Trigonometry Examples

$\cos (\frac{5 π}{12}) \cos (\frac{π}{4}) + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (\frac{5 π}{12})$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 1.1.1

Split $\frac{5 π}{12}$ into two angles where the values of the six trigonometric functions are known.

$\cos (\frac{π}{6} + \frac{π}{4}) \cos (\frac{π}{4}) + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.1.2

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

$(\cos (\frac{π}{6}) \cos (\frac{π}{4}) - \sin (\frac{π}{6}) \sin (\frac{π}{4})) \cos (\frac{π}{4}) + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.1.3

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

$(\frac{\sqrt{3}}{2} \cos (\frac{π}{4}) - \sin (\frac{π}{6}) \sin (\frac{π}{4})) \cos (\frac{π}{4}) + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.1.4

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

$(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (\frac{π}{6}) \sin (\frac{π}{4})) \cos (\frac{π}{4}) + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.1.5

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

$(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (\frac{π}{4})) \cos (\frac{π}{4}) + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.1.6

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

$(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \cos (\frac{π}{4}) + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.1.7

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

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Step 1.1.7.1

Simplify each term.

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Step 1.1.7.1.1

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

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Step 1.1.7.1.1.1

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

$(\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \cos (\frac{π}{4}) + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.1.7.1.1.2

Combine using the product rule for radicals.

$(\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \cos (\frac{π}{4}) + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.1.7.1.1.3

Multiply $3$ by $2$ .

$(\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \cos (\frac{π}{4}) + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.1.7.1.1.4

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \cos (\frac{π}{4}) + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.1.7.1.2

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

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Step 1.1.7.1.2.1

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

$(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) \cos (\frac{π}{4}) + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.1.7.1.2.2

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) \cos (\frac{π}{4}) + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.1.7.2

Combine the numerators over the common denominator.

$\frac{\sqrt{6} - \sqrt{2}}{4} \cos (\frac{π}{4}) + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.2

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

$\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{\sqrt{2}}{2} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.3

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{\sqrt{2}}{2}$ .

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Step 1.3.1

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{(\sqrt{6} - \sqrt{2}) \sqrt{2}}{4 \cdot 2} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.3.2

Multiply $4$ by $2$ .

$\frac{(\sqrt{6} - \sqrt{2}) \sqrt{2}}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.4

Apply the distributive property.

$\frac{\sqrt{6} \sqrt{2} - \sqrt{2} \sqrt{2}}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.5

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 2} - \sqrt{2} \sqrt{2}}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.6

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

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Step 1.6.1

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

$\frac{\sqrt{6 \cdot 2} - ({\sqrt{2}}^{1} \sqrt{2})}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.6.2

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

$\frac{\sqrt{6 \cdot 2} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.6.3

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

$\frac{\sqrt{6 \cdot 2} - {\sqrt{2}}^{1 + 1}}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.6.4

Add $1$ and $1$ .

$\frac{\sqrt{6 \cdot 2} - {\sqrt{2}}^{2}}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.7

Simplify each term.

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Step 1.7.1

Multiply $6$ by $2$ .

$\frac{\sqrt{12} - {\sqrt{2}}^{2}}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.7.2

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

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Step 1.7.2.1

Factor $4$ out of $12$ .

$\frac{\sqrt{4 (3)} - {\sqrt{2}}^{2}}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.7.2.2

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

$\frac{\sqrt{2^{2} \cdot 3} - {\sqrt{2}}^{2}}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.7.3

Pull terms out from under the radical.

$\frac{2 \sqrt{3} - {\sqrt{2}}^{2}}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.7.4

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

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Step 1.7.4.1

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

$\frac{2 \sqrt{3} - {(2^{\frac{1}{2}})}^{2}}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.7.4.2

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

$\frac{2 \sqrt{3} - 2^{\frac{1}{2} \cdot 2}}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.7.4.3

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

$\frac{2 \sqrt{3} - 2^{\frac{2}{2}}}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.7.4.4

Cancel the common factor of $2$ .

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Step 1.7.4.4.1

Cancel the common factor.

$\frac{2 \sqrt{3} - 2^{\frac{2}{2}}}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.7.4.4.2

Rewrite the expression.

$\frac{2 \sqrt{3} - 2^{1}}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.7.4.5

Evaluate the exponent.

$\frac{2 \sqrt{3} - 1 \cdot 2}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.7.5

Multiply $- 1$ by $2$ .

$\frac{2 \sqrt{3} - 2}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.8

Cancel the common factor of $2 \sqrt{3} - 2$ and $8$ .

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Step 1.8.1

Factor $2$ out of $2 \sqrt{3}$ .

$\frac{2 (\sqrt{3}) - 2}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.8.2

Factor $2$ out of $- 2$ .

$\frac{2 (\sqrt{3}) + 2 (- 1)}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.8.3

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

$\frac{2 (\sqrt{3} - 1)}{8} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.8.4

Cancel the common factors.

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Step 1.8.4.1

Factor $2$ out of $8$ .

$\frac{2 (\sqrt{3} - 1)}{2 (4)} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.8.4.2

Cancel the common factor.

$\frac{2 (\sqrt{3} - 1)}{2 \cdot 4} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.8.4.3

Rewrite the expression.

$\frac{\sqrt{3} - 1}{4} + \sin (\frac{5 π}{12}) \sin (\frac{π}{4})$

Step 1.9

The exact value of $\sin (\frac{5 π}{12})$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

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Step 1.9.1

Split $\frac{5 π}{12}$ into two angles where the values of the six trigonometric functions are known.

$\frac{\sqrt{3} - 1}{4} + \sin (\frac{π}{6} + \frac{π}{4}) \sin (\frac{π}{4})$

Step 1.9.2

Apply the sum of angles identity.

$\frac{\sqrt{3} - 1}{4} + (\sin (\frac{π}{6}) \cos (\frac{π}{4}) + \cos (\frac{π}{6}) \sin (\frac{π}{4})) \sin (\frac{π}{4})$

Step 1.9.3

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

$\frac{\sqrt{3} - 1}{4} + (\frac{1}{2} \cos (\frac{π}{4}) + \cos (\frac{π}{6}) \sin (\frac{π}{4})) \sin (\frac{π}{4})$

Step 1.9.4

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

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

Step 1.9.5

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

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

Step 1.9.6

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

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

Step 1.9.7

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

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Step 1.9.7.1

Simplify each term.

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Step 1.9.7.1.1

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

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Step 1.9.7.1.1.1

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

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

Step 1.9.7.1.1.2

Multiply $2$ by $2$ .

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

Step 1.9.7.1.2

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

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Step 1.9.7.1.2.1

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

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

Step 1.9.7.1.2.2

Combine using the product rule for radicals.

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

Step 1.9.7.1.2.3

Multiply $3$ by $2$ .

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

Step 1.9.7.1.2.4

Multiply $2$ by $2$ .

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

Step 1.9.7.2

Combine the numerators over the common denominator.

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

Step 1.10

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

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

Step 1.11

Multiply $\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{\sqrt{2}}{2}$ .

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Step 1.11.1

Multiply $\frac{\sqrt{2} + \sqrt{6}}{4}$ by $\frac{\sqrt{2}}{2}$ .

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

Step 1.11.2

Multiply $4$ by $2$ .

$\frac{\sqrt{3} - 1}{4} + \frac{(\sqrt{2} + \sqrt{6}) \sqrt{2}}{8}$

Step 1.12

Apply the distributive property.

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{2} \sqrt{2} + \sqrt{6} \sqrt{2}}{8}$

Step 1.13

Combine using the product rule for radicals.

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{2 \cdot 2} + \sqrt{6} \sqrt{2}}{8}$

Step 1.14

Combine using the product rule for radicals.

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{2 \cdot 2} + \sqrt{6 \cdot 2}}{8}$

Step 1.15

Simplify each term.

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Step 1.15.1

Multiply $2$ by $2$ .

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{4} + \sqrt{6 \cdot 2}}{8}$

Step 1.15.2

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

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{2^{2}} + \sqrt{6 \cdot 2}}{8}$

Step 1.15.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{3} - 1}{4} + \frac{2 + \sqrt{6 \cdot 2}}{8}$

Step 1.15.4

Multiply $6$ by $2$ .

$\frac{\sqrt{3} - 1}{4} + \frac{2 + \sqrt{12}}{8}$

Step 1.15.5

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

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Step 1.15.5.1

Factor $4$ out of $12$ .

$\frac{\sqrt{3} - 1}{4} + \frac{2 + \sqrt{4 (3)}}{8}$

Step 1.15.5.2

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

$\frac{\sqrt{3} - 1}{4} + \frac{2 + \sqrt{2^{2} \cdot 3}}{8}$

Step 1.15.6

Pull terms out from under the radical.

$\frac{\sqrt{3} - 1}{4} + \frac{2 + 2 \sqrt{3}}{8}$

Step 1.16

Cancel the common factor of $2 + 2 \sqrt{3}$ and $8$ .

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Step 1.16.1

Factor $2$ out of $2$ .

$\frac{\sqrt{3} - 1}{4} + \frac{2 \cdot 1 + 2 \sqrt{3}}{8}$

Step 1.16.2

Factor $2$ out of $2 \sqrt{3}$ .

$\frac{\sqrt{3} - 1}{4} + \frac{2 \cdot 1 + 2 (\sqrt{3})}{8}$

Step 1.16.3

Factor $2$ out of $2 \cdot 1 + 2 (\sqrt{3})$ .

$\frac{\sqrt{3} - 1}{4} + \frac{2 \cdot (1 + \sqrt{3})}{8}$

Step 1.16.4

Cancel the common factors.

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Step 1.16.4.1

Factor $2$ out of $8$ .

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

Step 1.16.4.2

Cancel the common factor.

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

Step 1.16.4.3

Rewrite the expression.

$\frac{\sqrt{3} - 1}{4} + \frac{1 + \sqrt{3}}{4}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{\sqrt{3} - 1 + 1 + \sqrt{3}}{4}$

Step 2.2

Add $\sqrt{3}$ and $\sqrt{3}$ .

$\frac{- 1 + 1 + 2 \sqrt{3}}{4}$

Step 2.3

Simplify by adding numbers.

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Step 2.3.1

Add $- 1$ and $1$ .

$\frac{0 + 2 \sqrt{3}}{4}$

Step 2.3.2

Add $0$ and $2 \sqrt{3}$ .

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

Step 2.4

Cancel the common factor of $2$ and $4$ .

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Step 2.4.1

Factor $2$ out of $2 \sqrt{3}$ .

$\frac{2 (\sqrt{3})}{4}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $2$ out of $4$ .

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

Step 2.4.2.2

Cancel the common factor.

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

Step 2.4.2.3

Rewrite the expression.

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.86602540 \dots$