Precalculus Examples

$6 \sin (\frac{π}{12}) \cos (\frac{π}{12})$

Step 1

The exact value of $\sin (\frac{π}{12})$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 1.1

Split $\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

$6 \sin (\frac{π}{4} - \frac{π}{6}) \cos (\frac{π}{12})$

Step 1.2

Apply the difference of angles identity.

$6 (\sin (\frac{π}{4}) \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6})) \cos (\frac{π}{12})$

Step 1.3

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$6 (\frac{\sqrt{2}}{2} \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6})) \cos (\frac{π}{12})$

Step 1.4

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

$6 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (\frac{π}{4}) \sin (\frac{π}{6})) \cos (\frac{π}{12})$

Step 1.5

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$6 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (\frac{π}{6})) \cos (\frac{π}{12})$

Step 1.6

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

$6 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{π}{12})$

Step 1.7

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

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

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Step 1.7.1.1.1

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

$6 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{π}{12})$

Step 1.7.1.1.2

Combine using the product rule for radicals.

$6 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{π}{12})$

Step 1.7.1.1.3

Multiply $2$ by $3$ .

$6 (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{π}{12})$

Step 1.7.1.1.4

Multiply $2$ by $2$ .

$6 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{π}{12})$

Step 1.7.1.2

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.7.1.2.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

$6 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) \cos (\frac{π}{12})$

Step 1.7.1.2.2

Multiply $2$ by $2$ .

$6 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) \cos (\frac{π}{12})$

Step 1.7.2

Combine the numerators over the common denominator.

$6 \frac{\sqrt{6} - \sqrt{2}}{4} \cos (\frac{π}{12})$

Step 2

Cancel the common factor of $2$ .

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Step 2.1

Factor $2$ out of $6$ .

$2 (3) \frac{\sqrt{6} - \sqrt{2}}{4} \cos (\frac{π}{12})$

Step 2.2

Factor $2$ out of $4$ .

$2 \cdot 3 \frac{\sqrt{6} - \sqrt{2}}{2 \cdot 2} \cos (\frac{π}{12})$

Step 2.3

Cancel the common factor.

$2 \cdot 3 \frac{\sqrt{6} - \sqrt{2}}{2 \cdot 2} \cos (\frac{π}{12})$

Step 2.4

Rewrite the expression.

$3 \frac{\sqrt{6} - \sqrt{2}}{2} \cos (\frac{π}{12})$

Step 3

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

$\frac{3 (\sqrt{6} - \sqrt{2})}{2} \cos (\frac{π}{12})$

Step 4

The exact value of $\cos (\frac{π}{12})$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 4.1

Split $\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

$\frac{3 (\sqrt{6} - \sqrt{2})}{2} \cos (\frac{π}{4} - \frac{π}{6})$

Step 4.2

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$\frac{3 (\sqrt{6} - \sqrt{2})}{2} (\cos (\frac{π}{4}) \cos (\frac{π}{6}) + \sin (\frac{π}{4}) \sin (\frac{π}{6}))$

Step 4.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{3 (\sqrt{6} - \sqrt{2})}{2} (\frac{\sqrt{2}}{2} \cos (\frac{π}{6}) + \sin (\frac{π}{4}) \sin (\frac{π}{6}))$

Step 4.4

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

$\frac{3 (\sqrt{6} - \sqrt{2})}{2} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (\frac{π}{4}) \sin (\frac{π}{6}))$

Step 4.5

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 4.6

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

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

Step 4.7

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 4.7.1

Simplify each term.

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Step 4.7.1.1

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

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Step 4.7.1.1.1

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

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

Step 4.7.1.1.2

Combine using the product rule for radicals.

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

Step 4.7.1.1.3

Multiply $2$ by $3$ .

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

Step 4.7.1.1.4

Multiply $2$ by $2$ .

$\frac{3 (\sqrt{6} - \sqrt{2})}{2} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 4.7.1.2

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 4.7.1.2.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{1}{2}$ .

$\frac{3 (\sqrt{6} - \sqrt{2})}{2} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2})$

Step 4.7.1.2.2

Multiply $2$ by $2$ .

$\frac{3 (\sqrt{6} - \sqrt{2})}{2} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4})$

Step 4.7.2

Combine the numerators over the common denominator.

$\frac{3 (\sqrt{6} - \sqrt{2})}{2} \cdot \frac{\sqrt{6} + \sqrt{2}}{4}$

Step 5

Multiply $\frac{3 (\sqrt{6} - \sqrt{2})}{2} \cdot \frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 5.1

Multiply $\frac{3 (\sqrt{6} - \sqrt{2})}{2}$ by $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

$\frac{3 (\sqrt{6} - \sqrt{2}) (\sqrt{6} + \sqrt{2})}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$\frac{3 (\sqrt{6} - \sqrt{2}) (\sqrt{6} + \sqrt{2})}{8}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{(3 \sqrt{6} + 3 (- \sqrt{2})) (\sqrt{6} + \sqrt{2})}{8}$

Step 6.2

Multiply $- 1$ by $3$ .

$\frac{(3 \sqrt{6} - 3 \sqrt{2}) (\sqrt{6} + \sqrt{2})}{8}$

Step 6.3

Expand $(3 \sqrt{6} - 3 \sqrt{2}) (\sqrt{6} + \sqrt{2})$ using the FOIL Method.

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Step 6.3.1

Apply the distributive property.

$\frac{3 \sqrt{6} (\sqrt{6} + \sqrt{2}) - 3 \sqrt{2} (\sqrt{6} + \sqrt{2})}{8}$

Step 6.3.2

Apply the distributive property.

$\frac{3 \sqrt{6} \sqrt{6} + 3 \sqrt{6} \sqrt{2} - 3 \sqrt{2} (\sqrt{6} + \sqrt{2})}{8}$

Step 6.3.3

Apply the distributive property.

$\frac{3 \sqrt{6} \sqrt{6} + 3 \sqrt{6} \sqrt{2} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4

Simplify and combine like terms.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

Multiply $3 \sqrt{6} \sqrt{6}$ .

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Step 6.4.1.1.1

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{3 ({\sqrt{6}}^{1} \sqrt{6}) + 3 \sqrt{6} \sqrt{2} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.1.2

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{3 ({\sqrt{6}}^{1} {\sqrt{6}}^{1}) + 3 \sqrt{6} \sqrt{2} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.1.3

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

$\frac{3 {\sqrt{6}}^{1 + 1} + 3 \sqrt{6} \sqrt{2} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.1.4

Add $1$ and $1$ .

$\frac{3 {\sqrt{6}}^{2} + 3 \sqrt{6} \sqrt{2} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.2

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 6.4.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{3 {(6^{\frac{1}{2}})}^{2} + 3 \sqrt{6} \sqrt{2} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.2.2

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

$\frac{3 \cdot 6^{\frac{1}{2} \cdot 2} + 3 \sqrt{6} \sqrt{2} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.2.3

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

$\frac{3 \cdot 6^{\frac{2}{2}} + 3 \sqrt{6} \sqrt{2} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.2.4

Cancel the common factor of $2$ .

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Step 6.4.1.2.4.1

Cancel the common factor.

$\frac{3 \cdot 6^{\frac{2}{2}} + 3 \sqrt{6} \sqrt{2} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.2.4.2

Rewrite the expression.

$\frac{3 \cdot 6^{1} + 3 \sqrt{6} \sqrt{2} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.2.5

Evaluate the exponent.

$\frac{3 \cdot 6 + 3 \sqrt{6} \sqrt{2} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.3

Multiply $3$ by $6$ .

$\frac{18 + 3 \sqrt{6} \sqrt{2} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.4

Multiply $3 \sqrt{6} \sqrt{2}$ .

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Step 6.4.1.4.1

Combine using the product rule for radicals.

$\frac{18 + 3 \sqrt{2 \cdot 6} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.4.2

Multiply $2$ by $6$ .

$\frac{18 + 3 \sqrt{12} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.5

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 6.4.1.5.1

Factor $4$ out of $12$ .

$\frac{18 + 3 \sqrt{4 (3)} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.5.2

Rewrite $4$ as $2^{2}$ .

$\frac{18 + 3 \sqrt{2^{2} \cdot 3} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.6

Pull terms out from under the radical.

$\frac{18 + 3 (2 \sqrt{3}) - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.7

Multiply $2$ by $3$ .

$\frac{18 + 6 \sqrt{3} - 3 \sqrt{2} \sqrt{6} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.8

Multiply $- 3 \sqrt{2} \sqrt{6}$ .

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Step 6.4.1.8.1

Combine using the product rule for radicals.

$\frac{18 + 6 \sqrt{3} - 3 \sqrt{6 \cdot 2} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.8.2

Multiply $6$ by $2$ .

$\frac{18 + 6 \sqrt{3} - 3 \sqrt{12} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.9

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 6.4.1.9.1

Factor $4$ out of $12$ .

$\frac{18 + 6 \sqrt{3} - 3 \sqrt{4 (3)} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.9.2

Rewrite $4$ as $2^{2}$ .

$\frac{18 + 6 \sqrt{3} - 3 \sqrt{2^{2} \cdot 3} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.10

Pull terms out from under the radical.

$\frac{18 + 6 \sqrt{3} - 3 (2 \sqrt{3}) - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.11

Multiply $2$ by $- 3$ .

$\frac{18 + 6 \sqrt{3} - 6 \sqrt{3} - 3 \sqrt{2} \sqrt{2}}{8}$

Step 6.4.1.12

Multiply $- 3 \sqrt{2} \sqrt{2}$ .

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Step 6.4.1.12.1

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{18 + 6 \sqrt{3} - 6 \sqrt{3} - 3 ({\sqrt{2}}^{1} \sqrt{2})}{8}$

Step 6.4.1.12.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{18 + 6 \sqrt{3} - 6 \sqrt{3} - 3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{8}$

Step 6.4.1.12.3

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

$\frac{18 + 6 \sqrt{3} - 6 \sqrt{3} - 3 {\sqrt{2}}^{1 + 1}}{8}$

Step 6.4.1.12.4

Add $1$ and $1$ .

$\frac{18 + 6 \sqrt{3} - 6 \sqrt{3} - 3 {\sqrt{2}}^{2}}{8}$

Step 6.4.1.13

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

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Step 6.4.1.13.1

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

$\frac{18 + 6 \sqrt{3} - 6 \sqrt{3} - 3 {(2^{\frac{1}{2}})}^{2}}{8}$

Step 6.4.1.13.2

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

$\frac{18 + 6 \sqrt{3} - 6 \sqrt{3} - 3 \cdot 2^{\frac{1}{2} \cdot 2}}{8}$

Step 6.4.1.13.3

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

$\frac{18 + 6 \sqrt{3} - 6 \sqrt{3} - 3 \cdot 2^{\frac{2}{2}}}{8}$

Step 6.4.1.13.4

Cancel the common factor of $2$ .

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Step 6.4.1.13.4.1

Cancel the common factor.

$\frac{18 + 6 \sqrt{3} - 6 \sqrt{3} - 3 \cdot 2^{\frac{2}{2}}}{8}$

Step 6.4.1.13.4.2

Rewrite the expression.

$\frac{18 + 6 \sqrt{3} - 6 \sqrt{3} - 3 \cdot 2^{1}}{8}$

Step 6.4.1.13.5

Evaluate the exponent.

$\frac{18 + 6 \sqrt{3} - 6 \sqrt{3} - 3 \cdot 2}{8}$

Step 6.4.1.14

Multiply $- 3$ by $2$ .

$\frac{18 + 6 \sqrt{3} - 6 \sqrt{3} - 6}{8}$

Step 6.4.2

Subtract $6$ from $18$ .

$\frac{12 + 6 \sqrt{3} - 6 \sqrt{3}}{8}$

Step 6.4.3

Subtract $6 \sqrt{3}$ from $6 \sqrt{3}$ .

$\frac{12 + 0}{8}$

Step 6.4.4

Add $12$ and $0$ .

$\frac{12}{8}$

Step 7

Cancel the common factor of $12$ and $8$ .

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Step 7.1

Factor $4$ out of $12$ .

$\frac{4 (3)}{8}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $4$ out of $8$ .

$\frac{4 \cdot 3}{4 \cdot 2}$

Step 7.2.2

Cancel the common factor.

$\frac{4 \cdot 3}{4 \cdot 2}$

Step 7.2.3

Rewrite the expression.

$\frac{3}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$

Mixed Number Form:

$1 \frac{1}{2}$