Question

How do you write the complex number in trigonometric form $6-7i$?

Answer 1

From the given question we are asked to convert the given complex number into a trigonometric number. For this question we will use the concept of trigonometry in complex numbers. we will use the formulae $r\left( \cos \theta +i\sin \theta \right)$ and explain its parameters and its conditions for the value of $\theta$ it takes and solve the given question. So, we proceed with the solution as follows.

Complete step by step answer:
To convert the complex number into trigonometry generally the formulae which is used will be as follows.
$\Rightarrow x+iy=r\left( \cos \theta +i\sin \theta \right)$
Here the term $r$ and the condition of the $\theta$ value will be as follows.
$\Rightarrow r=\sqrt{{{x}^{2}}+{{y}^{2}}}$
$\Rightarrow \theta ={{\tan }^{-1}}\left( \dfrac{y}{x} \right);-\pi <\theta \le \pi$
From the question we know that, given a complex number is $6-7i$.
After comparing the given complex number with the formulae, we get,
$\Rightarrow x=6,y=-7$.
So, the value of $r$ will be as follows.
$\Rightarrow r=\sqrt{{{x}^{2}}+{{y}^{2}}}$
$\Rightarrow r=\sqrt{{{6}^{2}}+{{\left( -7 \right)}^{2}}}$
$\Rightarrow r=\sqrt{36+49}$
$\Rightarrow r=\sqrt{85}$
$6-7i$ is in the fourth quadrant so we must ensure that $\theta$ is in the fourth quadrant.
$\Rightarrow \theta ={{\tan }^{-1}}\left( \dfrac{7}{6} \right)=0.862$
So, $\Rightarrow \theta =-0.862$ is in the fourth quadrant.
So, we got the values of the $r$ and the $\theta$ value as $\Rightarrow r=\sqrt{85}$ and $\Rightarrow \theta =-0.862$ respectively.
So, now we will use the substitution method and substitute these values in the formulae.
So, we get the equation reduced as follows.
$\Rightarrow x+iy=r\left( \cos \theta +i\sin \theta \right)$
$\Rightarrow 6-7i=\sqrt{85}\left( \cos \left( -0.862 \right)+i\sin \left( -0.862 \right) \right)$
We know that $\cos (-\theta )=\cos \theta$ and $\sin (-\theta )=-\sin \theta$
$\Rightarrow 6-7i=\sqrt{85}\left( \cos \left( 0.862 \right)-i\sin \left( 0.862 \right) \right)$

Note: Students must not do any calculation mistakes. Students must know the concept of trigonometry and complex numbers along with their applications. We must know the formulae like,
$\Rightarrow x+iy=r\left( \cos \theta +i\sin \theta \right)$
$\Rightarrow r=\sqrt{{{x}^{2}}+{{y}^{2}}}$
$\Rightarrow \theta ={{\tan }^{-1}}\left( \dfrac{y}{x} \right);-\pi <\theta \le \pi$,$\cos (-\theta )=\cos \theta$ and $\sin (-\theta )=-\sin \theta$ to solve the question.