Calculus Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

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

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

Step 3

Cancel the common factor of $2$ .

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Step 3.1

Factor $2$ out of $6$ .

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

Step 3.2

Cancel the common factor.

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

Step 3.3

Rewrite the expression.

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

Step 4

Combine $\sqrt{3}$ and $\frac{3}{π}$ .

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

Step 5

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

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

Step 6

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

Step 7

Simplify the denominator.

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Step 7.1

Apply the product rule to $\frac{π}{6}$ .

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

Step 7.2

Raise $6$ to the power of $2$ .

$\frac{3 \sqrt{3}}{π} - \frac{\frac{1}{2}}{\frac{π^{2}}{36}}$

Step 8

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 \sqrt{3}}{π} - (\frac{1}{2} \cdot \frac{36}{π^{2}})$

Step 9

Combine.

$\frac{3 \sqrt{3}}{π} - \frac{1 \cdot 36}{2 π^{2}}$

Step 10

Cancel the common factor of $36$ and $2$ .

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Step 10.1

Factor $2$ out of $1 \cdot 36$ .

$\frac{3 \sqrt{3}}{π} - \frac{2 (1 \cdot 18)}{2 π^{2}}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $2$ out of $2 π^{2}$ .

$\frac{3 \sqrt{3}}{π} - \frac{2 (1 \cdot 18)}{2 (π^{2})}$

Step 10.2.2

Cancel the common factor.

$\frac{3 \sqrt{3}}{π} - \frac{2 (1 \cdot 18)}{2 π^{2}}$

Step 10.2.3

Rewrite the expression.

$\frac{3 \sqrt{3}}{π} - \frac{1 \cdot 18}{π^{2}}$

Step 11

Multiply $18$ by $1$ .

$\frac{3 \sqrt{3}}{π} - \frac{18}{π^{2}}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{3}}{π} - \frac{18}{π^{2}}$

Decimal Form:

$- 0.16979461 \dots$