Final answer:
To write the complex number -3-3i√3 in trigonometric form, its magnitude is calculated to be 6 and its angle to be 4π/3. Thus, the trigonometric form is 6(cos(4π/3) + i sin(4π/3)).
Step-by-step explanation:
To write the complex number -3-3i√3 in trigonometric form, we need to find its magnitude and angle.
The magnitude (r) of a complex number a + bi is given by r = √(a² + b²).
Similarly, the angle (θ) can be found using the formula θ = arctan(b/a), where arctan signifies the inverse tangent function.
For the complex number -3-3i√3, we have a = -3 and b = -3√3. Calculating the magnitude:
- r = √((-3)² + (-3√3)²) = √(9 + 27) = √36 = 6
Calculating the angle, since both a and b are negative, θ is in the third quadrant:
- θ = arctan((-3√3)/(-3)) = arctan(√3) = π + π/3 (since in the third quadrant we add π to the principal value)
Now we can write the complex number in trigonometric form as:
r(cosθ + i sinθ) = 6(cos(4π/3) + i sin(4π/3))