Find the argument of the complex number -3sqrt3-9i in the interval 0

Question

Find the argument of the complex number  -3sqrt3-9i in the interval  0 <= theta < 2pi. Express your answer in terms of  pi. Answer:

Bytelearn · Accepted Answer

Calculate atan2 ratio: To find the argument of a complex number in the form a + bi, where a is the real part and b is the imaginary part, we use the formula theta = atan2(b, a). The complex number given is -3sqrt3 - 9i, so a = -3sqrt3 and b = -9.Determine quadrant: First, we calculate the arctangent of the ratio of the imaginary part to the real part using atan2, which takes into account the signs of a and b to determine the correct quadrant for the argument. theta = atan2(-9, -3sqrt3)Adjust angle range: Since both the real and imaginary parts are negative, the complex number lies in the third quadrant. The arctangent function will give us a principal value in the range (-pi, pi), but we need to adjust it to be in the range (0, 2pi) for the final answer.Convert to positive values: To adjust the angle to the correct range, we add pi to the principal value because the angle in the third quadrant is pi radians more than the angle in the first quadrant. theta = atan2(-9, -3sqrt3) + piSimplify arctangent: Using the properties of arctangent, we know that atan2(-9, -3sqrt3) is the same as atan2(9, 3sqrt3), but since we are in the third quadrant, we add pi to the result. theta = atan2(9, 3sqrt3) + piFinalize argument: The arctangent of the ratio 9 / (3sqrt3) simplifies to the arctangent of sqrt3, which is pi/3. theta = pi/3 + piAdding pi to pi/3 gives us 4pi/3, which is the argument of the complex number in the interval 0 <= theta < 2pi. theta = 4pi/3

Find the argument of the complex number $-3 \sqrt{3}-9 i$ in the interval 0 \leq \theta<2 \pi . Express your answer in terms of $\pi$ . $\newline$ Answer:

Full solution

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

Related topics

Find the argument of the complex number −33−9i -3 \sqrt{3}-9 i −33​−9i in the interval 0 \leq \theta<2 \pi . Express your answer in terms of π \pi π.\newlineAnswer:

Find the argument of the complex number $-3 \sqrt{3}-9 i$ in the interval 0 \leq \theta<2 \pi . Express your answer in terms of $\pi$ . $\newline$ Answer: