Use De Moivre's Theorem to write the complex number in trigonometric form
(cos(2pi)/7)+ i sin((2pi)/7))^5
To write the complex number (cos(2π/7) + i sin(2π/7))^5 in trigonometric form using De Moivre's theorem, follow these steps:
Step 1: Determine the magnitude (r) of the complex number:
The magnitude r is calculated using the formula r = √(a^2 + b^2), where a = cos(2π/7) and b = sin(2π/7). Therefore:
r = √((cos(2π/7))^2 + (sin(2π/7))^2)
Step 2: Determine the argument (θ) of the complex number:
The argument θ is calculated using the formula θ = arctan(b/a), where a = cos(2π/7) and b = sin(2π/7). Therefore:
θ = arctan(sin(2π/7)/cos(2π/7))
Step 3: Apply De Moivre's theorem:
De Moivre's theorem states that (cos(θ) + i sin(θ))^n = cos(nθ) + i sin(nθ). In this case, n = 5. Therefore:
(cos(2π/7) + i sin(2π/7))^5 = cos(5(2π/7)) + i sin(5(2π/7))
Step 4: Simplify the trigonometric expression:
Using the periodicity of sine and cosine functions, we can write:
cos(5(2π/7)) = cos(10π/7) = cos(8π/7 + 2π/7)
sin(5(2π/7)) = sin(10π/7) = sin(8π/7 + 2π/7)
Step 5: Substitute the simplified expressions back into the trigonometric form:
(cos(2π/7) + i sin(2π/7))^5 = cos(8π/7 + 2π/7) + i sin(8π/7 + 2π/7)
Therefore, the complex number (cos(2π/7) + i sin(2π/7))^5 can be written in trigonometric form as:
cos(8π/7 + 2π/7) + i sin(8π/7 + 2π/7)
To use De Moivre's Theorem to write the complex number in trigonometric form, we first express the given complex number in polar form.
The polar form of a complex number is given by z = r(cosθ + isinθ), where r represents the magnitude or modulus of the complex number, and θ represents the argument or angle of the complex number.
In this case, the complex number is (cos(2π/7) + isin(2π/7))^5. To simplify this, we can let r = 1 since the given complex number lies on the unit circle, where the radius is always 1.
Now, let's find the angle or argument of the complex number. The angle can be calculated using the formula θ = arctan(Im/Re), where Im represents the imaginary part (sin(2π/7) in this case), and Re represents the real part (cos(2π/7) in this case).
θ = arctan(sin(2π/7)/cos(2π/7))
Using a calculator in radians mode, we can evaluate this expression to find the value of θ.
Next, we apply De Moivre's Theorem, which states that for any complex number z = r(cosθ + isinθ) raised to the power of n, we have:
z^n = r^n (cos nθ + i sin nθ)
Using this theorem, we can raise our polar form (cos(2π/7) + isin(2π/7))^5 to the power of 5, while keeping the angle and magnitude we found earlier.
(z)^5 = (r(cosθ + isinθ))^5
= r^5 (cos 5θ + i sin 5θ)
By substituting the values of r and θ, we can obtain the trigonometric form of the given complex number.
(cos(2pi)/7)+ i sin((2pi)/7))^5
= 1^5(cos (10π/7) + i sin(10π/7)
= (cos (-4π/7) + i sin(-4π/7) , after subtracting 2π from the angle
= (cos (4π/7) - i sin(4π/7) )