + {7^2}\\{r^2} &= 49 + 49\\r &= \sqrt {98}\\ r &= 7\sqrt 2\\\end{aligned}\)
The angle can be obtained as follows,
\(\begin{aligned}\tan {\theta }}&=\Dfrac{{\cos {{\theta }}}}{{\sin {\theta }}}}\\ &=\Dfrac{{\frac{a}{r}}}{{\Dfrac{b}{r}}}\\&= \Dfrac{{\Dfrac{7}{{7\sqrt 2 }}}}{{ - \Dfrac{7}{{7\sqrt 2 }}}} \\&= - 1 \\ \end{aligned}\)
The angle lies in the fourth quadrant as value of \(\sin \theta\) is negative and value of \(\cos \thet\) a is positive.
The angle can be obtained as follows,
\(\begin{aligned}{\theta }}&= 2\pi- \frac{\pi }{4}\\&=\frac{{7\pi }}{4}\\\end{aligned}\)
The polar form of the complex number \(7 - 7i\) is \(\boxed{z = 7\sqrt 2 \left( {\cos \left( {\frac{{7\pi }}{4}} \right) - i\sin \left( {\frac{{7\pi }}{4}} \right)} \right)}.\)
Kindly refer to the image attached.
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Answer details:
Grade: Middle School
Subject: Mathematics
Chapter: Arithmetic Sequence
Keywords: complex numbers, imaginary roots, polar form, 7-7i, general form, argument, coordinate.
