Trigonometry Examples

- v \frac{1 + \frac{\sqrt{- 5}}{3}}{2}

$- v \frac{1 + \frac{\sqrt{- 5}}{3}}{2}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite

- 5

$- 5$ as

- 1 (5)

$- 1 (5)$ .

- v \frac{1 + \frac{\sqrt{- 1 (5)}}{3}}{2}

$- v \frac{1 + \frac{\sqrt{- 1 (5)}}{3}}{2}$

Step 1.2

Rewrite

\sqrt{- 1 (5)}

$\sqrt{- 1 (5)}$ as

\sqrt{- 1} \cdot \sqrt{5}

$\sqrt{- 1} \cdot \sqrt{5}$ .

- v \frac{1 + \frac{\sqrt{- 1} \cdot \sqrt{5}}{3}}{2}

$- v \frac{1 + \frac{\sqrt{- 1} \cdot \sqrt{5}}{3}}{2}$

Step 1.3

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

- v \frac{1 + \frac{i \sqrt{5}}{3}}{2}

$- v \frac{1 + \frac{i \sqrt{5}}{3}}{2}$

- v \frac{1 + \frac{i \sqrt{5}}{3}}{2}

$- v \frac{1 + \frac{i \sqrt{5}}{3}}{2}$

Step 2

Simplify the numerator.

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Step 2.1

Write

1

$1$ as a fraction with a common denominator.

- v \frac{\frac{3}{3} + \frac{i \sqrt{5}}{3}}{2}

$- v \frac{\frac{3}{3} + \frac{i \sqrt{5}}{3}}{2}$

Step 2.2

Combine the numerators over the common denominator.

- v \frac{\frac{3 + i \sqrt{5}}{3}}{2}

$- v \frac{\frac{3 + i \sqrt{5}}{3}}{2}$

- v \frac{\frac{3 + i \sqrt{5}}{3}}{2}

$- v \frac{\frac{3 + i \sqrt{5}}{3}}{2}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

- v (\frac{3 + i \sqrt{5}}{3} \cdot \frac{1}{2})

$- v (\frac{3 + i \sqrt{5}}{3} \cdot \frac{1}{2})$

Step 4

Multiply

\frac{3 + i \sqrt{5}}{3} \cdot \frac{1}{2}

$\frac{3 + i \sqrt{5}}{3} \cdot \frac{1}{2}$ .

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Step 4.1

Multiply

\frac{3 + i \sqrt{5}}{3}

$\frac{3 + i \sqrt{5}}{3}$ by

\frac{1}{2}

$\frac{1}{2}$ .

- v \frac{3 + i \sqrt{5}}{3 \cdot 2}

$- v \frac{3 + i \sqrt{5}}{3 \cdot 2}$

Step 4.2

Multiply

3

$3$ by

2

$2$ .

- v \frac{3 + i \sqrt{5}}{6}

$- v \frac{3 + i \sqrt{5}}{6}$

- v \frac{3 + i \sqrt{5}}{6}

$- v \frac{3 + i \sqrt{5}}{6}$

Step 5

Combine

\frac{3 + i \sqrt{5}}{6}

$\frac{3 + i \sqrt{5}}{6}$ and

v

$v$ .

- \frac{(3 + i \sqrt{5}) v}{6}

$- \frac{(3 + i \sqrt{5}) v}{6}$