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z=7+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=7+3i z=7+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=7+3i z=7+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 Complex Number: Represent the complex number in the form z=a+biz = a + bi, where aa is the real part and bb is the imaginary part.\newlineFor z=7+3iz = 7 + 3i, we have a=7a = 7 and b=3b = 3.
  2. Calculate Angle: Calculate the angle θ\theta using the arctangent function, which gives the angle in radians. The formula to find the angle is θ=arctan(ba)\theta = \text{arctan}(\frac{b}{a}).\newlineθ=arctan(37)\theta = \text{arctan}(\frac{3}{7})
  3. Convert to Degrees: Use a calculator to find the value of θ\theta in radians and then convert it to degrees.θ=arctan(37)≈arctan(0.4286)≈0.4189\theta = \text{arctan}(\frac{3}{7}) \approx \text{arctan}(0.4286) \approx 0.4189 radians To convert radians to degrees, multiply by 180π\frac{180}{\pi}.θ≈0.4189×(180π)≈24.0\theta \approx 0.4189 \times (\frac{180}{\pi}) \approx 24.0 degrees
  4. Check Quadrant: Since the real part a=7a = 7 is positive and the imaginary part b=3b = 3 is also positive, the angle θ\theta is in the first quadrant. Therefore, no further adjustments are needed for the angle.
  5. Express in Range: Express θ\theta between −180-180 degrees and 180180 degrees. Since the angle is already in the first quadrant and less than 9090 degrees, it is within the required range.

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