Trigonometry Examples

\sin (\frac{π}{12})

$\sin (\frac{π}{12})$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case,

\frac{π}{12}

$\frac{π}{12}$ can be split into

\frac{π}{3} - \frac{π}{4}

$\frac{π}{3} - \frac{π}{4}$ .

\sin (\frac{π}{3} - \frac{π}{4})

$\sin (\frac{π}{3} - \frac{π}{4})$

Step 2

Use the difference formula for sine to simplify the expression. The formula states that

\sin (A - B) = \sin (A) \cos (B) - \cos (A) \sin (B)

$\sin (A - B) = \sin (A) \cos (B) - \cos (A) \sin (B)$ .

\sin (\frac{π}{3}) \cos (\frac{π}{4}) - \cos (\frac{π}{3}) \sin (\frac{π}{4})

$\sin (\frac{π}{3}) \cos (\frac{π}{4}) - \cos (\frac{π}{3}) \sin (\frac{π}{4})$

Step 3

Remove parentheses.

\sin (\frac{π}{3}) \cos (\frac{π}{4}) - \cos (\frac{π}{3}) \sin (\frac{π}{4})

$\sin (\frac{π}{3}) \cos (\frac{π}{4}) - \cos (\frac{π}{3}) \sin (\frac{π}{4})$

Step 4

Simplify each term.

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Step 4.1

The exact value of

\sin (\frac{π}{3})

$\sin (\frac{π}{3})$ is

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

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

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

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

Step 4.2

The exact value of

\cos (\frac{π}{4})

$\cos (\frac{π}{4})$ is

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

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

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

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

Step 4.3

Multiply

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

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

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Step 4.3.1

Multiply

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

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

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

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

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

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

Step 4.3.2

Combine using the product rule for radicals.

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

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

Step 4.3.3

Multiply

3

$3$ by

2

$2$ .

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

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

Step 4.3.4

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

Step 4.4

The exact value of

\cos (\frac{π}{3})

$\cos (\frac{π}{3})$ is

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 4.5

The exact value of

\sin (\frac{π}{4})

$\sin (\frac{π}{4})$ is

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

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

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

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

Step 4.6

Multiply

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

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

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Step 4.6.1

Multiply

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

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

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 4.6.2

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

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

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

Step 5

Combine the numerators over the common denominator.

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

0.25881904 \dots

$0.25881904 \dots$