Question

what is the answer in de moivre teorm?
$(cos\frac{\pi }{10} +icos\frac{2\pi }{5} )^{10}$

Answer 1

Answer:

If

z=r(cosθ+isinθ)

and

z2=z⋅z

then:

z2z2z2=r(cosθ+isinθ)⋅r(cosθ+isinθ)=r2[cos(θ+θ)+isin(θ+θ)]=r2(cos2θ+isin2θ)

Likewise, if

z=r(cosθ+isinθ)

and

z3=z2⋅z

then:

z3z3z3=r2(cos2θ+isin2θ)⋅r(cosθ+isinθ)=r3[cos(2θ+θ)+isin(2θ+θ)]=r3(cos3θ+isin3θ)

Again, if

z=r(cosθ+isinθ)

and

z4=z3⋅z

then

z4=r4(cos4θ+isin4θ)

These examples suggest a general rule valid for all powers of

z

, or

n

. We offer this rule and assume its validity for all

n

without formal proof, leaving that for later studies. The general rule for raising a complex number in polar form to a power is called De Moivre’s Theorem, and has important applications in engineering, particularly circuit analysis. The rule is as follows:

zn=[r(cosθ+isinθ)]n=rn(cosnθ+isinnθ)

Where

z=r(cosθ+isinθ)

and let

n

be a positive integer.

Notice what this rule looks like geometrically. A complex number taken to the

n

th power has two motions: First, its distance from the origin is taken to the

n

th power; second, its angle is multiplied by

n

. Conversely, the roots of a number have angles that are evenly spaced about the origin.

Example 1: Find.

[2(cos120∘+isin120∘)]5

Solution:

θ=120∘=2π3 rad

, using De Moivre’s Theorem:

zn=[r(cosθ+isinθ)]n[2(cos120∘+isin120∘)]5=rn(cosnθ+isinnθ)=25[cos52π3+isin52π3]=32(cos10π3+isin10π3)=32(−12+−i3–√2)=−16−16i3–√