To find the fifth root of the given expression 243cos(300°) + i sin(300°), we first express it in a simpler form. 243 can be rewritten as 3^5, and the angle 300° corresponds to the point in the complex plane.
Using Euler’s formula, we can convert the expression into polar form:
The angle 300° is equivalent to 300° – 360° = -60°. This can be expressed as:
r(cos θ + i sin θ) = 243 (cos 300° + i sin 300°) = 243 (cos(-60°) + i sin(-60°))Calculating the cosine and sine values, we find that:
- cos(-60°) = 1/2
- sin(-60°) = -√3/2
Thus, our expression becomes:
243 (1/2 - i (√3/2)) = 243/2 - i(243√3/2)Now we compute the fifth root. The magnitude |z| of this complex number is:
|z| = 243 = 3^5
So, the fifth root of |z| is 3.
Next, we need to determine the angle. Since we are taking the fifth root, we divide the angle by 5. Since angles in the complex plane can be represented as:
θ = -60° + 360°k (where k = 0, 1, 2, 3, 4)This gives us five angles for the roots:
- k=0: θ = -60°/5 = -12°
- k=1: θ = (-60° + 360°) / 5 = 60°
- k=2: θ = (-60° + 720°) / 5 = 132°
- k=3: θ = (-60° + 1080°) / 5 = 204°
- k=4: θ = (-60° + 1440°) / 5 = 276°
So the fifth roots are:
3 (cos(-12°) + i sin(-12°))3 (cos(60°) + i sin(60°))3 (cos(132°) + i sin(132°))3 (cos(204°) + i sin(204°))3 (cos(276°) + i sin(276°))In conclusion, the fifth roots of the given expression are :
- 3 (cos(-12°) + i sin(-12°))
- 3 (cos(60°) + i sin(60°))
- 3 (cos(132°) + i sin(132°))
- 3 (cos(204°) + i sin(204°))
- 3 (cos(276°) + i sin(276°))