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z=-8+3i
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=

$z=-8+3 i$ $\newline$ 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+3 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 parts of $z$ : Identify the real and imaginary parts of the complex number $z$ . $z = -8 + 3i$ , where the real part is $-8$ and the imaginary part is $3$ .
Calculate angle theta: Calculate the angle $\theta$ using the arctangent function. $\newline$ The angle $\theta$ in the complex plane is given by the arctangent of the imaginary part divided by the real part, which is $\text{arctan}(\frac{3}{-8})$ . However, since the real part is negative and the imaginary part is positive, the angle lies in the second quadrant.
Use arctangent function: Use a calculator to find the arctangent of $\frac{3}{-8}$ . $\newline$ $\theta = \arctan\left(\frac{3}{-8}\right) \approx \arctan(-0.375)$
Calculate theta in radians: Calculate the approximate value of theta in radians. $\theta \approx -0.359$ (in radians, using a calculator)
Adjust angle in quadrant: Adjust the angle to lie in the correct quadrant. $\newline$ Since the angle is in the second quadrant, we need to add $\pi$ to the calculated value to get the angle between $-\pi$ and $\pi$ . $\newline$ $\theta = -0.359 + \pi$
Calculate final theta: Calculate the final value of theta. $\newline$ $\theta \approx -0.359 + 3.142$ (using the approximation $\pi \approx 3.142$ ) $\newline$ $\theta \approx 2.783$ (in radians)

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