Dick said:Sure. e^(i*2*theta)=(e^(i*theta))^2. Write down both sides and equate the imaginary parts.
Dick said:Let's just call theta t, ok? On one side you have cos(2t)+i*sin(2t). On the other side you have (cos(t)+i*sin(t))*(cos(t)+i*sin(t)). Multiply that out carefully. You are getting things all mushed up.
kathrynag said:cos(2t)+isin(2t)=(cost+isint)^2
cos(2t)+isin(2t)=(cost)^2+2isint-(sint)^2
isin2t=(cost)^2+2isint-(sint)^2-cos(2t)
Is there something I'm missing on what to do here?
kathrynag said:cos(2t)+isin(2t)=(cost+isint)^2
cos(2t)+isin(2t)=(cost)^2+2isintcost-(sint)^2
sin(2t)=2sintcost
Dick said:Yes. And cos(2t)=cos^2(t)-sin^2(t). Right?
DeMoivre's formula is a mathematical formula that relates complex numbers to trigonometric functions. It states that for any complex number z = r(cos θ + i sin θ), where r is the magnitude and θ is the angle, the nth power of z can be expressed as rn(cos nθ + i sin nθ).
DeMoivre's formula can be derived using Euler's formula, which states that eix = cos x + i sin x. By substituting x = nθ into Euler's formula and raising both sides to the nth power, we can obtain DeMoivre's formula.
DeMoivre's formula is significant because it allows us to easily calculate the powers of complex numbers, which are essential in many mathematical and scientific fields. It also has applications in physics, engineering, and signal processing.
Yes, DeMoivre's formula can be used to find n th roots of a complex number. By setting r to the nth root of the original complex number and solving for θ, we can find the n distinct roots of the original number.
DeMoivre's formula is limited to calculating powers and roots of complex numbers. It cannot be used for other operations such as addition, subtraction, multiplication, or division. Additionally, it only applies to complex numbers in polar form.