Question

How do you write the complex number in trigonometric form \[6-7i\]?

Answer 1

Hint: 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.