Trigonometry Examples

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

Step 1

The exact value of $\cos (\frac{π}{12})$ 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.

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

$(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (- \frac{π}{6}) + \sin (\frac{π}{12}) \sin (- \frac{π}{6})$

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

$(\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (- \frac{π}{6}) + \sin (\frac{π}{12}) \sin (- \frac{π}{6})$

Step 1.7.1.1.2

Combine using the product rule for radicals.

$(\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (- \frac{π}{6}) + \sin (\frac{π}{12}) \sin (- \frac{π}{6})$

Step 1.7.1.1.3

Multiply $2$ by $3$ .

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

Step 1.7.1.1.4

Multiply $2$ by $2$ .

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

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

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

Step 1.7.1.2.2

Multiply $2$ by $2$ .

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

Step 1.7.2

Combine the numerators over the common denominator.

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

Step 2

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{\sqrt{6} + \sqrt{2}}{4} \cos (\frac{11 π}{6}) + \sin (\frac{π}{12}) \sin (- \frac{π}{6})$

Step 3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

Step 4

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

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

Step 5

Multiply $\frac{\sqrt{6} + \sqrt{2}}{4} \cdot \frac{\sqrt{3}}{2}$ .

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Step 5.1

Multiply $\frac{\sqrt{6} + \sqrt{2}}{4}$ by $\frac{\sqrt{3}}{2}$ .

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

Step 5.2

Multiply $4$ by $2$ .

$\frac{(\sqrt{6} + \sqrt{2}) \sqrt{3}}{8} + \sin (\frac{π}{12}) \sin (- \frac{π}{6})$

Step 6

Apply the distributive property.

$\frac{\sqrt{6} \sqrt{3} + \sqrt{2} \sqrt{3}}{8} + \sin (\frac{π}{12}) \sin (- \frac{π}{6})$

Step 7

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 3} + \sqrt{2} \sqrt{3}}{8} + \sin (\frac{π}{12}) \sin (- \frac{π}{6})$

Step 8

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 3} + \sqrt{2 \cdot 3}}{8} + \sin (\frac{π}{12}) \sin (- \frac{π}{6})$

Step 9

Simplify each term.

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Step 9.1

Multiply $6$ by $3$ .

$\frac{\sqrt{18} + \sqrt{2 \cdot 3}}{8} + \sin (\frac{π}{12}) \sin (- \frac{π}{6})$

Step 9.2

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 9.2.1

Factor $9$ out of $18$ .

$\frac{\sqrt{9 (2)} + \sqrt{2 \cdot 3}}{8} + \sin (\frac{π}{12}) \sin (- \frac{π}{6})$

Step 9.2.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 2} + \sqrt{2 \cdot 3}}{8} + \sin (\frac{π}{12}) \sin (- \frac{π}{6})$

Step 9.3

Pull terms out from under the radical.

$\frac{3 \sqrt{2} + \sqrt{2 \cdot 3}}{8} + \sin (\frac{π}{12}) \sin (- \frac{π}{6})$

Step 9.4

Multiply $2$ by $3$ .

$\frac{3 \sqrt{2} + \sqrt{6}}{8} + \sin (\frac{π}{12}) \sin (- \frac{π}{6})$

Step 10

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

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Step 10.1

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

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

Step 10.2

Apply the difference of angles identity.

$\frac{3 \sqrt{2} + \sqrt{6}}{8} + (\sin (\frac{π}{4}) \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6})) \sin (- \frac{π}{6})$

Step 10.3

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

$\frac{3 \sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{2}}{2} \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6})) \sin (- \frac{π}{6})$

Step 10.4

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

$\frac{3 \sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (\frac{π}{4}) \sin (\frac{π}{6})) \sin (- \frac{π}{6})$

Step 10.5

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

$\frac{3 \sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (\frac{π}{6})) \sin (- \frac{π}{6})$

Step 10.6

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

$\frac{3 \sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (- \frac{π}{6})$

Step 10.7

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

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Step 10.7.1

Simplify each term.

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Step 10.7.1.1

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

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Step 10.7.1.1.1

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

$\frac{3 \sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (- \frac{π}{6})$

Step 10.7.1.1.2

Combine using the product rule for radicals.

$\frac{3 \sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (- \frac{π}{6})$

Step 10.7.1.1.3

Multiply $2$ by $3$ .

$\frac{3 \sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (- \frac{π}{6})$

Step 10.7.1.1.4

Multiply $2$ by $2$ .

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

Step 10.7.1.2

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

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Step 10.7.1.2.1

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

$\frac{3 \sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) \sin (- \frac{π}{6})$

Step 10.7.1.2.2

Multiply $2$ by $2$ .

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

Step 10.7.2

Combine the numerators over the common denominator.

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

Step 11

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{3 \sqrt{2} + \sqrt{6}}{8} + \frac{\sqrt{6} - \sqrt{2}}{4} \sin (\frac{11 π}{6})$

Step 12

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

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

Step 13

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

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

Step 14

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4} (- \frac{1}{2})$ .

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Step 14.1

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4}$ by $\frac{1}{2}$ .

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

Step 14.2

Multiply $4$ by $2$ .

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

Step 15

Combine the numerators over the common denominator.

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

Step 16

Rewrite $\frac{3 \sqrt{2} + \sqrt{6} - (\sqrt{6} - \sqrt{2})}{8}$ in a factored form.

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Step 16.1

Apply the distributive property.

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

Step 16.2

Multiply $- - \sqrt{2}$ .

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Step 16.2.1

Multiply $- 1$ by $- 1$ .

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

Step 16.2.2

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

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

Step 16.3

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

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

Step 16.4

Subtract $\sqrt{6}$ from $\sqrt{6}$ .

$\frac{4 \sqrt{2} + 0}{8}$

Step 16.5

Add $4 \sqrt{2}$ and $0$ .

$\frac{4 \sqrt{2}}{8}$

Step 16.6

Reduce the expression $\frac{4 \sqrt{2}}{8}$ by cancelling the common factors.

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Step 16.6.1

Factor $4$ out of $4 \sqrt{2}$ .

$\frac{4 (\sqrt{2})}{8}$

Step 16.6.2

Factor $4$ out of $8$ .

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

Step 16.6.3

Cancel the common factor.

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

Step 16.6.4

Rewrite the expression.

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

Step 17

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.70710678 \dots$