Question

If \[\dfrac{\pi }{5}\] and \[\dfrac{\pi }{3}\]are the arguments of \[{\bar z_1}\] and\[{z_2}\]then, find the value of arg(\[{z_1}\])+arg(\[{z_2}\])

Answer 1

Hint: The angle inclined from the real axis in the direction of the complex number represented on the complex plane is defined as the argument of the complex number. It is denoted by “θ” or “φ”. It can be measured in the standard unit called “radians”.
The argument function is denoted by arg(z), where z denotes the complex number, that is \[{\text{z }} = {\text{ x}} + {\text{ iy}}\]. The computation of the complex argument can be done by using the following formula:
\[{\text{arg }}\left( {\text{z}} \right){\text{ }} = {\text{ arg }}\left( {{\text{x}} + {\text{iy}}} \right){\text{ }} = {\text{ ta}}{{\text{n}}^{ - {\text{1}}}}\left( {{\text{y}}/{\text{x}}} \right)\]
\[{\text{arg }}\left( {\mathop z\limits^\_ } \right){\text{ }} = {\text{ arg }}\left( {{\text{x - iy}}} \right){\text{ }} = {\text{ ta}}{{\text{n}}^{ - {\text{1}}}}\left( {{\text{ - y}}/{\text{x}}} \right)\]

Complete step by step answer:
It is given that, \[\dfrac{\pi }{3}\] and \[\dfrac{\pi }{5}\]are the arguments of \[{\bar z_1}\] and \[{z_2}\] respectively.
That is \[\arg ({\bar z_1}) = \dfrac{\pi }{5}\] and \[\arg ({z_2}) = \dfrac{\pi }{3}\]
By the definition of arguments we come to a fact that,
arg(\[{\bar z_1}\]) =- arg(\[{z_1}\])
Here the equation is rewritten as
 -arg(\[{\bar z_1}\]) = arg(\[{z_1}\])
Now let us substitute the value of arg(\[{\bar z_1}\]) in the above equation so that, we get,
\[ - \arg ({\bar z_1})\]\[ = - \dfrac{\pi }{5}\]= \[\arg ({z_1})\]
Now, we have \[\arg ({z_2}) = \dfrac{\pi }{3}\] and \[\arg ({z_1}) = - \dfrac{\pi }{5}\] substitute the value in the required expression to be found then, we get,
arg(\[{z_1}\])+arg(\[{z_2}\])\[ = - \dfrac{\pi }{5} + \dfrac{\pi }{3}\]
Let us solve the values to get the final answer,
arg(\[{z_1}\])+arg(\[{z_2}\])\[ = \dfrac{{ - 3\pi + 5\pi }}{{15}} = \dfrac{{2\pi }}{{15}}\]
Hence we have found that the value of arg(\[{z_1}\])+arg(\[{z_2}\]) is given as
arg(\[{z_1}\])+arg(\[{z_2}\])\[ = \dfrac{{2\pi }}{{15}}\].
Note- We can substitute the value \[\pi = \dfrac{{22}}{7}\] in the answer then we get arg(\[{z_1}\])+arg(\[{z_2}\])\[ = \dfrac{2}{{15}} \times \dfrac{{22}}{7} = \dfrac{{44}}{{105}} = 0.419\]
Any real number can be \[r\]written as\[r + i.0\].
Then, \[{\text{arg }}\left( {\text{r}} \right){\text{ }} = {\text{ arg }}\left( {r + i.0} \right){\text{ }} = {\text{ ta}}{{\text{n}}^{ - {\text{1}}}}\left( 0 \right) = {0^ \circ }\]
So, all the real numbers lie on the real line and it is clear that the argument of a real number is always zero.