Suppose z is a complex number such that z^5 = -\sqrt{3} + i and z = \sqrt[5]{2} ( \cos n^{\circ} + i \sin n^{\circ}) , where 270 < n < 360 . Find the exact value of n .
To find the exact value of n, we need to compare the given expression for z with the polar form of a complex number.
The polar form of a complex number z is given by z = r (cosθ + i sinθ), where r is the magnitude of z and θ is the argument of z.
From the given expression for z = √[5]{2} (cos n° + i sin n°), we can see that the magnitude of z is √[5]{2}.
Now, let's compare this with the given expression for z^5:
z^5 = -(√3 + i)
To compare these two expressions, we equate the magnitudes and arguments separately.
Magnitude:
√[5]{2} = |-(√3 + i)| = √(√3^2 + 1^2) = √(3 + 1) = 2
From this, we can see that r = 2.
Argument:
cos (5n°) + i sin (5n°) = -(√3 + i) / √[5]{2}
To simplify the right side of the equation, we can multiply the numerator and denominator by the conjugate of the denominator:
cos (5n°) + i sin (5n°) = -[(√3 + i) / √[5]{2}] * [√[5]{2} / √[5]{2}]
cos (5n°) + i sin (5n°) = -(√3√[5]{2} + i√[5]{2}) / 2
Now, comparing the real and imaginary parts separately:
Real part:
cos (5n°) = -√3√[5]{2}/2
Imaginary part:
sin (5n°) = -√[5]{2}/2
To find the value of n, we need to determine the angle whose cosine is -√3√[5]{2}/2 and whose sine is -√[5]{2}/2 in the given range of 270° < n < 360°.
Using a calculator or trigonometric identities, we find the principal angle for which these conditions are satisfied:
Principal angle for cosine: cos^(-1)(-√3√[5]{2}/2) ≈ 150.98°
Principal angle for sine: sin^(-1)(-√[5]{2}/2) ≈ 333.43°
However, we need to ensure that the angle lies in the range of 270° < n < 360°. Since the principal angle for sine (333.43°) satisfies this condition, the exact value of n is:
n = 333.43°