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z=4+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=

$z=4+3 i$ $\newline$ 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=4+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 Complex Number: Identify the real and imaginary parts of the complex number $z = 4 + 3i$ . $\newline$ Real part (Re) = $4$ $\newline$ Imaginary part (Im) = $3$
Find Angle Theta: Use the arctangent function to find the angle $\theta$ , since $\tan(\theta) = \frac{\text{Im}}{\text{Re}}$ .
$\theta = \arctan\left(\frac{\text{Im}}{\text{Re}}\right)$
$\theta = \arctan\left(\frac{3}{4}\right)$
Calculate Theta Value: Calculate the value of theta using a calculator. $\newline$ $\theta = \text{arctan}(\frac{3}{4})$ $\newline$ $\theta \approx 0.643$ (rounded to the nearest thousandth)
Check Theta Range: Check the range of $\theta$ to ensure it is between $-\pi$ and $\pi$ . Since $0 < \theta < \frac{\pi}{2}$ , it is within the required range.

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