Question

How to add $1 + 7i$ and $- 4 - 2i$ and convert it to trigonometric form?

Answer 1

In this question we have to find the trigonometric form of the complex number when they are added. First to add the complex numbers first add the real part together and then add the imaginary part together, then we will get a complex number which is to be converted to trigonometric form. If $a + ib$ is a complex number given as,$z = r\left( {\cos \theta + i\sin \theta } \right)$, where, $r = \left| {a + ib} \right|$, is the modulus of $z$, i.e., $r = \left| {a + ib} \right| = \sqrt {{a^2} + {b^2}}$, and $\tan \theta = \dfrac{b}{a}$, where $\theta$ is called the argument of $z$, now finding the values and substituting the values we will get the required trigonometric form of the complex number.

Complete step-by-step solution:
Given complex numbers are $1 + 7i$ and $- 4 - 2i$, now we have to add then complex numbers,
To add the complex numbers first we have to add/subtract the real part of the complex number then add/subtract the imaginary part of the complex number.
Now adding the complex numbers we get,
$\Rightarrow 1 + 7i + \left( { - 4 - 2i} \right)$,
Now grouping the real part of the complex numbers and grouping the imaginary part of the complex numbers, we get,
$\Rightarrow \left( {1 - 4} \right) + i\left( {7 - 2} \right)$,
Now simplifying we get,
$\Rightarrow - 3 + 5i$,
So now we have to convert this complex number into trigonometric form,
The trigonometric form of the complex number $a + ib$ is given as, $z = r\left( {\cos \theta + i\sin \theta } \right)$, where, $r = \left| {a + ib} \right|$,is the modulus of $z$, i.e., $r = \left| {a + ib} \right| = \sqrt {{a^2} + {b^2}}$, and $\tan \theta = \dfrac{b}{a}$, where $\theta$ is called the argument of $z$, we require $0 \leqslant \theta < 2\pi$,
So here, $a = - 3$ and $b = 5$, now substituting the values we get,
$\Rightarrow r = \sqrt {{{\left( { - 3} \right)}^2} + {5^2}}$,
Now simplifying we get,
$\Rightarrow r = \sqrt {9 + 25}$,
Now adding the values in the square root we get,
$\Rightarrow r = \sqrt {34}$,
Now we know that $\tan \theta = \dfrac{b}{a}$, substituting the values we get,
$\Rightarrow \tan \theta = \dfrac{5}{{ - 3}}$,
Now simplifying we get,
$\Rightarrow \tan \theta = - \dfrac{5}{3}$,
Now taking tan to the other side we get,
$\Rightarrow \theta = {\tan ^{ - 1}}\left( { - \dfrac{5}{3}} \right)$,
Now finding the value of ${\tan ^{ - 1}}\left( { - \dfrac{5}{3}} \right)$ which is equal to $2.1112$, now substituting the values in the trigonometric form of the complex number i.e.,$z = r\left( {\cos \theta + i\sin \theta } \right)$, we get,
$\Rightarrow z = \sqrt {34} \left( {\cos \left( {2.1112} \right) + i\sin \left( {2.1112} \right)} \right)$.
The trigonometric form is equal to $z = \sqrt {34} \left( {\cos \left( {2.1112} \right) + i\sin \left( {2.1112} \right)} \right)$.

$\therefore$ The trigonometric form when $1 + 7i$ and $- 4 - 2i$ are added is equal to $z = \sqrt {34} \left( {\cos \left( {2.1112} \right) + i\sin \left( {2.1112} \right)} \right)$.

Note: The trigonometric form of a complex number is also called the polar form. Because there are infinitely many choices for $\theta$, the trigonometric form of a complex number is not unique. Normally, $\theta$ is restricted to the interval $0 \leqslant \theta < 2\pi$, although on occasion it is convenient to use $\theta < 0$.