Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

z=-2-5i
Find the angle
theta (in radians) that
z makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
theta between
-pi and
pi.

theta=

$z=-2-5 i$ $\newline$ Find the angle $\theta$ (in radians) that $z$ makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express $\theta$ between $-\pi$ and $\pi$ . $\newline$ $\theta=$

View step-by-step help

View step-by-step help

Find trigonometric ratios using a Pythagorean or reciprocal identity

Full solution

Q.

z=-2-5 i

\newline

Find the angle

\theta

(in radians) that

z

makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express

\theta

between

-\pi

and

\pi

.

\newline

\theta=

Calculate Argument of z: To find the angle $\theta$ , we need to calculate the argument of the complex number $z=-2-5i$ . The argument is the angle the complex number makes with the positive x-axis in the complex plane.
Determine Quadrant for Angle: The argument of a complex number $a+bi$ is given by $\arctan(\frac{b}{a})$ , but since our $a$ is negative and $b$ is negative, the angle will be in the third quadrant. We need to add $\pi$ to the $\arctan$ value to get the correct angle in the range of $-\pi$ to $\pi$ .
Calculate Arctan Value: First, calculate the arctan value: $\arctan(\frac{5}{2})$ . This is the angle in the first quadrant, but we need the angle in the third quadrant.
Adjust Angle for Third Quadrant: Using a calculator, we find $\arctan(\frac{5}{2}) \approx 1.190$ .
Add Pi to Arctan Value: Since the angle is in the third quadrant, we add $\pi$ to get the angle in the correct range: $1.190 + \pi$ .
Calculate Final Angle: Adding $1.190$ to $\pi$ gives us approximately $1.190 + 3.142 = 4.332$ .
Ensure Angle Range: However, we need to ensure the angle is between $-\pi$ and $\pi$ . Since $4.332$ is greater than $\pi$ , we subtract $2\pi$ to get the angle in the correct range.
Final Angle Calculation: Subtracting $2\pi$ from $4.332$ gives us $4.332 - 2\times3.142 = -1.952$ .

More problems from Find trigonometric ratios using a Pythagorean or reciprocal identity

Question

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

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

\sec(\theta)

? Write your answer in simplified, rationalized form.

\sec(\theta)=

Posted 3 months ago

Question

\cos(x) = -0.3

\sin(90^\circ - x)

Posted 3 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 3 months ago

Question

Is the sine function even or odd?

\newline

\newline

\text{[A]even}

\newline

\text{[B]odd}

Posted 3 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 2 months 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 4 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 4 months ago

Question

h^2 - 4h = 0

Posted 2 months 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 2 months ago