Find the roots of z^6 + 1 and hence resolve z^6 + 1into read quadratic factors; deduce that
cos3x = 4[cos(x) -cos(pi/6)][(cos(x) -cos(pi/2)][(cos(x) -cos(5pi/6)]
To find the roots of the equation z^6 + 1 = 0, we can use the fact that any complex number can be written in polar form as z = r * e^(iθ), where r is the magnitude and θ is the argument of the complex number.
Let's consider the equation z^6 + 1 = 0:
z^6 = -1
Now, we can write -1 in polar form as e^(iπ).
So, we have:
z^6 = e^(iπ)
To find the roots, we need to find the 6th roots of e^(iπ). We can do this by considering the equation:
z^6 = e^(iπ)
We can write e^(iπ) as e^(iπ/6) * e^(iπ/6) * e^(iπ/6) * e^(iπ/6) * e^(iπ/6) * e^(iπ/6)
Now, let's simplify each term:
e^(iπ/6) = cos(π/6) + i * sin(π/6)
e^(iπ/3) = cos(π/3) + i * sin(π/3)
e^(iπ/2) = cos(π/2) + i * sin(π/2)
Using these values, we can find the 6th roots of e^(iπ):
z1 = e^(iπ/6) = cos(π/6) + i * sin(π/6)
z2 = e^(iπ/3) = cos(π/3) + i * sin(π/3)
z3 = e^(iπ/2) = cos(π/2) + i * sin(π/2)
z4 = e^(i5π/6) = cos(5π/6) + i * sin(5π/6)
z5 = e^(i2π/3) = cos(2π/3) + i * sin(2π/3)
z6 = e^(i5π/3) = cos(5π/3) + i * sin(5π/3)
Now, we can resolve z^6 + 1 into quadratic factors:
z^6 + 1 = (z - z1)(z - z2)(z - z3)(z - z4)(z - z5)(z - z6)
Let's plug in the values we found:
z^6 + 1 = [z - (cos(π/6) + i * sin(π/6))][z - (cos(π/3) + i * sin(π/3))][z - (cos(π/2) + i * sin(π/2))][z - (cos(5π/6) + i * sin(5π/6))][z - (cos(2π/3) + i * sin(2π/3))][z - (cos(5π/3) + i * sin(5π/3))]
Expanding this expression will give us the polynomial factors of z^6 + 1.
Now, let's deduce the given expression:
cos3x = 4[cos(x) -cos(π/6)][(cos(x) -cos(π/2)][(cos(x) -cos(5π/6)]
Using the identity cos(A) - cos(B) = -2sin((A+B)/2)sin((A-B)/2), we can simplify the expression:
cos3x = 4[-2sin((x + π/6)/2)sin((x - π/6)/2)][-2sin((x + π/2)/2)sin((x - π/2)/2)][-2sin((x + 5π/6)/2)sin((x - 5π/6)/2)]
Further simplifying, we have:
cos3x = 16sin((x + π/6)/2)sin((x - π/6)/2)sin((x + π/2)/2)sin((x - π/2)/2)sin((x + 5π/6)/2)sin((x - 5π/6)/2)
Therefore, we have deduced the given expression using the trigonometric identity and the properties of complex numbers.