Precalculus Examples

$\frac{6 (\cos (5 π) + i \sin (5 π))}{3 (\cos (2 π) + i \sin (2 π))}$

Step 1

Cancel the common factor of $6$ and $3$ .

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Step 1.1

Factor $3$ out of $6 (\cos (5 π) + i \sin (5 π))$ .

$\frac{3 (2 (\cos (5 π) + i \sin (5 π)))}{3 (\cos (2 π) + i \sin (2 π))}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Cancel the common factor.

$\frac{3 (2 (\cos (5 π) + i \sin (5 π)))}{3 (\cos (2 π) + i \sin (2 π))}$

Step 1.2.2

Rewrite the expression.

$\frac{2 (\cos (5 π) + i \sin (5 π))}{\cos (2 π) + i \sin (2 π)}$

Step 2

Multiply the numerator and denominator of $\frac{2 (\cos (5 π) + i \sin (5 π))}{10 i}$ by the conjugate of $10 i$ to make the denominator real.

$\frac{2 (\cos (5 π) + i \sin (5 π))}{10 i} \cdot \frac{1 + 0 i}{1 + 0 i}$

Step 3

Multiply.

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Step 3.1

Combine.

$\frac{2 (\cos (5 π) + i \sin (5 π)) (1 + 0 i)}{(10 i) (1 + 0 i)}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{2 (\cos (π) + i \sin (5 π)) (1 + 0 i)}{(10 i) (1 + 0 i)}$

Step 3.2.1.2

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.

$\frac{2 (- \cos (0) + i \sin (5 π)) (1 + 0 i)}{(10 i) (1 + 0 i)}$

Step 3.2.1.3

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

$\frac{2 (- 1 \cdot 1 + i \sin (5 π)) (1 + 0 i)}{(10 i) (1 + 0 i)}$

Step 3.2.1.4

Multiply $- 1$ by $1$ .

$\frac{2 (- 1 + i \sin (5 π)) (1 + 0 i)}{(10 i) (1 + 0 i)}$

Step 3.2.1.5

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{2 (- 1 + i \sin (π)) (1 + 0 i)}{(10 i) (1 + 0 i)}$

Step 3.2.1.6

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

$\frac{2 (- 1 + i \sin (0)) (1 + 0 i)}{(10 i) (1 + 0 i)}$

Step 3.2.1.7

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

$\frac{2 (- 1 + i \cdot 0) (1 + 0 i)}{(10 i) (1 + 0 i)}$

Step 3.2.1.8

Multiply $i$ by $0$ .

$\frac{2 (- 1 + 0) (1 + 0 i)}{(10 i) (1 + 0 i)}$

Step 3.2.2

Add $- 1$ and $0$ .

$\frac{2 \cdot - 1 (1 + 0 i)}{(10 i) (1 + 0 i)}$

Step 3.2.3

Multiply $2$ by $- 1$ .

$\frac{- 2 (1 + 0 i)}{(10 i) (1 + 0 i)}$

Step 3.2.4

Apply the distributive property.

$\frac{- 2 \cdot 1 - 2 (0 i)}{(10 i) (1 + 0 i)}$

Step 3.2.5

Multiply $- 2$ by $1$ .

$\frac{- 2 - 2 (0 i)}{(10 i) (1 + 0 i)}$

Step 3.2.6

Multiply $0$ by $- 2$ .

$\frac{- 20 i}{(10 i) (1 + 0 i)}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Expand $(10 i) (1 + 0 i)$ using the FOIL Method.

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Step 3.3.1.1

Apply the distributive property.

$\frac{- 20 i}{1 (1 + 0 i) 0 i (1 + 0 i)}$

Step 3.3.1.2

Apply the distributive property.

$\frac{- 20 i}{1 \cdot 1 + 1 (0 i) 0 i (1 + 0 i)}$

Step 3.3.1.3

Apply the distributive property.

$\frac{- 20 i}{1 \cdot 1 + 1 (0 i) 0 i \cdot 10 i (0 i)}$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Multiply $1$ by $1$ .

$\frac{- 20 i}{1 + 1 (0 i) 0 i \cdot 10 i (0 i)}$

Step 3.3.2.2

Multiply $0$ by $1$ .

$\frac{- 20 i}{1 + 0 i 0 i \cdot 10 i (0 i)}$

Step 3.3.2.3

Multiply $0$ by $1$ .

$\frac{- 20 i}{1 + 0 i 0 i 0 i (0 i)}$

Step 3.3.2.4

Multiply $0$ by $0$ .

$\frac{- 20 i}{1 + 0 i 0 i + 0 i i}$

Step 3.3.2.5

Raise $i$ to the power of $1$ .

$\frac{- 20 i}{1 + 0 i 0 i + 0 (i^{1} i)}$

Step 3.3.2.6

Raise $i$ to the power of $1$ .

$\frac{- 20 i}{1 + 0 i 0 i + 0 (i^{1} i^{1})}$

Step 3.3.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- 20 i}{1 + 0 i 0 i + 0 i^{1 + 1}}$

Step 3.3.2.8

Add $1$ and $1$ .

$\frac{- 20 i}{1 + 0 i 0 i + 0 i^{2}}$

Step 3.3.2.9

Multiply $0$ by $i^{2}$ .

$\frac{- 20 i}{1 + 0 i 0 i + 0}$

Step 3.3.2.10

Add $1$ and $0$ .

$\frac{- 20 i}{1 + 0 i 0 i}$

Step 3.3.2.11

Subtract $0 i$ from $0 i$ .

$\frac{- 20 i}{1 + 0 i}$

Step 3.3.3

Multiply $0$ by $i$ .

$\frac{- 20 i}{1 + 0}$

Step 3.3.4

Add $1$ and $0$ .

$\frac{- 20 i}{1}$

Step 4

Divide $- 20 i$ by $1$ .

$- 20 i$

Step 5

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 6

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 7

Substitute the actual values of $a = - 2$ and $b = 0$ .

$| z | = \sqrt{{(0)}^{2} + {(- 2)}^{2}}$

Step 8

Find $| z |$ .

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Step 8.1

Raise $0$ to the power of $2$ .

$| z | = \sqrt{0 + {(- 2)}^{2}}$

Step 8.2

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

$| z | = \sqrt{0 + 4}$

Step 8.3

Add $0$ and $4$ .

$| z | = \sqrt{4}$

Step 8.4

Rewrite $4$ as $2^{2}$ .

$| z | = \sqrt{2^{2}}$

Step 8.5

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

$| z | = 2$

Step 9

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{0}{- 2})$

Step 10

Since inverse tangent of $\frac{0}{- 2}$ produces an angle in the third quadrant, the value of the angle is $3.14159265$ .

$θ = 3.14159265$

Step 11

Substitute the values of $θ = 3.14159265$ and $| z | = 2$ .

$2 (\cos (3.14159265) + i \sin (3.14159265))$