Trigonometry Examples

$\sin^{2} (\frac{π}{12})$

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.

$\sin^{2} (\frac{π}{4} - \frac{π}{6})$

Step 1.2

Apply the difference of angles identity.

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

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

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

Step 1.7.1.1.2

Combine using the product rule for radicals.

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

Step 1.7.1.1.3

Multiply $2$ by $3$ .

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

Step 1.7.1.1.4

Multiply $2$ by $2$ .

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

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 $\frac{\sqrt{2}}{2}$ .

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

Step 1.7.1.2.2

Multiply $2$ by $2$ .

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

Step 1.7.2

Combine the numerators over the common denominator.

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

Step 2

Simplify the expression.

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Step 2.1

Apply the product rule to $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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

Step 2.2

Raise $4$ to the power of $2$ .

$\frac{{(\sqrt{6} - \sqrt{2})}^{2}}{16}$

Step 2.3

Rewrite ${(\sqrt{6} - \sqrt{2})}^{2}$ as $(\sqrt{6} - \sqrt{2}) (\sqrt{6} - \sqrt{2})$ .

$\frac{(\sqrt{6} - \sqrt{2}) (\sqrt{6} - \sqrt{2})}{16}$

Step 3

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

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{6} (\sqrt{6} - \sqrt{2}) - \sqrt{2} (\sqrt{6} - \sqrt{2})}{16}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} (\sqrt{6} - \sqrt{2})}{16}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 6} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 4.1.2

Multiply $6$ by $6$ .

$\frac{\sqrt{36} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 4.1.3

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{6^{2}} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 4.1.4

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

$\frac{6 + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 4.1.5

Multiply $\sqrt{6} (- \sqrt{2})$ .

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Step 4.1.5.1

Combine using the product rule for radicals.

$\frac{6 - \sqrt{6 \cdot 2} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 4.1.5.2

Multiply $6$ by $2$ .

$\frac{6 - \sqrt{12} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 4.1.6

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

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Step 4.1.6.1

Factor $4$ out of $12$ .

$\frac{6 - \sqrt{4 (3)} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 4.1.6.2

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

$\frac{6 - \sqrt{2^{2} \cdot 3} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 4.1.7

Pull terms out from under the radical.

$\frac{6 - (2 \sqrt{3}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 4.1.8

Multiply $2$ by $- 1$ .

$\frac{6 - 2 \sqrt{3} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 4.1.9

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

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Step 4.1.9.1

Combine using the product rule for radicals.

$\frac{6 - 2 \sqrt{3} - \sqrt{6 \cdot 2} - \sqrt{2} (- \sqrt{2})}{16}$

Step 4.1.9.2

Multiply $6$ by $2$ .

$\frac{6 - 2 \sqrt{3} - \sqrt{12} - \sqrt{2} (- \sqrt{2})}{16}$

Step 4.1.10

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

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Step 4.1.10.1

Factor $4$ out of $12$ .

$\frac{6 - 2 \sqrt{3} - \sqrt{4 (3)} - \sqrt{2} (- \sqrt{2})}{16}$

Step 4.1.10.2

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

$\frac{6 - 2 \sqrt{3} - \sqrt{2^{2} \cdot 3} - \sqrt{2} (- \sqrt{2})}{16}$

Step 4.1.11

Pull terms out from under the radical.

$\frac{6 - 2 \sqrt{3} - (2 \sqrt{3}) - \sqrt{2} (- \sqrt{2})}{16}$

Step 4.1.12

Multiply $2$ by $- 1$ .

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} - \sqrt{2} (- \sqrt{2})}{16}$

Step 4.1.13

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

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Step 4.1.13.1

Multiply $- 1$ by $- 1$ .

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 1 \sqrt{2} \sqrt{2}}{16}$

Step 4.1.13.2

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

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{2} \sqrt{2}}{16}$

Step 4.1.13.3

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

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1} \sqrt{2}}{16}$

Step 4.1.13.4

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

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1} {\sqrt{2}}^{1}}{16}$

Step 4.1.13.5

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

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1 + 1}}{16}$

Step 4.1.13.6

Add $1$ and $1$ .

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{2}}{16}$

Step 4.1.14

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

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Step 4.1.14.1

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

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {(2^{\frac{1}{2}})}^{2}}{16}$

Step 4.1.14.2

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

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{\frac{1}{2} \cdot 2}}{16}$

Step 4.1.14.3

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

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{\frac{2}{2}}}{16}$

Step 4.1.14.4

Cancel the common factor of $2$ .

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Step 4.1.14.4.1

Cancel the common factor.

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{\frac{2}{2}}}{16}$

Step 4.1.14.4.2

Rewrite the expression.

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{1}}{16}$

Step 4.1.14.5

Evaluate the exponent.

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2}{16}$

Step 4.2

Add $6$ and $2$ .

$\frac{8 - 2 \sqrt{3} - 2 \sqrt{3}}{16}$

Step 4.3

Subtract $2 \sqrt{3}$ from $- 2 \sqrt{3}$ .

$\frac{8 - 4 \sqrt{3}}{16}$

Step 5

Cancel the common factor of $8 - 4 \sqrt{3}$ and $16$ .

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Step 5.1

Factor $4$ out of $8$ .

$\frac{4 (2) - 4 \sqrt{3}}{16}$

Step 5.2

Factor $4$ out of $- 4 \sqrt{3}$ .

$\frac{4 (2) + 4 (- \sqrt{3})}{16}$

Step 5.3

Factor $4$ out of $4 (2) + 4 (- \sqrt{3})$ .

$\frac{4 (2 - \sqrt{3})}{16}$

Step 5.4

Cancel the common factors.

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Step 5.4.1

Factor $4$ out of $16$ .

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

Step 5.4.2

Cancel the common factor.

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

Step 5.4.3

Rewrite the expression.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.06698729 \dots$