z=1+4i Find the angle theta (in degrees) that z makes in the complex plane. Round your answer, if necessary, to the nearest tenth. Express theta between -180^(@) and 180^(@). theta=◻" 。 "

Identify Complex Number: Identify the real and imaginary parts of the complex number z = 1 + 4i. Real part (Re) = 1 Imaginary part (Im) = 4Calculate Angle in Radians: Calculate the angle theta using the arctangent function, which gives the angle in radians. theta = arctan(Im/Re) theta = arctan(4/1) theta = arctan(4)Convert Angle to Degrees: Convert the angle from radians to degrees using the conversion factor 180/pi. theta (in degrees) = arctan(4) * (180/pi) Use a calculator to find the value of arctan(4) in degrees. theta (in degrees) ≈ 76.0°Check Angle Range: Check if the angle theta is within the specified range of -180° to 180°. Since 76.0° is within the range, no further adjustments are needed.

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Let’s check out your problem:

z=1+4i
Find the angle
theta (in degrees) that
z makes in the complex plane.
Round your answer, if necessary, to the nearest tenth. Express
theta between
-180^(@) and
180^(@).

theta=◻" 。 "

$z=1+4 i$ $\newline$ Find the angle $\theta$ (in degrees) that $z$ makes in the complex plane. $\newline$ Round your answer, if necessary, to the nearest tenth. Express $\theta$ between $-180^{\circ}$ and $180^{\circ}$ . $\newline$ $\theta=\square^{\circ}$

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Full solution

z=1+4 i

\newline

Find the angle

\theta

(in degrees) that

z

makes in the complex plane.

\newline

Round your answer, if necessary, to the nearest tenth. Express

\theta

between

-180^{\circ}

and

180^{\circ}

\newline

\theta=\square^{\circ}

Identify Complex Number: Identify the real and imaginary parts of the complex number $z = 1 + 4i$ . $\newline$ Real part (Re) = $1$ $\newline$ Imaginary part (Im) = $4$
Calculate Angle in Radians: Calculate the angle $\theta$ using the arctangent function, which gives the angle in radians. $\theta = \arctan(\frac{\text{Im}}{\text{Re}})$ $\theta = \arctan(\frac{4}{1})$ $\theta = \arctan(4)$
Convert Angle to Degrees: Convert the angle from radians to degrees using the conversion factor $180/\pi$ . $\newline$ $\theta$ (in degrees) = $\arctan(4) \times (180/\pi)$ $\newline$ Use a calculator to find the value of $\arctan(4)$ in degrees. $\newline$ $\theta$ (in degrees) $\approx 76.0^\circ$
Check Angle Range: Check if the angle $\theta$ is within the specified range of $-180°$ to $180°$ . Since $76.0°$ is within the range, no further adjustments are needed.