1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
artcher [175]
3 years ago
11

Express the complex number in trigonometric form -6 + 6sqrt3i

Mathematics
1 answer:
marusya05 [52]3 years ago
4 0
\large\begin{array}{l}\\\\ \textsf{This is a complex number in }\mathsf{a+bi}\textsf{ form:}\\\\ \mathsf{z=-6+6\sqrt{3}\,i}\\\\ \textsf{where }\mathsf{a=-6~~and~~b=6\sqrt{3}.} \end{array}


\large\begin{array}{l}\\\\ \bullet~~\textsf{1}\mathsf{^{st}}\textsf{ step: Find the absolute value of z (complex modulus of z):}\\\\ \mathsf{|z|=\sqrt{a^2+b^2}}\\\\ \mathsf{|z|=\sqrt{(-6)^2+(6\sqrt{3})^2}}\\\\ \mathsf{|z|=\sqrt{36+36\cdot 3}}\\\\ \mathsf{|z|=\sqrt{36\cdot 108}}\\\\ \mathsf{|z|=\sqrt{144}}\\\\ \mathsf{|z|=12} \end{array}


\large\begin{array}{l}\bullet~~\textsf{2}\mathsf{^{nd}}\textsf{ step: Find the argument }\mathsf{\theta}\textsf{ of z:}\\\\ \mathsf{arg(z)=\theta}\\\\ \textsf{is an angle that satisfies the following conditions:}\\\\ \begin{cases} \mathsf{cos\,\theta=\dfrac{a}{|z|}=\dfrac{a}{\sqrt{a^2+b^2}}}\\\\ \mathsf{sin\,\theta=\dfrac{b}{|z|}=\dfrac{b}{\sqrt{a^2+b^2}}} \end{cases}\qquad\quad-\pi


\large\begin{array}{l}\textsf{For }\mathsf{z=-6+6\sqrt{3}\,i,}\\\\ \begin{cases} \mathsf{cos\,\theta=\dfrac{-6}{12}=-\,\dfrac{1}{2}}\\\\ \mathsf{sin\,\theta=\dfrac{6\sqrt{3}}{12}=\dfrac{\sqrt{3}}{2}} \end{cases}\\\\\\ \textsf{Since}\\\\ \mathsf{cos\,\theta0,}\\\\ \theta\textsf{ lies in the 2}\mathsf{^{nd}}\textsf{ quadrant.}\\\\\\\textsf{Therefore,}\\\\ \mathsf{\theta=\dfrac{2\pi}{3}~rad~~(120^\circ)} \end{array}


\large\begin{array}{l}\bullet~~\textsf{3}\mathsf{^{rd}}\textsf{ step: Express z in the trigonometric form:}\\\\ \begin{array}{rcl} \mathsf{z}&\!\!=\!\!&\mathsf{|z|\cdot (cos\,\theta+i\,sin\,\theta)}\\\\ \mathsf{-6+6\sqrt{3}\,i}&\!\!=\!\!&\mathsf{12\cdot \left(cos\, \dfrac{2\pi}{3}+i\,sin\,\dfrac{2\pi}{3}\right)\qquad\quad\checkmark} \end{array} \end{array}


If you're having problems understanding this answer, try seeing it through your browser: brainly.com/question/2154167


\large\textsf{I hope it helps.}


Tags: <em>complex number trigonometric form trig absolute value modulus argument angle</em>

You might be interested in
What is the value of x?<br><br> Enter your answer in the box.<br> (____)
andrew-mc [135]
X = 90 - 64
x = 26 degrees
4 0
2 years ago
Read 2 more answers
Simplify this expression <br> 2/3 (6c+9)−(8c−5)
Ludmilka [50]

Answer:

This is the right answer −4c+11

Step-by-step explanation:

3 0
2 years ago
The graphs below are both absolute value functions. The equation of the redgraph is f(x) = |xl. Which of these is the equation o
grigory [225]

ANSWER

\Rightarrow y=\frac{1}{2}|x|

EXPLANATION

We want to find the absolute value function for the line in blue.

The general form of an absolute value function is:

y=a|x-h|+k

where (h, k) = vertex

From the line, the vertex of the graph in blue is:

(0,0)

To find a, we have to pick a point (x, y) on the line and input it into the general function.

Let us pick (2, 1).

Therefore, we have:

\begin{gathered} 1=a|2-0|+0 \\ 1=a|2|=2a \\ \Rightarrow a=\frac{1}{2} \end{gathered}

Therefore, the absolute value function is:

\begin{gathered} y=\frac{1}{2}|x-0|+0 \\ \Rightarrow y=\frac{1}{2}|x| \end{gathered}

3 0
1 year ago
You earn $9 per hour babysitting. This is $2 more than what you earned per hour last year. What did you earn last year?
slamgirl [31]
$7 is the correct answer for *per hour*, as 9-2=7. Notice how it says how much did you earn last year, implying total earned. How many hours did you work last year?
4 0
3 years ago
Read 2 more answers
Plz help wit number 5 will mark Brainiest
Nana76 [90]
The function is y=2x
this means that the solutions will have the form (x, y)=(x,2x).
For example: (0,0),(1,2),(2,4),(3,6) are solutions.
The first answer choice satisfies the above conditions.
The second answer choice contains (2,2), and (2,6), both of which do not satisfy the (x,2x) condition, so it's not correct.
The third answer choice contains (4,2), which again does not satisfy (x,2x), so not the right answer.

3 0
2 years ago
Other questions:
  • 2.5 times .76 show the work
    9·1 answer
  • Find the product of (2x + 3) and (x + 5)
    10·1 answer
  • It n is odd, which represents the first consecutive even inter after n
    11·1 answer
  • PLEASE HELP! A) 9 B) 8.6 C) 26.3 D) 5.7
    10·1 answer
  • Simplify using a table. <br> (5w+2)(8w+5)
    6·2 answers
  • The question is down bellow
    13·1 answer
  • PLS HELP DO NOT JUST PUT A RANDOM ANSWER PLEASE 15 POINTS!!!
    5·1 answer
  • Please help me with this ​
    10·2 answers
  • Given the following information, calculate the
    8·1 answer
  • The line passes through ​(​8,-7​) and is parallel to the line whose equation is y=2x+2.
    6·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!