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z=8-9i
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=

$z=8-9 i$ $\newline$ 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 $\pi$ . $\newline$ $\theta=$

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Find trigonometric ratios using a Pythagorean or reciprocal identity

Full solution

Q.

z=8-9 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=

Identify Complex Number: Identify the real and imaginary parts of the complex number $z$ . $z = 8 - 9i$ , where $8$ is the real part and $-9$ is the imaginary part.
Calculate Angle Theta: Calculate the angle theta using the arctangent function. The angle theta in the complex plane is given by the arctangent of the imaginary part divided by the real part, which is $\arctan\left(\frac{-9}{8}\right)$ .
Find Value of Theta: Use a calculator to find the value of theta. $\theta = \arctan\left(-\frac{9}{8}\right) \approx -0.844$ radians (rounded to the nearest thousandth).
Adjust Theta Range: Adjust the angle $\theta$ to be within the range $-\pi$ to $\pi$ . Since -\pi < \theta < \pi and $-0.844$ is already within this range, no adjustment is necessary.

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