Trigonometry Examples

$\tan (\frac{π}{6}) - \cos (\frac{π}{6})$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

To write $\frac{\sqrt{3}}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3

To write $- \frac{\sqrt{3}}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

Multiply $3$ by $2$ .

$\frac{\sqrt{3} \cdot 2}{6} - \frac{\sqrt{3}}{2} \cdot \frac{3}{3}$

Step 4.3

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

$\frac{\sqrt{3} \cdot 2}{6} - \frac{\sqrt{3} \cdot 3}{2 \cdot 3}$

Step 4.4

Multiply $2$ by $3$ .

$\frac{\sqrt{3} \cdot 2}{6} - \frac{\sqrt{3} \cdot 3}{6}$

Step 5

Combine the numerators over the common denominator.

$\frac{\sqrt{3} \cdot 2 - \sqrt{3} \cdot 3}{6}$

Step 6

Simplify the numerator.

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Step 6.1

Move $2$ to the left of $\sqrt{3}$ .

$\frac{2 \cdot \sqrt{3} - \sqrt{3} \cdot 3}{6}$

Step 6.2

Multiply $3$ by $- 1$ .

$\frac{2 \sqrt{3} - 3 \sqrt{3}}{6}$

Step 6.3

Subtract $3 \sqrt{3}$ from $2 \sqrt{3}$ .

$\frac{- \sqrt{3}}{6}$

Step 7

Move the negative in front of the fraction.

$- \frac{\sqrt{3}}{6}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{3}}{6}$

Decimal Form:

$- 0.28867513 \dots$