Trigonometry Examples

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

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

Step 1

Simplify the numerator.

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Step 1.1

The exact value of

sin (\frac{π}{3})

$\sin (\frac{π}{3})$ is

\frac{\sqrt{3}}{2}

$\frac{\sqrt{3}}{2}$ .

\frac{\frac{\sqrt{3}}{2} - cos (\frac{π}{6})}{cos (\frac{π}{6}) + cos (\frac{π}{3})}

$\frac{\frac{\sqrt{3}}{2} - \cos (\frac{π}{6})}{\cos (\frac{π}{6}) + \cos (\frac{π}{3})}$

Step 1.2

The exact value of

cos (\frac{π}{6})

$\cos (\frac{π}{6})$ is

\frac{\sqrt{3}}{2}

$\frac{\sqrt{3}}{2}$ .

\frac{\frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2}}{cos (\frac{π}{6}) + cos (\frac{π}{3})}

$\frac{\frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2}}{\cos (\frac{π}{6}) + \cos (\frac{π}{3})}$

Step 1.3

Combine the numerators over the common denominator.

\frac{\frac{\sqrt{3} - \sqrt{3}}{2}}{cos (\frac{π}{6}) + cos (\frac{π}{3})}

$\frac{\frac{\sqrt{3} - \sqrt{3}}{2}}{\cos (\frac{π}{6}) + \cos (\frac{π}{3})}$

Step 1.4

Rewrite

\frac{\sqrt{3} - \sqrt{3}}{2}

$\frac{\sqrt{3} - \sqrt{3}}{2}$ in a factored form.

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Step 1.4.1

Subtract

\sqrt{3}

$\sqrt{3}$ from

\sqrt{3}

$\sqrt{3}$ .

\frac{\frac{0}{2}}{cos (\frac{π}{6}) + cos (\frac{π}{3})}

$\frac{\frac{0}{2}}{\cos (\frac{π}{6}) + \cos (\frac{π}{3})}$

Step 1.4.2

Divide

0

$0$ by

2

$2$ .

\frac{0}{cos (\frac{π}{6}) + cos (\frac{π}{3})}

$\frac{0}{\cos (\frac{π}{6}) + \cos (\frac{π}{3})}$

\frac{0}{cos (\frac{π}{6}) + cos (\frac{π}{3})}

$\frac{0}{\cos (\frac{π}{6}) + \cos (\frac{π}{3})}$

\frac{0}{cos (\frac{π}{6}) + cos (\frac{π}{3})}

$\frac{0}{\cos (\frac{π}{6}) + \cos (\frac{π}{3})}$

Step 2

Simplify the denominator.

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Step 2.1

The exact value of

cos (\frac{π}{6})

$\cos (\frac{π}{6})$ is

\frac{\sqrt{3}}{2}

$\frac{\sqrt{3}}{2}$ .

\frac{0}{\frac{\sqrt{3}}{2} + cos (\frac{π}{3})}

$\frac{0}{\frac{\sqrt{3}}{2} + \cos (\frac{π}{3})}$

Step 2.2

The exact value of

cos (\frac{π}{3})

$\cos (\frac{π}{3})$ is

\frac{1}{2}

$\frac{1}{2}$ .

\frac{0}{\frac{\sqrt{3}}{2} + \frac{1}{2}}

$\frac{0}{\frac{\sqrt{3}}{2} + \frac{1}{2}}$

Step 2.3

Combine the numerators over the common denominator.

\frac{0}{\frac{\sqrt{3} + 1}{2}}

$\frac{0}{\frac{\sqrt{3} + 1}{2}}$

\frac{0}{\frac{\sqrt{3} + 1}{2}}

$\frac{0}{\frac{\sqrt{3} + 1}{2}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$0 \frac{2}{\sqrt{3} + 1}$

Step 4

Multiply

$\frac{2}{\sqrt{3} + 1}$ by

$\frac{\sqrt{3} - 1}{\sqrt{3} - 1}$ .

$0 (\frac{2}{\sqrt{3} + 1} \cdot \frac{\sqrt{3} - 1}{\sqrt{3} - 1})$

Step 5

Multiply

$\frac{2}{\sqrt{3} + 1}$ by

$\frac{\sqrt{3} - 1}{\sqrt{3} - 1}$ .

$0 \frac{2 (\sqrt{3} - 1)}{(\sqrt{3} + 1) (\sqrt{3} - 1)}$

Step 6

Expand the denominator using the FOIL method.

$0 \frac{2 (\sqrt{3} - 1)}{{\sqrt{3}}^{2} + \sqrt{3} \cdot - 1 + \sqrt{3} - 1}$

Step 7

Simplify.

$0 \frac{2 (\sqrt{3} - 1)}{2}$

Step 8

Cancel the common factor of

$2$ .

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Step 8.1

Cancel the common factor.

$0 \frac{2 (\sqrt{3} - 1)}{2}$

Step 8.2

Divide

$\sqrt{3} - 1$ by

$1$ .

$0 (\sqrt{3} - 1)$

Step 9

Multiply

$0$ by

$\sqrt{3} - 1$ .

$0$