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z=-5+3i 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=−5+3i z=-5+3 i \newlineFind the angle θ \theta (in degrees) that z z makes in the complex plane. Round 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=−5+3i z=-5+3 i \newlineFind the angle θ \theta (in degrees) that z z makes in the complex plane. Round your answer, if necessary, to the nearest tenth. Express θ \theta between −180∘ -180^{\circ} and 180∘ 180^{\circ} .\newlineθ=□∘ \theta=\square^{\circ}
  1. Identify parts of zz: Identify the real and imaginary parts of the complex number zz.z=−5+3iz = -5 + 3i, where the real part is −5-5 and the imaginary part is 33.
  2. Calculate argument of z: Calculate the argument of z, which is the angle θ\theta in the complex plane.\newlineThe argument of z is given by θ=arctan⁡(imaginary partreal part)=arctan⁡(3−5)\theta = \arctan(\frac{\text{imaginary part}}{\text{real part}}) = \arctan(\frac{3}{-5}).
  3. Find theta in radians: Use a calculator to find the value of theta in radians. θ=arctan(3−5)≈arctan(−0.6)\theta = \text{arctan}(\frac{3}{-5}) \approx \text{arctan}(-0.6).
  4. Convert angle to degrees: Convert the angle from radians to degrees. θ≈arctan⁡(−0.6)×(180/π)\theta \approx \arctan(-0.6) \times (180 / \pi) degrees.
  5. Ensure angle range: Calculate the angle in degrees and ensure it is within the specified range of −180∘-180^\circ to 180∘180^\circ.\newlineSince the complex number is in the second quadrant (real part is negative, imaginary part is positive), we add 180∘180^\circ to the calculated angle to get the angle in the correct range.\newlineθ≈arctan⁡(−0.6)×(180/π)+180\theta \approx \arctan(-0.6) \times (180 / \pi) + 180 degrees.
  6. Find final theta in degrees: Use a calculator to find the final value of theta in degrees. θ≈arctan⁡(−0.6)×(180/π)+180≈−30.96+180≈149.04\theta \approx \arctan(-0.6) \times (180 / \pi) + 180 \approx -30.96 + 180 \approx 149.04 degrees.

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