What is (7-7i) in polar form

Mathematics

2 answers:

KonstantinChe [14]2 years ago

7 0

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 $\boxed{z = 7\sqrt 2 \left( {\cos \left( {\frac{{7\pi }}{4}} \right) - i\sin \left( {\frac{{7\pi }}{4}} \right)} \right)}.$