Question

Z= -8 - 5i find the angle θtheta (in degrees) that z makes in the complex plane.

a) -63.43 degrees
b) -126.87 degrees
c) -53.13 degrees
d) -116.57 degrees

Answer 1

The angle θ theta that z = -8 - 5i makes in the complex plane is approximately -116.57 degrees when measured from the positive real axis in the complex plane. Option D is correct.

Step-by-step explanation:

To find the angle θ (theta) that the complex number z = -8 - 5i makes in the complex plane, we can use the argument of a complex number, which is the angle the number line makes with the positive real axis in the complex plane. The argument (θ) can be found using the formula θ = atan2(y, x), where x is the real part and y is the imaginary part of the complex number.

For the complex number z = -8 - 5i, x = -8 and y = -5. Plugging these into the formula:

θ = atan2(-5, -8)

Using a calculator, we find:

θ ≈ -148.41°

However, angles in the complex plane are typically measured from the positive real axis in the counter-clockwise direction. So, to convert this to the conventional measurement, we add 360° to our result:

θ ≈ -148.41° + 360°

θ ≈ 211.59°

But we need the angle in degrees between -180° and 180°. Thus, we can also represent θ as -116.57 (since 211.59° is coterminal with -116.57