Trigonometry Examples

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

Step 1

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

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Step 1.1

Factor $3$ out of $6 (\cos (\frac{π}{3}) + i \sin (\frac{π}{3}))$ .

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

Step 1.2

Cancel the common factors.

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 2

Multiply the numerator and denominator of $\frac{2 (\cos (\frac{π}{3}) + i \sin (\frac{π}{3}))}{0.8660254 + 0.5 i}$ by the conjugate of $0.8660254 + 0.5 i$ to make the denominator real.

$\frac{2 (\cos (\frac{π}{3}) + i \sin (\frac{π}{3}))}{0.8660254 + 0.5 i} \cdot \frac{0.8660254 - 0.5 i}{0.8660254 - 0.5 i}$

Step 3

Multiply.

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Step 3.1

Combine.

$\frac{2 (\cos (\frac{π}{3}) + i \sin (\frac{π}{3})) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

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

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{2 (\frac{1}{2} + i \sin (\frac{π}{3})) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.1.2

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{2 (\frac{1}{2} + i \frac{\sqrt{3}}{2}) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.1.3

Combine $i$ and $\frac{\sqrt{3}}{2}$ .

$\frac{2 (\frac{1}{2} + \frac{i \sqrt{3}}{2}) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.2

Apply the distributive property.

$\frac{(2 (\frac{1}{2}) + 2 \frac{i \sqrt{3}}{2}) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.3

Cancel the common factor of $2$ .

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Step 3.2.3.1

Cancel the common factor.

$\frac{(2 (\frac{1}{2}) + 2 \frac{i \sqrt{3}}{2}) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.3.2

Rewrite the expression.

$\frac{(1 + 2 \frac{i \sqrt{3}}{2}) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Cancel the common factor.

$\frac{(1 + 2 \frac{i \sqrt{3}}{2}) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.4.2

Rewrite the expression.

$\frac{(1 + i \sqrt{3}) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.5

Expand $(1 + i \sqrt{3}) (0.8660254 - 0.5 i)$ using the FOIL Method.

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Step 3.2.5.1

Apply the distributive property.

$\frac{1 (0.8660254 - 0.5 i) + i \sqrt{3} (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.5.2

Apply the distributive property.

$\frac{1 \cdot 0.8660254 + 1 (- 0.5 i) + i \sqrt{3} (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.5.3

Apply the distributive property.

$\frac{1 \cdot 0.8660254 + 1 (- 0.5 i) + i \sqrt{3} \cdot 0.8660254 + i \sqrt{3} (- 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.6

Simplify and combine like terms.

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Step 3.2.6.1

Simplify each term.

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Step 3.2.6.1.1

Multiply $0.8660254$ by $1$ .

$\frac{0.8660254 + 1 (- 0.5 i) + i \sqrt{3} \cdot 0.8660254 + i \sqrt{3} (- 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.6.1.2

Multiply $- 0.5 i$ by $1$ .

$\frac{0.8660254 - 0.5 i + i \sqrt{3} \cdot 0.8660254 + i \sqrt{3} (- 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.6.1.3

Multiply $0.8660254$ by $\sqrt{3}$ .

$\frac{0.8660254 - 0.5 i + i \cdot 1.5 + i \sqrt{3} (- 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.6.1.4

Move $1.5$ to the left of $i$ .

$\frac{0.8660254 - 0.5 i + 1.5 \cdot i + i \sqrt{3} (- 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.6.1.5

Multiply $i \sqrt{3} (- 0.5 i)$ .

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Step 3.2.6.1.5.1

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

$\frac{0.8660254 - 0.5 i + 1.5 i + \sqrt{3} (- 0.5 (i^{1} i))}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.6.1.5.2

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

$\frac{0.8660254 - 0.5 i + 1.5 i + \sqrt{3} (- 0.5 (i^{1} i^{1}))}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.6.1.5.3

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

$\frac{0.8660254 - 0.5 i + 1.5 i + \sqrt{3} (- 0.5 i^{1 + 1})}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.6.1.5.4

Add $1$ and $1$ .

$\frac{0.8660254 - 0.5 i + 1.5 i + \sqrt{3} (- 0.5 i^{2})}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.6.1.5.5

Multiply $- 0.5$ by $\sqrt{3}$ .

$\frac{0.8660254 - 0.5 i + 1.5 i - 0.8660254 i^{2}}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.6.1.6

Rewrite $i^{2}$ as $- 1$ .

$\frac{0.8660254 - 0.5 i + 1.5 i - 0.8660254 \cdot - 1}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.6.1.7

Multiply $- 0.8660254$ by $- 1$ .

$\frac{0.8660254 - 0.5 i + 1.5 i + 0.8660254}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.6.2

Add $0.8660254$ and $0.8660254$ .

$\frac{1.7320508 - 0.5 i + 1.5 i}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.2.6.3

Add $- 0.5 i$ and $1.5 i$ .

$\frac{1.7320508 + 1 i}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Expand $(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)$ using the FOIL Method.

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Step 3.3.1.1

Apply the distributive property.

$\frac{1.7320508 + 1 i}{0.8660254 (0.8660254 - 0.5 i) + 0.5 i (0.8660254 - 0.5 i)}$

Step 3.3.1.2

Apply the distributive property.

$\frac{1.7320508 + 1 i}{0.8660254 \cdot 0.8660254 + 0.8660254 (- 0.5 i) + 0.5 i (0.8660254 - 0.5 i)}$

Step 3.3.1.3

Apply the distributive property.

$\frac{1.7320508 + 1 i}{0.8660254 \cdot 0.8660254 + 0.8660254 (- 0.5 i) + 0.5 i \cdot 0.8660254 + 0.5 i (- 0.5 i)}$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Multiply $0.8660254$ by $0.8660254$ .

$\frac{1.7320508 + 1 i}{0.75 + 0.8660254 (- 0.5 i) + 0.5 i \cdot 0.8660254 + 0.5 i (- 0.5 i)}$

Step 3.3.2.2

Multiply $- 0.5$ by $0.8660254$ .

$\frac{1.7320508 + 1 i}{0.75 - 0.4330127 i + 0.5 i \cdot 0.8660254 + 0.5 i (- 0.5 i)}$

Step 3.3.2.3

Multiply $0.8660254$ by $0.5$ .

$\frac{1.7320508 + 1 i}{0.75 - 0.4330127 i + 0.4330127 i + 0.5 i (- 0.5 i)}$

Step 3.3.2.4

Multiply $- 0.5$ by $0.5$ .

$\frac{1.7320508 + 1 i}{0.75 - 0.4330127 i + 0.4330127 i - 0.25 i i}$

Step 3.3.2.5

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

$\frac{1.7320508 + 1 i}{0.75 - 0.4330127 i + 0.4330127 i - 0.25 (i^{1} i)}$

Step 3.3.2.6

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

$\frac{1.7320508 + 1 i}{0.75 - 0.4330127 i + 0.4330127 i - 0.25 (i^{1} i^{1})}$

Step 3.3.2.7

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

$\frac{1.7320508 + 1 i}{0.75 - 0.4330127 i + 0.4330127 i - 0.25 i^{1 + 1}}$

Step 3.3.2.8

Add $1$ and $1$ .

$\frac{1.7320508 + 1 i}{0.75 - 0.4330127 i + 0.4330127 i - 0.25 i^{2}}$

Step 3.3.2.9

Add $- 0.4330127 i$ and $0.4330127 i$ .

$\frac{1.7320508 + 1 i}{0.75 + 0 i - 0.25 i^{2}}$

Step 3.3.3

Simplify each term.

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Step 3.3.3.1

Multiply $0$ by $i$ .

$\frac{1.7320508 + 1 i}{0.75 + 0 - 0.25 i^{2}}$

Step 3.3.3.2

Rewrite $i^{2}$ as $- 1$ .

$\frac{1.7320508 + 1 i}{0.75 + 0 - 0.25 \cdot - 1}$

Step 3.3.3.3

Multiply $- 0.25$ by $- 1$ .

$\frac{1.7320508 + 1 i}{0.75 + 0 + 0.25}$

Step 3.3.4

Add $0.75$ and $0$ .

$\frac{1.7320508 + 1 i}{0.75 + 0.25}$

Step 3.3.5

Add $0.75$ and $0.25$ .

$\frac{1.7320508 + 1 i}{1}$

Step 4

Rewrite $1.7320508$ as $1 (1.7320508)$ .

$\frac{1 (1.7320508) + 1 i}{1}$

Step 5

Factor $1$ out of $1 i$ .

$\frac{1 (1.7320508) + 1 (1 i)}{1}$

Step 6

Factor $1$ out of $1 (1.7320508) + 1 (1 i)$ .

$\frac{1 (1.7320508 + 1 i)}{1}$

Step 7

Factor $1$ out of $1$ .

$\frac{1 (1.7320508 + 1 i)}{1 (1)}$

Step 8

Separate fractions.

$\frac{1}{1} \cdot \frac{1.7320508 + 1 i}{1}$

Step 9

Simplify the expression.

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Step 9.1

Divide $1$ by $1$ .

$1 \frac{1.7320508 + 1 i}{1}$

Step 9.2

Divide $1.7320508 + 1 i$ by $1$ .

$1 (1.7320508 + 1 i)$

Step 10

Apply the distributive property.

$1 \cdot 1.7320508 + 1 (1 i)$

Step 11

Multiply.

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Step 11.1

Multiply $1$ by $1.7320508$ .

$1.7320508 + 1 (1 i)$

Step 11.2

Multiply $1$ by $1$ .

$1.7320508 + 1 i$

Step 12

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 13

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 14

Substitute the actual values of $a = 1.7320508$ and $b = 1$ .

$| z | = \sqrt{1^{2} + {1.7320508}^{2}}$

Step 15

Find $| z |$ .

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Step 15.1

Raise $1$ to the power of $2$ .

$| z | = \sqrt{1 + {1.7320508}^{2}}$

Step 15.2

Raise $1.7320508$ to the power of $2$ .

$| z | = \sqrt{1 + 3}$

Step 15.3

Add $1$ and $3$ .

$| z | = \sqrt{4}$

Step 15.4

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

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

Step 15.5

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

$| z | = 2$

Step 16

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

$θ = \arctan (\frac{1}{1.7320508})$

Step 17

Since inverse tangent of $\frac{1}{1.7320508}$ produces an angle in the first quadrant, the value of the angle is $\frac{π}{6}$ .

$θ = \frac{π}{6}$

Step 18

Substitute the values of $θ = \frac{π}{6}$ and $| z | = 2$ .

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