Question

How do you write $-3+4i$ in trigonometric form?

Answer 1

Hint: For converting a complex number in the trigonometric form, we need two values; the modulus of the complex number from the origin, and the argument. The distance can be calculated by using the formula $r=\sqrt{{{a}^{2}}+{{b}^{2}}}$ and the argument is given by $\tan \theta =\dfrac{b}{a}$. Finally, on substituting the modulus and the argument into the standard trigonometric form $z=r\left( \cos \theta +i\sin \theta \right)$, we will obtain the trigonometric form of the given complex number.

Complete step by step solution:
The complex number given to us in the above question is
$\Rightarrow z=-3+4i........\left( i \right)$
We know that the trigonometric form of a complex number has two components, the modulus, and the argument. The modulus of a complex number $a+ib$ is given by
$\Rightarrow r=\sqrt{{{a}^{2}}+{{b}^{2}}}.......\left( ii \right)$
And the argument is given by
\[\Rightarrow \tan \theta =\dfrac{b}{a}........\left( iii \right)\]
From (i) we can see that in this case we have $a=-3$ and $b=4$. So from (ii) the modulus of the complex number is given by
$\begin{align}
  & \Rightarrow r=\sqrt{{{\left( -3 \right)}^{2}}+{{\left( 4 \right)}^{2}}} \\
 & \Rightarrow r=\sqrt{9+16} \\
 & \Rightarrow r=\sqrt{25} \\
 & \Rightarrow r=5.......\left( iv \right) \\
\end{align}$
And from (iii) the argument of the complex number is given by
$\Rightarrow \tan \theta =\dfrac{4}{-3}$
On solving the above equation, we get
$\begin{align}
  & \Rightarrow \theta ={{180}^{\circ }}-{{53}^{\circ }} \\
 & \Rightarrow \theta ={{127}^{\circ }}.......\left( v \right) \\
\end{align}$
Now, we know that the trigonometric form of a complex number is given by
$\Rightarrow z=r\left( \cos \theta +i\sin \theta \right)$
On substituting the equations (iv) and (v) in the above expression, we finally get
\[\Rightarrow z=5\left( \cos {{127}^{\circ }}+i\sin {{127}^{\circ }} \right)\]

Hence, we have finally obtained the trigonometric form of the given complex number as \[5\left( \cos {{127}^{\circ }}+i\sin {{127}^{\circ }} \right)\].

Note: The trigonometric form of a complex number is also referred to as the polar form of the complex number. The polar form of a complex number can also be written as $z=r{{e}^{i\theta }}$, where $\theta $ must be in radians. The value of $\theta $ must be chosen such that it lies in the same quadrant as that of the complex number. In this case, a was negative and b was positive, which means that the given complex number was in the second quadrant, and so is the angle \[\theta ={{127}^{\circ }}\].