To find the fifth root of the complex number given by 243cos(260°) + i sin(260°), we can first express the number in polar form. The general form of a complex number is represented as r(cos θ + i sin θ), where r is the modulus and θ is the argument.
Here, we see that the modulus r is 243, and the argument θ is 260°. To find the fifth root, we apply De Moivre’s Theorem. This theorem states that the nth root of a complex number can be found by taking the nth root of the modulus and dividing the argument by n. In our case, n = 5.
First, calculate the modulus:
- Fifth root of 243: ∛(243) = 3. (Since 35 = 243)
Next, calculate the argument:
- Argument after division: 260° / 5 = 52°.
Now we write the fifth root in polar form:
3(cos(52°) + i sin(52°))
Finally, we can express this in rectangular form using the values of cosine and sine:
- cos(52°) ≈ 0.6157
- sin(52°) ≈ 0.7880
So, we have:
3(0.6157 + 0.7880i) ≈ 1.8471 + 2.3640i
Therefore, the fifth root of 243cos(260°) + i sin(260°) is approximately 1.8471 + 2.3640i.