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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+4i z=1+4 i \newlineFind the angle θ \theta (in degrees) that z z makes in the complex plane.\newlineRound your answer, if necessary, to the nearest tenth. Express θ \theta between −180∘ -180^{\circ} and 180∘ 180^{\circ} .\newlineθ= \theta= \newline□ \square

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Full solution

Q. z=1+4i z=1+4 i \newlineFind the angle θ \theta (in degrees) that z z makes in the complex plane.\newlineRound your answer, if necessary, to the nearest tenth. Express θ \theta between −180∘ -180^{\circ} and 180∘ 180^{\circ} .\newlineθ= \theta= \newline□ \square
  1. Calculate arctan⁡(4)\arctan(4): Now we need to calculate arctan⁡(4)\arctan(4) using a calculator or a trigonometry table.\newlineCalculation: θ≈arctan⁡(4)≈76\theta \approx \arctan(4) \approx 76 degrees.
  2. Check angle range: We need to check if the angle is in the correct range between −180-180 degrees and 180180 degrees.\newlineSince 7676 degrees is within the range, we don't need to adjust the angle.
  3. Round to nearest tenth: Finally, we round the angle to the nearest tenth, if necessary.\newlineCalculation: θ≈76.0\theta \approx 76.0 degrees.

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