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z=1-2i 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=1−2i z=1-2 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 = \square ^{\circ}

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Q. z=1−2i z=1-2 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 = \square ^{\circ}
  1. Calculate Argument: To find the angle θ\theta that the complex number z=1−2iz = 1 - 2i makes in the complex plane, we need to calculate the argument of the complex number. The argument is the angle formed by the real axis and the line segment representing the complex number in the complex plane. We can use the arctangent function to find this angle, which is given by θ=atan2(imaginary part,real part)\theta = \text{atan2}(\text{imaginary part}, \text{real part}).
  2. Identify Real and Imaginary Parts: First, identify the real part xx and the imaginary part yy of the complex number z=1−2iz = 1 - 2i. Here, the real part xx is 11, and the imaginary part yy is −2-2.
  3. Use Arctangent Function: Now, use the arctangent function with the two parts. The function atan2(y,x)\text{atan2}(y, x) takes into account the signs of both xx and yy to determine the correct quadrant for the angle. Calculate θ=atan2(−2,1)\theta = \text{atan2}(-2, 1).
  4. Convert Radians to Degrees: Perform the calculation using a calculator or a software tool that can compute atan2\text{atan2}. The result will be in radians, so we need to convert it to degrees by multiplying by 180π\frac{180}{\pi}.
  5. Final Result: After performing the calculation, we find that θ≈atan2(−2,1)×(180π)≈−63.4\theta \approx \text{atan2}(-2, 1) \times (\frac{180}{\pi}) \approx -63.4 degrees. This is the angle that zz makes in the complex plane, and it is already expressed between −180-180 degrees and 180180 degrees.

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