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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+3i z=4+3 i \newlineFind the angle θ \theta (in radians) that z z makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express θ \theta between −π -\pi and π \pi .\newlineθ= \theta=

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Q. z=4+3i z=4+3 i \newlineFind the angle θ \theta (in radians) that z z makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express θ \theta between −π -\pi and π \pi .\newlineθ= \theta=
  1. Identify Complex Number: Identify the real and imaginary parts of the complex number z=4+3iz = 4 + 3i.\newlineReal part (Re) = 44\newlineImaginary part (Im) = 33
  2. Find Angle Theta: Use the arctangent function to find the angle θ\theta, since tan⁡(θ)=ImRe\tan(\theta) = \frac{\text{Im}}{\text{Re}}.
    θ=arctan⁡(ImRe)\theta = \arctan\left(\frac{\text{Im}}{\text{Re}}\right)
    θ=arctan⁡(34)\theta = \arctan\left(\frac{3}{4}\right)
  3. Calculate Theta Value: Calculate the value of theta using a calculator. \newlineθ=arctan(34)\theta = \text{arctan}(\frac{3}{4})\newlineθ≈0.643\theta \approx 0.643 (rounded to the nearest thousandth)
  4. Check Theta Range: Check the range of θ\theta to ensure it is between −π-\pi and π\pi. Since 0 < \theta < \frac{\pi}{2}, it is within the required range.

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