Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the argument of the complex number
3+sqrt3i in the interval
0 <= theta < 2pi.
Express your answer in terms of
pi.
Answer:

Find the argument of the complex number $3+\sqrt{3} i$ in the interval $0 \leq \theta<2 \pi$ . $\newline$ Express your answer in terms of $\pi$ . $\newline$ Answer:

View step-by-step help

View step-by-step help

Find trigonometric ratios using multiple identities

Full solution

Q. Find the argument of the complex number

3+\sqrt{3} i

in the interval

0 \leq \theta<2 \pi

.

\newline

Express your answer in terms of

\pi

.

\newline

Answer:

Calculate Angle $\theta$ : To find the argument of the complex number $3+\sqrt{3}i$ , we need to calculate the angle $\theta$ that the line connecting the origin to the point $(3, \sqrt{3})$ makes with the positive x-axis in the complex plane. The argument is given by $\theta = \arctan(\frac{\text{imaginary part}}{\text{real part}})$ .
Find Arctan: Calculate the $\arctan$ of the imaginary part divided by the real part: $\arctan(\sqrt{3}/3)$ .
Identify Triangle Angle: Recognize that $\arctan(\sqrt{3}/3)$ corresponds to the angle whose tangent is $\sqrt{3}/3$ . This is a well-known angle in a $30$ - $60$ - $90$ right triangle, where the angle opposite the side with length $\sqrt{3}$ is $60$ degrees or $\pi/3$ radians.
Determine Final Argument: Since the complex number $3+\sqrt{3}i$ is in the first quadrant (both real and imaginary parts are positive), the argument $\theta$ is simply $\pi/3$ .

More problems from Find trigonometric ratios using multiple identities

Question

\cos(\theta)=\frac{8}{17}

0^\circ<\theta<90^\circ

\sec(\theta)

? Write your answer in simplified, rationalized form.

\sec(\theta)=

Posted 2 months ago

Question

\cos(x) = -0.3

\sin(90^\circ - x)

Posted 2 months ago

Question

\csc(\theta) = \frac{17}{8}

0^\circ < \theta < 90^\circ

\tan(\theta)

\newline

Write your answer in simplified, rationalized form.

\newline

\tan(\theta) =

\newline

Posted 2 months ago

Question

Is the sine function even or odd?

\newline

\newline

\text{[A]even}

\newline

\text{[B]odd}

Posted 2 months ago

Question

Write the sentence as an equation.

\newline

89

is equal to the quotient of

y

17

\newline

/

) if you want to use a division sign.

\newline

Posted 1 month ago

Question

Write the sentence as an equation.

\newline

45

is equal to the quotient of

z

5

\newline

/

) if you want to use a division sign.

\newline

Posted 3 months ago

Question

h

\newline

h^2 + 24h = 0

\newline

Write each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas.

\newline

h =

Posted 3 months ago

Question

h^2 - 4h = 0

Posted 1 month ago

Question

Combine the like terms to create an equivalent expression:

\newline

-k-(-8k)=\square

Posted 4 months ago

Question

Which of the following degree measures is equal to

5 \pi

\newline

(The number of degrees of arc in a circle is

360

. The number of radians of arc in a circle is

2 \pi

\newline

1

\newline

144^{\circ}

\newline

900^{\circ}

\newline

1,080^{\circ}

\newline

1,800^{\circ}

Posted 1 month ago