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=◻" 。 "

Question

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=◻" 。 "

Bytelearn · Accepted Answer

Calculate Argument using Arctangent: To find the angle theta that the complex number z = 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 the formula theta = atan2(imaginary part, real part).Identify Real and Imaginary Parts: First, identify the real part (x) and the imaginary part (y) of the complex number z = 1 - 2i. Here, the real part x is 1, and the imaginary part y is -2.Use atan2 Function: Now, use the atan2 function to find the angle in radians. The atan2 function takes into account the signs of both the real and imaginary parts to determine the correct quadrant for the angle. The formula is theta = atan2(y, x), so we have theta = atan2(-2, 1).Calculate Angle in Radians: Calculate the angle using a calculator or a programming language that has the atan2 function. The angle in radians is theta = atan2(-2, 1).Convert Radians to Degrees: Convert the angle from radians to degrees, since the question asks for the angle in degrees. To convert radians to degrees, multiply the angle in radians by 180/pi. If we assume the angle in radians from the previous step is correct, the conversion to degrees would be theta_degrees = theta_radians * (180/pi).Adjust Angle if Necessary: After calculating the angle in degrees, make sure that it falls within the specified range of -180 degrees to 180 degrees. If the calculated angle is outside this range, adjust it by adding or subtracting 360 degrees until it falls within the range.Perform Calculation for Degrees: Perform the calculation to find the angle in degrees. Assuming the angle in radians is correct, we would have theta_degrees = atan2(-2, 1) * (180/pi). Using a calculator, we find that theta_degrees ≈ -63.4 degrees.Check Angle Range: Check if the calculated angle is within the range of -180 degrees to 180 degrees. Since -63.4 degrees is within this range, no further adjustments are needed.

$z=1-2 i$ $\newline$ Find the angle $\theta$ (in degrees) that $z$ makes in the complex plane. $\newline$ Round your answer, if necessary, to the nearest tenth. Express $\theta$ between $-180^{\circ}$ and $180^{\circ}$ . $\newline$ $\theta=\square^{\circ}$

Full solution

More problems from Find trigonometric ratios using a Pythagorean or reciprocal identity

Related topics