Answer:
Explanation:
(a) Using de Moivre's Theorem, we have that:
z^n = cos(nθ) + i sin(nθ)
1/z^n = cos(-nθ) + i sin(-nθ)
So, we can find z^n + 1/z^n as:
z^n + 1/z^n = (cos(nθ) + i sin(nθ)) + (cos(-nθ) + i sin(-nθ)) = 2 cos(nθ)
And z^n - 1/z^n as:
z^n - 1/z^n = (cos(nθ) + i sin(nθ)) - (cos(-nθ) + i sin(-nθ)) = 2i sin(nθ)
(b) To express the equation 2z^4 - 5z^3 + 7z^2 - 5z + 2 = 0 in terms of cos(θ), we can substitute z with cos(θ) + i sin(θ):
2(cos(4θ) + i sin(4θ)) - 5(cos(3θ) + i sin(3θ)) + 7(cos(2θ) + i sin(2θ)) - 5(cos(θ) + i sin(θ)) + 2 = 0
Expanding the right hand side using the trigonometric identities, we get:
2 cos(4θ) - 5 cos(3θ) + 7 cos(2θ) - 5 cos(θ) + 2 = 0
Note that the real and imaginary parts are separate, so we can simplify this equation to a real equation by equating the real and imaginary parts to 0.
(c) To solve the equation 2 cos(4θ) - 5 cos(3θ) + 7 cos(2θ) - 5 cos(θ) + 2 = 0, we can use numerical methods or analytical methods such as the roots of unity.
Once the roots are found, they can be represented on an Argand diagram by plotting the real and imaginary parts of each root as a point in the complex plane. The magnitude of each point represents the magnitude of the complex number, and the argument (angle) represents the argument of the complex number.