Linear Algebra Examples

3 (cos (π) + i sin (π))

$3 (\cos (π) + i \sin (π))$

Step 1

Calculate the distance from

(a, b)

$(a, b)$ to the origin using the formula

r = \sqrt{a^{2} + b^{2}}

$r = \sqrt{a^{2} + b^{2}}$ .

r = \sqrt{{(3 cos (π))}^{2} + {(sin (π) \cdot 3)}^{2}}

$r = \sqrt{{(3 \cos (π))}^{2} + {(\sin (π) \cdot 3)}^{2}}$

Step 2

Simplify

\sqrt{{(3 cos (π))}^{2} + {(sin (π) \cdot 3)}^{2}}

$\sqrt{{(3 \cos (π))}^{2} + {(\sin (π) \cdot 3)}^{2}}$ .

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Step 2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

r = \sqrt{{(3 (- cos (0)))}^{2} + {(sin (π) \cdot 3)}^{2}}

$r = \sqrt{{(3 (- \cos (0)))}^{2} + {(\sin (π) \cdot 3)}^{2}}$

Step 2.2

The exact value of

cos (0)

$\cos (0)$ is

1

$1$ .

r = \sqrt{{(3 (- 1 \cdot 1))}^{2} + {(sin (π) \cdot 3)}^{2}}

$r = \sqrt{{(3 (- 1 \cdot 1))}^{2} + {(\sin (π) \cdot 3)}^{2}}$

Step 2.3

Multiply

3 (- 1 \cdot 1)

$3 (- 1 \cdot 1)$ .

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Step 2.3.1

Multiply

- 1

$- 1$ by

1

$1$ .

r = \sqrt{{(3 \cdot - 1)}^{2} + {(sin (π) \cdot 3)}^{2}}

$r = \sqrt{{(3 \cdot - 1)}^{2} + {(\sin (π) \cdot 3)}^{2}}$

Step 2.3.2

Multiply

3

$3$ by

- 1

$- 1$ .

r = \sqrt{{(- 3)}^{2} + {(sin (π) \cdot 3)}^{2}}

$r = \sqrt{{(- 3)}^{2} + {(\sin (π) \cdot 3)}^{2}}$

r = \sqrt{{(- 3)}^{2} + {(sin (π) \cdot 3)}^{2}}

$r = \sqrt{{(- 3)}^{2} + {(\sin (π) \cdot 3)}^{2}}$

Step 2.4

Raise

- 3

$- 3$ to the power of

2

$2$ .

r = \sqrt{9 + {(sin (π) \cdot 3)}^{2}}

$r = \sqrt{9 + {(\sin (π) \cdot 3)}^{2}}$

Step 2.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

r = \sqrt{9 + {(sin (0) \cdot 3)}^{2}}

$r = \sqrt{9 + {(\sin (0) \cdot 3)}^{2}}$

Step 2.6

The exact value of

sin (0)

$\sin (0)$ is

0

$0$ .

r = \sqrt{9 + {(0 \cdot 3)}^{2}}

$r = \sqrt{9 + {(0 \cdot 3)}^{2}}$

Step 2.7

Multiply

0

$0$ by

3

$3$ .

r = \sqrt{9 + 0^{2}}

$r = \sqrt{9 + 0^{2}}$

Step 2.8

Raising

0

$0$ to any positive power yields

0

$0$ .

r = \sqrt{9 + 0}

$r = \sqrt{9 + 0}$

Step 2.9

Add

9

$9$ and

0

$0$ .

r = \sqrt{9}

$r = \sqrt{9}$

Step 2.10

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

r = \sqrt{3^{2}}

$r = \sqrt{3^{2}}$

Step 2.11

Pull terms out from under the radical, assuming positive real numbers.

r = 3

$r = 3$

r = 3

$r = 3$

Step 3

Calculate reference angle

θˆ = arctan (∣ ∣ ∣ \frac{b}{a} ∣ ∣ ∣)

$θ̂ = \arctan (| \frac{b}{a} |)$ .

θˆ = arctan (∣ ∣ ∣ \frac{sin (π) \cdot 3}{3 cos (π)} ∣ ∣ ∣)

$θ̂ = \arctan (| \frac{\sin (π) \cdot 3}{3 \cos (π)} |)$

Step 4

Simplify

arctan (∣ ∣ ∣ \frac{sin (π) \cdot 3}{3 cos (π)} ∣ ∣ ∣)

$\arctan (| \frac{\sin (π) \cdot 3}{3 \cos (π)} |)$ .

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Step 4.1

Cancel the common factor of

3

$3$ .

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Step 4.1.1

Cancel the common factor.

$θ̂ = \arctan (| \frac{\sin (π) \cdot 3}{3 \cos (π)} |)$

Step 4.1.2

Rewrite the expression.

$θ̂ = \arctan (| \frac{\sin (π)}{\cos (π)} |)$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$θ̂ = \arctan (| \frac{\sin (0)}{\cos (π)} |)$

Step 4.2.2

The exact value of

$\sin (0)$ is

$0$ .

$θ̂ = \arctan (| \frac{0}{\cos (π)} |)$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$θ̂ = \arctan (| \frac{0}{- \cos (0)} |)$

Step 4.3.2

The exact value of

$\cos (0)$ is

$1$ .

$θ̂ = \arctan (| \frac{0}{- 1 \cdot 1} |)$

Step 4.3.3

Multiply

$- 1$ by

$1$ .

$θ̂ = \arctan (| \frac{0}{- 1} |)$

Step 4.4

Simplify the expression.

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Step 4.4.1

Move the negative one from the denominator of

$\frac{0}{- 1}$ .

$θ̂ = \arctan (| - 1 \cdot 0 |)$

Step 4.4.2

Multiply

$- 1$ by

$0$ .

$θ̂ = \arctan (| 0 |)$

Step 4.5

The absolute value is the distance between a number and zero. The distance between

$0$ and

$0$ is

$0$ .

$θ̂ = \arctan (0)$

Step 4.6

The exact value of

$\arctan (0)$ is

$0$ .

$θ̂ = 0$

Step 5

Find the quadrant.

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Step 5.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$(3 (- \cos (0)), \sin (π) \cdot 3)$

Step 5.2

The exact value of

$\cos (0)$ is

$1$ .

$(3 (- 1 \cdot 1), \sin (π) \cdot 3)$

Step 5.3

Multiply

$3 (- 1 \cdot 1)$ .

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Step 5.3.1

Multiply

$- 1$ by

$1$ .

$(3 \cdot - 1, \sin (π) \cdot 3)$

Step 5.3.2

Multiply

$3$ by

$- 1$ .

$(- 3, \sin (π) \cdot 3)$

Step 5.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$(- 3, \sin (0) \cdot 3)$

Step 5.5

The exact value of

$\sin (0)$ is

$0$ .

$(- 3, 0 \cdot 3)$

Step 5.6

Multiply

$0$ by

$3$ .

$(- 3, 0)$

Step 5.7

Since the x-coordinate is negative and the y-coordinate is

$0$ , the point is located on x-axis between the second and third quadrants. The quadrants are labeled in counter-clockwise order, starting in the upper-right.

Between Quadrant

$2$ and

$3$

Between Quadrant

$2$ and

$3$

Step 6

Use the formula to find the roots of the complex number.

${(a + b i)}^{\frac{1}{n}} = r^{\frac{1}{n}} c i s (\frac{θ + 2 π k}{n})$ ,

$k = 0, 1, \dots, n - 1$

Step 7

Substitute

$r$ ,

$n$ , and

$θ$ into the formula.

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Step 7.1

Combine

${(3)}^{\frac{1}{3}}$ and

$\frac{θ + 2 π k}{3}$ .

$c i s \frac{{(3)}^{\frac{1}{3}} (θ + 2 π k)}{3}$

Step 7.2

Combine

$c$ and

$\frac{{(3)}^{\frac{1}{3}} (θ + 2 π k)}{3}$ .

$i s \frac{c ({(3)}^{\frac{1}{3}} (θ + 2 π k))}{3}$

Step 7.3

Combine

$i$ and

$\frac{c ({(3)}^{\frac{1}{3}} (θ + 2 π k))}{3}$ .

$s \frac{i (c ({(3)}^{\frac{1}{3}} (θ + 2 π k)))}{3}$

Step 7.4

Combine

$s$ and

$\frac{i (c ({(3)}^{\frac{1}{3}} (θ + 2 π k)))}{3}$ .

$\frac{s (i (c ({(3)}^{\frac{1}{3}} (θ + 2 π k))))}{3}$

Step 7.5

Remove parentheses.

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Step 7.5.1

Remove parentheses.

$\frac{s (i (c (3^{\frac{1}{3}} (θ + 2 π k))))}{3}$

Step 7.5.2

Remove parentheses.

$\frac{s (i (c \cdot 3^{\frac{1}{3}} (θ + 2 π k)))}{3}$

Step 7.5.3

Remove parentheses.

$\frac{s (i (c \cdot 3^{\frac{1}{3}}) (θ + 2 π k))}{3}$

Step 7.5.4

Remove parentheses.

$\frac{s (i c \cdot 3^{\frac{1}{3}} (θ + 2 π k))}{3}$

Step 7.5.5

Remove parentheses.

$\frac{s (i c \cdot 3^{\frac{1}{3}}) (θ + 2 π k)}{3}$

Step 7.5.6

Remove parentheses.

$\frac{s (i c) \cdot 3^{\frac{1}{3}} (θ + 2 π k)}{3}$

Step 7.5.7

Remove parentheses.

$\frac{s i c \cdot 3^{\frac{1}{3}} (θ + 2 π k)}{3}$

Step 8

Substitute

$k = 0$ into the formula and simplify.

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Step 8.1

Remove parentheses.

$k = 0 : 3^{\frac{1}{3}} c i s (\frac{θ + 2 π (0)}{3})$

Step 8.2

Multiply

$2 π (0)$ .

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Step 8.2.1

Multiply

$0$ by

$2$ .

$k = 0 : 3^{\frac{1}{3}} c i s (\frac{θ + 0 π}{3})$

Step 8.2.2

Multiply

$0$ by

$π$ .

$k = 0 : 3^{\frac{1}{3}} c i s (\frac{θ + 0}{3})$

Step 9

Substitute

$k = 1$ into the formula and simplify.

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Step 9.1

Remove parentheses.

$k = 1 : 3^{\frac{1}{3}} c i s (\frac{θ + 2 π (1)}{3})$

Step 9.2

Multiply

$2$ by

$1$ .

$k = 1 : 3^{\frac{1}{3}} c i s (\frac{θ + 2 π}{3})$

Step 10

Substitute

$k = 2$ into the formula and simplify.

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Step 10.1

Remove parentheses.

$k = 2 : 3^{\frac{1}{3}} c i s (\frac{θ + 2 π (2)}{3})$

Step 10.2

Multiply

$2$ by

$2$ .

$k = 2 : 3^{\frac{1}{3}} c i s (\frac{θ + 4 π}{3})$

Step 11

List the solutions.

$k = 0 : 3^{\frac{1}{3}} c i s (\frac{θ + 0}{3})$

$k = 1 : 3^{\frac{1}{3}} c i s (\frac{θ + 2 π}{3})$

$k = 2 : 3^{\frac{1}{3}} c i s (\frac{θ + 4 π}{3})$