Trigonometry Examples

sin (\frac{π}{12})

$\sin (\frac{π}{12})$

Step 1

Split

\frac{π}{12}

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

sin (\frac{π}{4} - \frac{π}{6})

$\sin (\frac{π}{4} - \frac{π}{6})$

Step 2

Apply the difference of angles identity.

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

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

Step 3

The exact value of

sin (\frac{π}{4})

$\sin (\frac{π}{4})$ is

\frac{\sqrt{2}}{2}

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

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

Step 4

The exact value of

$\cos (\frac{π}{6})$ is

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

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

Step 5

The exact value of

$\cos (\frac{π}{4})$ is

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

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

Step 6

The exact value of

$\sin (\frac{π}{6})$ is

$\frac{1}{2}$ .

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

Step 7

Simplify

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

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply

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

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Step 7.1.1.1

Multiply

$\frac{\sqrt{2}}{2}$ by

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

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

Step 7.1.1.2

Combine using the product rule for radicals.

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

Step 7.1.1.3

Multiply

$2$ by

$3$ .

$\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Step 7.1.1.4

Multiply

$2$ by

$2$ .

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

Step 7.1.2

Multiply

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

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Step 7.1.2.1

Multiply

$\frac{1}{2}$ by

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

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

Step 7.1.2.2

Multiply

$2$ by

$2$ .

$\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$

Step 7.2

Combine the numerators over the common denominator.

$\frac{\sqrt{6} - \sqrt{2}}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6} - \sqrt{2}}{4}$

Decimal Form:

$0.25881904 \dots$