Question

How do you find the trigonometric form of $z = 8 - 8i$

Answer 1

Hint : Here in this question we have to convert the complex number into the trigonometric number or form. As we know that $x + iy = r[\cos \theta + i\sin \theta ]$, hence we have to determine the value of r and the value of theta and hence by substituting we obtain the required solution for the given question.

Complete step-by-step answer :
The complex number can be converted into a trigonometric form.
Now consider the given equation $z = 8 - 8i$, generally the complex number is expressed as $z = x + iy$, so the value of x is 8 and the value of -8. When we consider the graph for this point the point will lie in the 4th quadrant.
We have formula for the writing the complex number into trigonometric form that is
$x + iy = r[\cos \theta + i\sin \theta ]$, the formula for r is given by $r = \sqrt {{x^2} + {y^2}}$ and we also know the formula $\tan \theta = \dfrac{y}{x}$
Now let we find the value of $r$ and $\theta$
Therefore, we have
$r = \sqrt {{x^2} + {y^2}}$
On substituting the value of x and y we get
$\Rightarrow r = \sqrt {{{(8)}^2} + {{( - 8)}^2}}$
On simplifying we get
$\Rightarrow r = \sqrt {64 + 64}$
$\Rightarrow r = \sqrt {2 \times 64}$
64 is a perfect square and the square root of 64 is 8.
therefore the value of r is
$\Rightarrow r = 8\sqrt 2$
now let we find the value of $\theta$
$\tan \theta = \dfrac{y}{x}$
On substituting the value of y and the value of x we get
$\Rightarrow \tan \theta = \dfrac{{ - 8}}{8}$
on simplifying we get
$\Rightarrow \tan \theta = - 1$
We know that the tangent trigonometry ratio is negative in the 2nd quadrant and the 4th quadrant. By the given data the points lie in the 4th quadrant. therefore the value of $\theta$ is $\theta = \dfrac{{3\pi }}{2} + \dfrac{\pi }{4} = \dfrac{{7\pi }}{4}$
Hence the value of $r = 8\sqrt 2$ and $\theta = \dfrac{{7\pi }}{4}$
Therefore $8 - i8 = 8\sqrt 2 \left[ {\cos \left( {\dfrac{{7\pi }}{4}} \right) + i\sin \left( {\dfrac{{7\pi }}{4}} \right)} \right]$
Hence we have converted the complex number into trigonometric form.
So, the correct answer is “ $8 - i8 = 8\sqrt 2 \left[ {\cos \left( {\dfrac{{7\pi }}{4}} \right) + i\sin \left( {\dfrac{{7\pi }}{4}} \right)} \right]$”.

Note : To convert the complex number into a trigonometric function or form we must know 3 formulas i.e., $x + iy = r[\cos \theta + i\sin \theta ]$, $r = \sqrt {{x^2} + {y^2}}$ and $\tan \theta = \dfrac{y}{x}$. We must know about the table of trigonometry ratios for standard angles and about ASTC rule.