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z=6+5i 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=6+5i z=6+5 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=6+5i z=6+5 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 zz.z=6+5iz = 6 + 5i, where the real part is 66 and the imaginary part is 55.
  2. Calculate Angle Theta: Calculate the angle theta using the arctangent function.\newlineThe angle theta in the complex plane is given by the arctangent of the imaginary part over the real part.\newlineθ=arctan⁡(imaginary partreal part)\theta = \arctan(\frac{\text{imaginary part}}{\text{real part}})\newlineθ=arctan⁡(56)\theta = \arctan(\frac{5}{6})
  3. Find Theta in Radians: Use a calculator to find the value of θ\theta in radians.θ=arctan(56)≈arctan(0.8333)\theta = \text{arctan}(\frac{5}{6}) \approx \text{arctan}(0.8333)Using a calculator, we find that:θ≈0.694\theta \approx 0.694 radians
  4. Check Angle Range: Check if the calculated angle θ\theta is within the specified range. The range specified is between −π-\pi and π\pi. Since 0.6940.694 is within this range, we do not need to adjust the angle.
  5. Round to Nearest Thousandth: Round the answer to the nearest thousandth, if necessary.\newlineθ≈0.694\theta \approx 0.694 radians (rounded to the nearest thousandth)

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