Top answers

Use de Moivre's theorem to calculate an expression for sin(5x) in terms of sin(x) only.

de Moivre's theorem gives us that (cos(x) + i sin(x)) n = cos(nx) + i sin(nx), for integers n and real values x.Therefore cos(5x) + i sin(5x) = (cos(x) + i sin(x)) 5 = cos 5 (x) + 5i cos 4 (x)sin(x) - 10 cos...
CB
38555 Views

Use de Moivre’s theorem to show that, (sin(x))^5 = A sin(5x) + Bsin(3x) + Csin(x), where A , B and C are constants to be found.

State de Moivre's theorem. Use n =5 and solve. I'll show this on the whiteboard.
RM
6763 Views

Find all of the roots of unity, Zn, in the case that (Zn)^6=1

Here we use the complex exponential form of 1 which is e^(i 2n pi). Applying the sixth root and substituting in for integer values of n gives all roots in complex exponential form.These can be converted into...
CR
3322 Views

How can we describe complex numbers ?

The simplest way to describe a complex number is by its real and imaginary part, z=x+yi, this may be wrote as Re(z)=x and Im(z)=y. These complex numbers follow the same rules as normal algebra, that is we ca...
TL
3284 Views

Evaluate ∫sin⁴(x) dx by expressing sin⁴(x) in terms of multiple angles

First we remember that sinθ can be expressed in terms of powers of z, where z=cos(θ)+isin(θ), using the following:2isin(nθ)=zⁿ-z⁻ⁿ and 2cos(nθ)=zⁿ+z⁻ⁿ so, [2isin(θ)]⁴=[z¹-z⁻¹]⁴ 16sin(θ)=(z)⁴(-z⁻¹)⁰+4(z)³(-z⁻...
NH
3946 Views