Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

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

Step 1.3

Combine the numerators over the common denominator.

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

Step 1.4

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

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Step 1.4.1

Subtract $\sqrt{3}$ from $\sqrt{3}$ .

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

Step 1.4.2

Divide $0$ by $2$ .

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

Step 2

Simplify the denominator.

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Step 2.1

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

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

Step 2.2

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

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

Step 2.3

Combine the numerators over the common denominator.

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

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 $\sqrt{3} - 1$ .

$0$