De Moivre's Theorem and Finding Roots of Complex Numbers - Trigonometry

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Question

Simplify using De Moivre's Theorem:

$\small (\cos x+i\sin $x)^3$$

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Answer

We can use DeMoivre's formula which states:

Now plugging in our values of $\small n=3$ and we get the desired result.

$\small (\cos x+i\sin $x)^n$=(\cos nx+i\sin nx)$

$\cos 3x +i \sin 3x$

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Question

Evaluate using De Moivre's Theorem:

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Answer

First, convert this complex number to polar form.

$r= $\sqrt{ $a^2$ + $b^2$ }$ = $$\sqrt{1^2$ $+(-1)^2$ }$=$\sqrt{2}$$

$\sin \theta = $\frac{-1}{\sqrt2}$$

$\theta = $\frac{5 \pi }{4}$ \enspace or \enspace $\frac{7 \pi }{4}$$

Since the point has a positive real part and a negative imaginary part, it is located in quadrant IV, so the angle is $\frac{7\pi}{4}$ .

This gives us $($\sqrt{2}$ ( \cos $\frac{7 \pi }{4}$ + i \sin $\frac{ 7 \pi }{4}$ )) ^ 8$

To evaluate, use DeMoivre's Theorem:

DeMoivre's Theorem is

We apply it to our situation to get:

$(\sqrt2 $)^8$ ( \cos $\frac{ 7 \pi }{4}$ \cdot 8 + i \sin $\frac{ 7 \pi }{4}$ \cdot 8 )$ simplifying

$2 ^ 4 ( \cos 14 \pi + i \sin 14 \pi )$ , $14 \pi$ is coterminal with since it is an even multiple of $\pi$

$2^ 4 (\cos 0 + i \sin 0 ) = 16(1 + 0i) = 16$

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Question

Use De Moivre's Theorem to evaluate $(3 \sqrt3 -3i) ^ 6$ .

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Answer

First convert this point to polar form:

$r = $$\sqrt{(3\sqrt3)^2$ + $(-3)^2$ }$ = $\sqrt{9 \cdot 3 + 9 }$ = $\sqrt{36}$ = 6$

$\cos \theta = $\frac{ 3\sqrt3}{6 }$ = $\frac{\sqrt3}{2}$$

$\theta = \cos ^{-1} ($\frac{\sqrt3}{2}$ ) = $\frac{ \pi }{6}$ \enspace \or \enspace $\frac{ 11 \pi }{6}$$

Since this number has a negative imaginary part and a positive real part, it is in quadrant IV, so the angle is $\frac{ 11 \pi }{6}$

We are evaluating $(6 \cos $\frac{11 \pi }{6}$ + i \sin $\frac{ 11 \pi }{6}$ ) ^ 6$

Using DeMoivre's Theorem:

DeMoivre's Theorem is

We apply it to our situation to get:

$$\frac{11 \pi }{6}$ \cdot 6 = 11 \pi$ which is coterminal with $\pi$ since it is an odd multiplie

$46,656(\cos \pi + i \sin \pi ) = 46,656 (-1 + 0i) = -46,656$

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Question

Use De Moivre's Theorem to evaluate .

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Answer

First, convert the complex number to polar form:

$r = \sqrt{ 2^2 + 2^2 } = \sqrt{4 + 4 } = \sqrt{2 \cdot 4 } = 2 \sqrt 2$

$\sin \theta = \frac{ 2 }{ 2 \sqrt 2 } = \frac{ 1 }{ \sqrt2 }$

$\theta = \sin ^{-1 } (\frac{ 1}{ \sqrt 2 }) = \frac{ \pi }{4} \enspace or \enspace \frac{ 3 \pi }{4}$

Since both the real and the imaginary parts are positive, the angle is in quadrant I, so it is $\frac{ \pi }{4}$

This means we're evaluating

$(2 \sqrt2 (\cos \frac{ \pi }{4} + i \sin \frac{ \pi }{4} ))^ 9$

Using DeMoivre's Theorem:

DeMoivre's Theorem is

$(\cos(x)+i\sin(x))^n=\cos(nx)+i\sin(nx)$

We apply it to our situation to get.

$(2 \sqrt2 )^9 (\cos \frac{\pi }{4} \cdot 9 + i \sin \frac{ \pi }{4} \cdot 9 )$

First, evaluate $(2\sqrt2)^9$ . We can split this into $2^9 \cdot (\sqrt2)^9$ which is equivalent to $2^9 \cdot (\sqrt2)^8 \cdot \sqrt2 = 2^9 \cdot 2^4 \cdot \sqrt 2$

\[We can re-write the middle exponent since $\sqrt 2$ is equivalent to $2^{\frac{1}{2}}$ \]

This comes to $8,192 \sqrt 2$

Evaluating sine and cosine at $\frac{\pi }{4} \cdot 9 = \frac{ 9 \pi }{4}$ is equivalent to evaluating them at $\frac{ \pi }{4}$ since $\frac{ 9 \pi }{4} - 2 \pi = \frac{9 \pi }{4 } - \frac{ 8 \pi }{4} = \frac{ \pi }{4}$

This means our expression can be written as:

$8,192 \sqrt2 ( \cos \frac{\pi }{4} + i \sin \frac{ \pi }{4} ) = 8,192 \sqrt2 (\frac{1}{\sqrt2} + i \frac{ 1}{\sqrt2}) = 8,192 + 8,192i$

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Question

Find all fifth roots of .

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Answer

Begin by converting the complex number to polar form:

$3+4i=5(\cos53.13^\circ+i\sin53.13^\circ)$

Next, put this in its generalized form, using k which is any integer, including zero:

$5[\cos(53.13^\circ+k360^\circ)+i\sin(53.13^\circ+k360^\circ)]$

Using De Moivre's theorem, a fifth root of is given by:

$\left [5[\cos(53.13^\circ+k360^\circ)+i\sin(53.13^\circ+k360^\circ)] \right ]^$\frac{1}{5}$$

$=5^$\frac{1}{5}$\left (\cos$\frac{53.13^\circ+k360^\circ}{5}$+i\sin$\frac{53.13^\circ+k360^\circ}{5}$ \right )$

$=\sqrt[5]{5}\left [\cos(10.63^\circ+k72^\circ)+i\sin(10.63^\circ+k72^\circ) \right ]$

Assigning the values will allow us to find the following roots. In general, use the values .

$k=0: \sqrt[5]{5}\left (\cos10.63^\circ+i\sin10.63^\circ \right )=R_1$

$k=1: \sqrt[5]{5}\left (\cos82.63^\circ+i\sin82.63^\circ \right )=R_2$

$k=2: \sqrt[5]{5}\left (\cos154.63^\circ+i\sin154.63^\circ \right )=R_3$

$k=3: \sqrt[5]{5}\left (\cos226.63^\circ+i\sin226.63^\circ \right )=R_4$

$k=4: \sqrt[5]{5}\left (\cos298.63^\circ+i\sin298.63^\circ \right )=R_5$

These are the fifth roots of .

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Question

Find all cube roots of 1.

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Answer

Begin by converting the complex number to polar form:

$1=\cos0^\circ+i\sin0^\circ$

Next, put this in its generalized form, using k which is any integer, including zero:

$1=\cos(0^\circ+k360^\circ)+i\sin(0^\circ+k360^\circ)$

Using De Moivre's theorem, a fifth root of 1 is given by:

$[\cos(k360^\circ)+i\sin(k360^\circ)] ^$\frac{1}{3}$$

$=1^$\frac{1}{3}$\cdot \left [\cos\left ($\frac{k360^\circ}{3}$ \right )+i\sin\left ($\frac{k360^\circ}{3}$ \right ) \right ]$

$=\cos k 120^\circ+i\sin k120^\circ$

Assigning the values will allow us to find the following roots. In general, use the values .

$k=0: \cos0^\circ+i\sin0^\circ=1=R_1$

$k=1: \cos120^\circ+i\sin120^\circ=-$\frac{1}{2}$+i$\frac{1}{2}$$\sqrt{3}$=R_2$

$k=2: \cos240^\circ+i\sin240^\circ=-$\frac{1}{2}$-i$\frac{1}{2}$$\sqrt{3}$=R_3$

These are the cube roots of 1.

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Question

Find all fourth roots of $-8-8i$\sqrt{3}$$ .

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Answer

Begin by converting the complex number to polar form:

$-8-8i$\sqrt{3}$=16(\cos240^\circ+i\sin240^\circ)$

Next, put this in its generalized form, using k which is any integer, including zero:

$16[\cos(240^\circ+k360^\circ)+i\sin(240^\circ+k360^\circ)]$

Using De Moivre's theorem, a fifth root of $-8-8i$\sqrt{3}$$ is given by:

$\left [16\cos(240^\circ+k360^\circ)+i\sin(240^\circ+k360^\circ) \right ]^$\frac{1}{4}$$

$=16^$\frac{1}{4}$\left [\cos\left ($\frac{240^\circ}{4}$+\frac{k360^\circ}{4}$ \right)+i\sin\left($\frac{240^\circ}{4}$+\frac{k360^\circ}{4}$ \right ) \right ]$

$=2\left [\cos\left (60^\circ+k90^\circ \right )+i\sin\left (60^\circ+k90^\circ \right ) \right ]$

Assigning the values will allow us to find the following roots. In general, use the values .

$\k=0: 2\left (\cos60^\circ+i\sin60^\circ \right )=2\left ( $\frac{1}{2}$+i$\frac{1}{2}$$\sqrt{3}$ \right)=1+i$\sqrt{3}$=R_1\ k=1: 2\left(\cos150^\circ+i\sin150^\circ \right)=2\left( -$\frac{1}{2}$$\sqrt{3}$+\frac{1}{2}$i \right)=-$\sqrt{3}$+i=R_2\ k=2: 2\left(\cos240^\circ+i\sin240^\circ \right)=2\left( -$\frac{1}{2}$-i$\frac{1}{2}$$\sqrt{3}$ \right )=-1-i$\sqrt{3}$=R_3\ k=3: 2\left (\cos330^\circ+i\sin330^\circ \right )=2\left ( $\frac{1}{2}$$\sqrt{3}$-$\frac{1}{2}$i \right)=$\sqrt{3}$-i=R_4$

These are the fifth roots of $-8-8i$\sqrt{3}$$ .

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Question

Simplify using De Moivre's Theorem:

$\small (\cos x+i\sin $x)^3$$

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Answer

We can use DeMoivre's formula which states:

Now plugging in our values of $\small n=3$ and we get the desired result.

$\small (\cos x+i\sin $x)^n$=(\cos nx+i\sin nx)$

$\cos 3x +i \sin 3x$

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Question

Evaluate using De Moivre's Theorem:

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Answer

First, convert this complex number to polar form.

$r= $\sqrt{ $a^2$ + $b^2$ }$ = $$\sqrt{1^2$ $+(-1)^2$ }$=$\sqrt{2}$$

$\sin \theta = $\frac{-1}{\sqrt2}$$

$\theta = $\frac{5 \pi }{4}$ \enspace or \enspace $\frac{7 \pi }{4}$$

Since the point has a positive real part and a negative imaginary part, it is located in quadrant IV, so the angle is $\frac{7\pi}{4}$ .

This gives us $($\sqrt{2}$ ( \cos $\frac{7 \pi }{4}$ + i \sin $\frac{ 7 \pi }{4}$ )) ^ 8$

To evaluate, use DeMoivre's Theorem:

DeMoivre's Theorem is

We apply it to our situation to get:

$(\sqrt2 $)^8$ ( \cos $\frac{ 7 \pi }{4}$ \cdot 8 + i \sin $\frac{ 7 \pi }{4}$ \cdot 8 )$ simplifying

$2 ^ 4 ( \cos 14 \pi + i \sin 14 \pi )$ , $14 \pi$ is coterminal with since it is an even multiple of $\pi$

$2^ 4 (\cos 0 + i \sin 0 ) = 16(1 + 0i) = 16$

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Question

Use De Moivre's Theorem to evaluate $(3 \sqrt3 -3i) ^ 6$ .

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Answer

First convert this point to polar form:

$r = $$\sqrt{(3\sqrt3)^2$ + $(-3)^2$ }$ = $\sqrt{9 \cdot 3 + 9 }$ = $\sqrt{36}$ = 6$

$\cos \theta = $\frac{ 3\sqrt3}{6 }$ = $\frac{\sqrt3}{2}$$

$\theta = \cos ^{-1} ($\frac{\sqrt3}{2}$ ) = $\frac{ \pi }{6}$ \enspace \or \enspace $\frac{ 11 \pi }{6}$$

Since this number has a negative imaginary part and a positive real part, it is in quadrant IV, so the angle is $\frac{ 11 \pi }{6}$

We are evaluating $(6 \cos $\frac{11 \pi }{6}$ + i \sin $\frac{ 11 \pi }{6}$ ) ^ 6$

Using DeMoivre's Theorem:

DeMoivre's Theorem is

We apply it to our situation to get:

$$\frac{11 \pi }{6}$ \cdot 6 = 11 \pi$ which is coterminal with $\pi$ since it is an odd multiplie

$46,656(\cos \pi + i \sin \pi ) = 46,656 (-1 + 0i) = -46,656$

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Question

Use De Moivre's Theorem to evaluate .

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Answer

First, convert the complex number to polar form:

$r = \sqrt{ 2^2 + 2^2 } = \sqrt{4 + 4 } = \sqrt{2 \cdot 4 } = 2 \sqrt 2$

$\sin \theta = \frac{ 2 }{ 2 \sqrt 2 } = \frac{ 1 }{ \sqrt2 }$

$\theta = \sin ^{-1 } (\frac{ 1}{ \sqrt 2 }) = \frac{ \pi }{4} \enspace or \enspace \frac{ 3 \pi }{4}$

Since both the real and the imaginary parts are positive, the angle is in quadrant I, so it is $\frac{ \pi }{4}$

This means we're evaluating

$(2 \sqrt2 (\cos \frac{ \pi }{4} + i \sin \frac{ \pi }{4} ))^ 9$

Using DeMoivre's Theorem:

DeMoivre's Theorem is

$(\cos(x)+i\sin(x))^n=\cos(nx)+i\sin(nx)$

We apply it to our situation to get.

$(2 \sqrt2 )^9 (\cos \frac{\pi }{4} \cdot 9 + i \sin \frac{ \pi }{4} \cdot 9 )$

First, evaluate $(2\sqrt2)^9$ . We can split this into $2^9 \cdot (\sqrt2)^9$ which is equivalent to $2^9 \cdot (\sqrt2)^8 \cdot \sqrt2 = 2^9 \cdot 2^4 \cdot \sqrt 2$

\[We can re-write the middle exponent since $\sqrt 2$ is equivalent to $2^{\frac{1}{2}}$ \]

This comes to $8,192 \sqrt 2$

Evaluating sine and cosine at $\frac{\pi }{4} \cdot 9 = \frac{ 9 \pi }{4}$ is equivalent to evaluating them at $\frac{ \pi }{4}$ since $\frac{ 9 \pi }{4} - 2 \pi = \frac{9 \pi }{4 } - \frac{ 8 \pi }{4} = \frac{ \pi }{4}$

This means our expression can be written as:

$8,192 \sqrt2 ( \cos \frac{\pi }{4} + i \sin \frac{ \pi }{4} ) = 8,192 \sqrt2 (\frac{1}{\sqrt2} + i \frac{ 1}{\sqrt2}) = 8,192 + 8,192i$

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Question

Find all fifth roots of .

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Answer

Begin by converting the complex number to polar form:

$3+4i=5(\cos53.13^\circ+i\sin53.13^\circ)$

Next, put this in its generalized form, using k which is any integer, including zero:

$5[\cos(53.13^\circ+k360^\circ)+i\sin(53.13^\circ+k360^\circ)]$

Using De Moivre's theorem, a fifth root of is given by:

$\left [5[\cos(53.13^\circ+k360^\circ)+i\sin(53.13^\circ+k360^\circ)] \right ]^$\frac{1}{5}$$

$=5^$\frac{1}{5}$\left (\cos$\frac{53.13^\circ+k360^\circ}{5}$+i\sin$\frac{53.13^\circ+k360^\circ}{5}$ \right )$

$=\sqrt[5]{5}\left [\cos(10.63^\circ+k72^\circ)+i\sin(10.63^\circ+k72^\circ) \right ]$

Assigning the values will allow us to find the following roots. In general, use the values .

$k=0: \sqrt[5]{5}\left (\cos10.63^\circ+i\sin10.63^\circ \right )=R_1$

$k=1: \sqrt[5]{5}\left (\cos82.63^\circ+i\sin82.63^\circ \right )=R_2$

$k=2: \sqrt[5]{5}\left (\cos154.63^\circ+i\sin154.63^\circ \right )=R_3$

$k=3: \sqrt[5]{5}\left (\cos226.63^\circ+i\sin226.63^\circ \right )=R_4$

$k=4: \sqrt[5]{5}\left (\cos298.63^\circ+i\sin298.63^\circ \right )=R_5$

These are the fifth roots of .

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Question

Find all cube roots of 1.

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Answer

Begin by converting the complex number to polar form:

$1=\cos0^\circ+i\sin0^\circ$

Next, put this in its generalized form, using k which is any integer, including zero:

$1=\cos(0^\circ+k360^\circ)+i\sin(0^\circ+k360^\circ)$

Using De Moivre's theorem, a fifth root of 1 is given by:

$[\cos(k360^\circ)+i\sin(k360^\circ)] ^$\frac{1}{3}$$

$=1^$\frac{1}{3}$\cdot \left [\cos\left ($\frac{k360^\circ}{3}$ \right )+i\sin\left ($\frac{k360^\circ}{3}$ \right ) \right ]$

$=\cos k 120^\circ+i\sin k120^\circ$

Assigning the values will allow us to find the following roots. In general, use the values .

$k=0: \cos0^\circ+i\sin0^\circ=1=R_1$

$k=1: \cos120^\circ+i\sin120^\circ=-$\frac{1}{2}$+i$\frac{1}{2}$$\sqrt{3}$=R_2$

$k=2: \cos240^\circ+i\sin240^\circ=-$\frac{1}{2}$-i$\frac{1}{2}$$\sqrt{3}$=R_3$

These are the cube roots of 1.

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Question

Find all fourth roots of $-8-8i$\sqrt{3}$$ .

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Answer

Begin by converting the complex number to polar form:

$-8-8i$\sqrt{3}$=16(\cos240^\circ+i\sin240^\circ)$

Next, put this in its generalized form, using k which is any integer, including zero:

$16[\cos(240^\circ+k360^\circ)+i\sin(240^\circ+k360^\circ)]$

Using De Moivre's theorem, a fifth root of $-8-8i$\sqrt{3}$$ is given by:

$\left [16\cos(240^\circ+k360^\circ)+i\sin(240^\circ+k360^\circ) \right ]^$\frac{1}{4}$$

$=16^$\frac{1}{4}$\left [\cos\left ($\frac{240^\circ}{4}$+\frac{k360^\circ}{4}$ \right)+i\sin\left($\frac{240^\circ}{4}$+\frac{k360^\circ}{4}$ \right ) \right ]$

$=2\left [\cos\left (60^\circ+k90^\circ \right )+i\sin\left (60^\circ+k90^\circ \right ) \right ]$

Assigning the values will allow us to find the following roots. In general, use the values .

$\k=0: 2\left (\cos60^\circ+i\sin60^\circ \right )=2\left ( $\frac{1}{2}$+i$\frac{1}{2}$$\sqrt{3}$ \right)=1+i$\sqrt{3}$=R_1\ k=1: 2\left(\cos150^\circ+i\sin150^\circ \right)=2\left( -$\frac{1}{2}$$\sqrt{3}$+\frac{1}{2}$i \right)=-$\sqrt{3}$+i=R_2\ k=2: 2\left(\cos240^\circ+i\sin240^\circ \right)=2\left( -$\frac{1}{2}$-i$\frac{1}{2}$$\sqrt{3}$ \right )=-1-i$\sqrt{3}$=R_3\ k=3: 2\left (\cos330^\circ+i\sin330^\circ \right )=2\left ( $\frac{1}{2}$$\sqrt{3}$-$\frac{1}{2}$i \right)=$\sqrt{3}$-i=R_4$

These are the fifth roots of $-8-8i$\sqrt{3}$$ .

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Question

Simplify using De Moivre's Theorem:

$\small (\cos x+i\sin $x)^3$$

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Answer

We can use DeMoivre's formula which states:

Now plugging in our values of $\small n=3$ and we get the desired result.

$\small (\cos x+i\sin $x)^n$=(\cos nx+i\sin nx)$

$\cos 3x +i \sin 3x$

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Question

Evaluate using De Moivre's Theorem:

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Answer

First, convert this complex number to polar form.

$r= $\sqrt{ $a^2$ + $b^2$ }$ = $$\sqrt{1^2$ $+(-1)^2$ }$=$\sqrt{2}$$

$\sin \theta = $\frac{-1}{\sqrt2}$$

$\theta = $\frac{5 \pi }{4}$ \enspace or \enspace $\frac{7 \pi }{4}$$

Since the point has a positive real part and a negative imaginary part, it is located in quadrant IV, so the angle is $\frac{7\pi}{4}$ .

This gives us $($\sqrt{2}$ ( \cos $\frac{7 \pi }{4}$ + i \sin $\frac{ 7 \pi }{4}$ )) ^ 8$

To evaluate, use DeMoivre's Theorem:

DeMoivre's Theorem is

We apply it to our situation to get:

$(\sqrt2 $)^8$ ( \cos $\frac{ 7 \pi }{4}$ \cdot 8 + i \sin $\frac{ 7 \pi }{4}$ \cdot 8 )$ simplifying

$2 ^ 4 ( \cos 14 \pi + i \sin 14 \pi )$ , $14 \pi$ is coterminal with since it is an even multiple of $\pi$

$2^ 4 (\cos 0 + i \sin 0 ) = 16(1 + 0i) = 16$

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Question

Use De Moivre's Theorem to evaluate $(3 \sqrt3 -3i) ^ 6$ .

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Answer

First convert this point to polar form:

$r = $$\sqrt{(3\sqrt3)^2$ + $(-3)^2$ }$ = $\sqrt{9 \cdot 3 + 9 }$ = $\sqrt{36}$ = 6$

$\cos \theta = $\frac{ 3\sqrt3}{6 }$ = $\frac{\sqrt3}{2}$$

$\theta = \cos ^{-1} ($\frac{\sqrt3}{2}$ ) = $\frac{ \pi }{6}$ \enspace \or \enspace $\frac{ 11 \pi }{6}$$

Since this number has a negative imaginary part and a positive real part, it is in quadrant IV, so the angle is $\frac{ 11 \pi }{6}$

We are evaluating $(6 \cos $\frac{11 \pi }{6}$ + i \sin $\frac{ 11 \pi }{6}$ ) ^ 6$

Using DeMoivre's Theorem:

DeMoivre's Theorem is

We apply it to our situation to get:

$$\frac{11 \pi }{6}$ \cdot 6 = 11 \pi$ which is coterminal with $\pi$ since it is an odd multiplie

$46,656(\cos \pi + i \sin \pi ) = 46,656 (-1 + 0i) = -46,656$

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Question

Use De Moivre's Theorem to evaluate .

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Answer

First, convert the complex number to polar form:

$r = \sqrt{ 2^2 + 2^2 } = \sqrt{4 + 4 } = \sqrt{2 \cdot 4 } = 2 \sqrt 2$

$\sin \theta = \frac{ 2 }{ 2 \sqrt 2 } = \frac{ 1 }{ \sqrt2 }$

$\theta = \sin ^{-1 } (\frac{ 1}{ \sqrt 2 }) = \frac{ \pi }{4} \enspace or \enspace \frac{ 3 \pi }{4}$

Since both the real and the imaginary parts are positive, the angle is in quadrant I, so it is $\frac{ \pi }{4}$

This means we're evaluating

$(2 \sqrt2 (\cos \frac{ \pi }{4} + i \sin \frac{ \pi }{4} ))^ 9$

Using DeMoivre's Theorem:

DeMoivre's Theorem is

$(\cos(x)+i\sin(x))^n=\cos(nx)+i\sin(nx)$

We apply it to our situation to get.

$(2 \sqrt2 )^9 (\cos \frac{\pi }{4} \cdot 9 + i \sin \frac{ \pi }{4} \cdot 9 )$

First, evaluate $(2\sqrt2)^9$ . We can split this into $2^9 \cdot (\sqrt2)^9$ which is equivalent to $2^9 \cdot (\sqrt2)^8 \cdot \sqrt2 = 2^9 \cdot 2^4 \cdot \sqrt 2$

\[We can re-write the middle exponent since $\sqrt 2$ is equivalent to $2^{\frac{1}{2}}$ \]

This comes to $8,192 \sqrt 2$

Evaluating sine and cosine at $\frac{\pi }{4} \cdot 9 = \frac{ 9 \pi }{4}$ is equivalent to evaluating them at $\frac{ \pi }{4}$ since $\frac{ 9 \pi }{4} - 2 \pi = \frac{9 \pi }{4 } - \frac{ 8 \pi }{4} = \frac{ \pi }{4}$

This means our expression can be written as:

$8,192 \sqrt2 ( \cos \frac{\pi }{4} + i \sin \frac{ \pi }{4} ) = 8,192 \sqrt2 (\frac{1}{\sqrt2} + i \frac{ 1}{\sqrt2}) = 8,192 + 8,192i$

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Question

Find all fifth roots of .

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Answer

Begin by converting the complex number to polar form:

$3+4i=5(\cos53.13^\circ+i\sin53.13^\circ)$

Next, put this in its generalized form, using k which is any integer, including zero:

$5[\cos(53.13^\circ+k360^\circ)+i\sin(53.13^\circ+k360^\circ)]$

Using De Moivre's theorem, a fifth root of is given by:

$\left [5[\cos(53.13^\circ+k360^\circ)+i\sin(53.13^\circ+k360^\circ)] \right ]^$\frac{1}{5}$$

$=5^$\frac{1}{5}$\left (\cos$\frac{53.13^\circ+k360^\circ}{5}$+i\sin$\frac{53.13^\circ+k360^\circ}{5}$ \right )$

$=\sqrt[5]{5}\left [\cos(10.63^\circ+k72^\circ)+i\sin(10.63^\circ+k72^\circ) \right ]$

Assigning the values will allow us to find the following roots. In general, use the values .

$k=0: \sqrt[5]{5}\left (\cos10.63^\circ+i\sin10.63^\circ \right )=R_1$

$k=1: \sqrt[5]{5}\left (\cos82.63^\circ+i\sin82.63^\circ \right )=R_2$

$k=2: \sqrt[5]{5}\left (\cos154.63^\circ+i\sin154.63^\circ \right )=R_3$

$k=3: \sqrt[5]{5}\left (\cos226.63^\circ+i\sin226.63^\circ \right )=R_4$

$k=4: \sqrt[5]{5}\left (\cos298.63^\circ+i\sin298.63^\circ \right )=R_5$

These are the fifth roots of .

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Question

Find all cube roots of 1.

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Answer

Begin by converting the complex number to polar form:

$1=\cos0^\circ+i\sin0^\circ$

Next, put this in its generalized form, using k which is any integer, including zero:

$1=\cos(0^\circ+k360^\circ)+i\sin(0^\circ+k360^\circ)$

Using De Moivre's theorem, a fifth root of 1 is given by:

$[\cos(k360^\circ)+i\sin(k360^\circ)] ^$\frac{1}{3}$$

$=1^$\frac{1}{3}$\cdot \left [\cos\left ($\frac{k360^\circ}{3}$ \right )+i\sin\left ($\frac{k360^\circ}{3}$ \right ) \right ]$

$=\cos k 120^\circ+i\sin k120^\circ$

Assigning the values will allow us to find the following roots. In general, use the values .

$k=0: \cos0^\circ+i\sin0^\circ=1=R_1$

$k=1: \cos120^\circ+i\sin120^\circ=-$\frac{1}{2}$+i$\frac{1}{2}$$\sqrt{3}$=R_2$

$k=2: \cos240^\circ+i\sin240^\circ=-$\frac{1}{2}$-i$\frac{1}{2}$$\sqrt{3}$=R_3$

These are the cube roots of 1.

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