Final answer:
To find the angle that the complex number z = -3 - 6i makes in the complex plane, we calculate the arctan of the imaginary part over the real part and adjust for the negative signs, resulting in an angle of approximately θ = -(5/6)π.
Step-by-step explanation:
The goal is to find the angle θ (in radians) that the complex number z = -3 - 6i makes in the complex plane. To find this angle, we can use the argument of the complex number, which is the arctan of the imaginary part over the real part. However, since the real part is negative and the imaginary part is also negative, z lies in the third quadrant. In the third quadrant, the corresponding positive angle would be θ = 180 degrees + arctan(|y/x|), but since we want the angle in radians and in the range from -π to 0, we should add π to the arctan result to get the negative angle in the third quadrant.
For our complex number:
- Real part, x = -3
- Imaginary part, y = -6
- arctan(y/x) = arctan(6/3) = arctan(2)
arctan(2) is approximately θ = 1.107 radians. To adjust for the third quadrant, we calculate:
θ = π + 1.107 which is approximately 4.249 radians. However, because we seek the angle in the range -π to 0, we subtract 2π to get θ = 4.249 - 2π, which is approximately θ = -1.034. This value is nearest to -(5/6)π (~-1.047), so the correct answer is (c) θ = -(5/6)π.