Trigonometry Examples

\cos (\frac{π}{12})

$\cos (\frac{π}{12})$

Step 1

Split

\frac{π}{12}

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

\cos (\frac{π}{4} - \frac{π}{6})

$\cos (\frac{π}{4} - \frac{π}{6})$

Step 2

Apply the difference of angles identity

\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)

$\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

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

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

Step 3

The exact value of

\cos (\frac{π}{4})

$\cos (\frac{π}{4})$ is

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

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

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

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

Step 4

The exact value of

\cos (\frac{π}{6})

$\cos (\frac{π}{6})$ is

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

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

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

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

Step 5

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

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

Step 6

The exact value of

\sin (\frac{π}{6})

$\sin (\frac{π}{6})$ is

\frac{1}{2}

$\frac{1}{2}$ .

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

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

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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}

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

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Step 7.1.1.1

Multiply

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

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

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

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

\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{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}

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

Step 7.1.1.3

Multiply

2

$2$ by

3

$3$ .

\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}

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

Step 7.1.1.4

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

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Step 7.1.2.1

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 7.1.2.2

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 7.2

Combine the numerators over the common denominator.

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

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

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

0.96592582 \dots

$0.96592582 \dots$