Linear Algebra Examples

$3 (\cos (π) + i \sin (π))$

Step 1

Calculate the distance from $(a, b)$ to the origin using the formula $r = \sqrt{a^{2} + b^{2}}$ .

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

Step 2

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

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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.

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

Step 2.2

The exact value of $\cos (0)$ is $1$ .

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

Step 2.3

Multiply $3 (- 1 \cdot 1)$ .

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Step 2.3.1

Multiply $- 1$ by $1$ .

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

Step 2.3.2

Multiply $3$ by $- 1$ .

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

Step 2.4

Raise $- 3$ to the power of $2$ .

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

Step 2.5

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

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

Step 2.6

The exact value of $\sin (0)$ is $0$ .

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

Step 2.7

Multiply $0$ by $3$ .

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

Step 2.8

Raising $0$ to any positive power yields $0$ .

$r = \sqrt{9 + 0}$

Step 2.9

Add $9$ and $0$ .

$r = \sqrt{9}$

Step 2.10

Rewrite $9$ as $3^{2}$ .

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

Step 2.11

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

$r = 3$

Step 3

Calculate reference angle $θ̂ = \arctan (| \frac{b}{a} |)$ .

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

Step 4

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

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Step 4.1

Cancel the common factor of $3$ .

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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 $3$

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Remove parentheses.

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Step 7.5.1

Remove parentheses.

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

Step 7.5.2

Remove parentheses.

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

Step 7.5.3

Remove parentheses.

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

Step 7.5.4

Remove parentheses.

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

Step 7.5.5

Remove parentheses.

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

Step 7.5.6

Remove parentheses.

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

Step 7.5.7

Remove parentheses.

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

Step 8

Substitute $k = 0$ into the formula and simplify.

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Step 8.1

Remove parentheses.

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

Step 8.2

Multiply $2 π (0)$ .

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Step 8.2.1

Multiply $0$ by $2$ .

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

Step 8.2.2

Multiply $0$ by $π$ .

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

Step 9

Substitute $k = 1$ into the formula and simplify.

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Step 9.1

Remove parentheses.

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

Step 9.2

Multiply $2$ by $1$ .

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

Step 10

List the solutions.

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

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