Calculus Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $6$ .

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Step 1.1

Multiply $\frac{\sin (\frac{π}{3}) - \sin (\frac{π}{6})}{\frac{π}{3} - \frac{π}{6}}$ by $\frac{6}{6}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $3$ .

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Step 3.1.1

Factor $3$ out of $6$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $6$ .

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Step 3.2.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $6$ out of $6 \sin (\frac{π}{3}) + 6 (- \sin (\frac{π}{6}))$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 5

Simplify terms.

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Step 5.1

Subtract $π$ from $2 π$ .

$\frac{6 \frac{\sqrt{3} - 1}{2}}{π}$

Step 5.2

Combine $6$ and $\frac{\sqrt{3} - 1}{2}$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Reduce the expression $\frac{6 (\sqrt{3} - 1)}{2}$ by cancelling the common factors.

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Step 6.1.1

Factor $2$ out of $6 (\sqrt{3} - 1)$ .

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

Step 6.1.2

Factor $2$ out of $2$ .

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

Step 6.1.3

Cancel the common factor.

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

Step 6.1.4

Rewrite the expression.

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

Step 6.2

Divide $3 (\sqrt{3} - 1)$ by $1$ .

$\frac{3 (\sqrt{3} - 1)}{π}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{3 (\sqrt{3} - 1)}{π}$

Decimal Form:

$0.69905702 \dots$