Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

$\frac{1}{\sqrt{3}} - \cos (\frac{π}{6})$

Step 1.2

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} - \cos (\frac{π}{6})$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} - \cos (\frac{π}{6})$

Step 1.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} - \cos (\frac{π}{6})$

Step 1.3.4

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

$\frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} - \cos (\frac{π}{6})$

Step 1.3.5

Add $1$ and $1$ .

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

Step 1.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} - \cos (\frac{π}{6})$

Step 1.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} - \cos (\frac{π}{6})$

Step 1.3.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

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

Step 1.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{3}}{3^{1}} - \cos (\frac{π}{6})$

Step 1.3.6.5

Evaluate the exponent.

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

Step 1.4

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:

$- 0.28867513 \dots$