Use de-Moivre's theorem to find the reciprocal of each number below.3−iGiven 3−i we need to find the reciprocal of it using de-Moivre's theorem.13−i=1(cos⁡0c+isin⁡0c)2(32−i2)=12⋅cos⁡(π6)−isin⁡(π6)(32−i2)=12⋅cos⁡(π6)−isin⁡(π6)=12⋅(32+i12)=34+i4

De Moivre's Theorem states that:[r(cos⁡θ+isin⁡θ)]n=rn(cos⁡(nθ)+isin⁡(nθ))To find the reciprocal, take n=-1z=3−i→r=2,θ=tan−1({−13})=−π6Hence z−1=2−1(cos⁡{(−1)(−π6)}+isin⁡{(−1)(−π6)})=12(cos⁡π6+isin⁡π6)=34+14ias you achieved.

Step by step solution to "Use de-Moivre's theorem to find the reciprocal of..."

Lillianna Sandoval

Answered question

2022-04-11

Use de-Moivre's theorem to find the reciprocal of each number below.

\sqrt{3} - i

Given

\sqrt{3} - i

we need to find the reciprocal of it using de-Moivre's theorem.

\frac{1}{\sqrt{3} - i}

= \frac{1 ({\cos 0}^{c} + i {\sin 0}^{c})}{2 (\frac{\sqrt{3}}{2} - \frac{i}{2})}

= \frac{1}{2} \cdot \frac{\cos (\frac{π}{6}) - i \sin (\frac{π}{6})}{(\frac{\sqrt{3}}{2} - \frac{i}{2})}

= \frac{1}{2} \cdot \cos (\frac{π}{6}) - i \sin (\frac{π}{6})

= \frac{1}{2} \cdot (\frac{\sqrt{3}}{2} + i \frac{1}{2})

= \frac{\sqrt{3}}{4} + \frac{i}{4}

Answer & Explanation

pobijedi6wro

Beginner2022-04-12Added 15 answers

De Moivre's Theorem states that:

{[r (\cos θ + i \sin θ)]}^{n} = r^{n} (\cos (n θ) + i \sin (n θ))

To find the reciprocal, take n=-1

z = \sqrt{3} - i \to r = 2, θ = \tan^{- 1} ({\frac{- 1}{\sqrt{3}}}) = \frac{- π}{6}

Hence

z^{- 1} = 2^{- 1} (\cos {(- 1) (\frac{- π}{6})} + i \sin {(- 1) (\frac{- π}{6})}) = \frac{1}{2} (\cos \frac{π}{6} + i \sin \frac{π}{6})

= \frac{\sqrt{3}}{4} + \frac{1}{4} i

as you achieved.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Precalculus

Describe all solutions of Ax=0 in parametric vector form, where A is row equivalent to the given matrix
$[\begin{matrix} 1 & 3 & 0 & - 4 \\ 2 & 6 & 0 & - 8 \end{matrix}]$
Find, correct to the nearest degree, the three angles of the triangle with the given vertices
A(1, 0, -1), B(3, -2, 0), C(1, 3, 3)
Whether f is a function from Z to R if
a) $f (n) = \pm n$ .
b) $f (n) = \sqrt{n^{2} + 1}$ .
c) $f (n) = \frac{1}{n^{2} - 4}$ .
?
How to write the expression $6^{\frac{3}{2}}$ in radical form?
How to evaluate $\sin (\frac{- 5 π}{4})$ ?
What is the derivative of $\cot^{2} x$ ?
How to verify the identity: $\frac{\cos (x) - \cos (y)}{\sin (x) + \sin (y)} + \frac{\sin (x) - \sin (y)}{\cos (x) + \cos (y)} = 0$ ?
Find $\tan (22 . 5^{\circ})$ using the half-angle formula.
How to find the exact values of $\cos 22.5 °$ using the half-angle formula?
How to express the complex number in trigonometric form: 5-5i?
The solution set of $\tan θ = 3 \cot θ$ is
How to find the angle between the vector and $x -$ axis?
Find the probability of getting 5 Mondays in the month of february in a leap year.
How to find the inflection points for the given function $f (x) = x^{3} - 3 x^{2} + 6 x$ ?
How do I find the value of sec(3pi/4)?

Didn't find what you were looking for?