Trigonometry Examples

$\cos (\frac{11 π}{12}) \cos (\frac{2 π}{3}) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (\frac{11 π}{12})$ is $- \frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 1.1.1

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

$- \cos (\frac{π}{12}) \cos (\frac{2 π}{3}) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.1.2

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

$- \cos (\frac{π}{4} - \frac{π}{6}) \cos (\frac{2 π}{3}) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.1.3

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

$- (\cos (\frac{π}{4}) \cos (\frac{π}{6}) + \sin (\frac{π}{4}) \sin (\frac{π}{6})) \cos (\frac{2 π}{3}) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.1.4

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

$- (\frac{\sqrt{2}}{2} \cos (\frac{π}{6}) + \sin (\frac{π}{4}) \sin (\frac{π}{6})) \cos (\frac{2 π}{3}) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.1.5

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

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (\frac{π}{4}) \sin (\frac{π}{6})) \cos (\frac{2 π}{3}) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.1.6

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

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (\frac{π}{6})) \cos (\frac{2 π}{3}) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.1.7

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

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{2 π}{3}) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.1.8

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

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Step 1.1.8.1

Simplify each term.

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Step 1.1.8.1.1

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

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Step 1.1.8.1.1.1

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

$- (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{2 π}{3}) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.1.8.1.1.2

Combine using the product rule for radicals.

$- (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{2 π}{3}) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.1.8.1.1.3

Multiply $2$ by $3$ .

$- (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{2 π}{3}) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.1.8.1.1.4

Multiply $2$ by $2$ .

$- (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{2 π}{3}) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.1.8.1.2

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

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Step 1.1.8.1.2.1

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

$- (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) \cos (\frac{2 π}{3}) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.1.8.1.2.2

Multiply $2$ by $2$ .

$- (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) \cos (\frac{2 π}{3}) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.1.8.2

Combine the numerators over the common denominator.

$- \frac{\sqrt{6} + \sqrt{2}}{4} \cos (\frac{2 π}{3}) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.2

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

$- \frac{\sqrt{6} + \sqrt{2}}{4} (- \cos (\frac{π}{3})) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.3

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$- \frac{\sqrt{6} + \sqrt{2}}{4} (- \frac{1}{2}) + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.4

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

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Step 1.4.1

Multiply $- 1$ by $- 1$ .

$1 \frac{\sqrt{6} + \sqrt{2}}{4} \frac{1}{2} + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.4.2

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

$\frac{\sqrt{6} + \sqrt{2}}{4} \cdot \frac{1}{2} + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.4.3

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

$\frac{\sqrt{6} + \sqrt{2}}{4 \cdot 2} + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.4.4

Multiply $4$ by $2$ .

$\frac{\sqrt{6} + \sqrt{2}}{8} + \sin (\frac{11 π}{12}) \sin (\frac{2 π}{3})$

Step 1.5

The exact value of $\sin (\frac{11 π}{12})$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

Apply the difference of angles identity.

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

Step 1.5.4

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

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

Step 1.5.5

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

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

Step 1.5.6

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

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

Step 1.5.7

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

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

Step 1.5.8

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

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Step 1.5.8.1

Simplify each term.

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Step 1.5.8.1.1

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

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Step 1.5.8.1.1.1

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

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

Step 1.5.8.1.1.2

Combine using the product rule for radicals.

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

Step 1.5.8.1.1.3

Multiply $2$ by $3$ .

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

Step 1.5.8.1.1.4

Multiply $2$ by $2$ .

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

Step 1.5.8.1.2

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

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Step 1.5.8.1.2.1

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

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

Step 1.5.8.1.2.2

Multiply $2$ by $2$ .

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

Step 1.5.8.2

Combine the numerators over the common denominator.

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

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Step 1.8.1

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

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

Step 1.8.2

Multiply $4$ by $2$ .

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

Step 1.9

Apply the distributive property.

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

Step 1.10

Combine using the product rule for radicals.

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

Step 1.11

Multiply $- \sqrt{2} \sqrt{3}$ .

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Step 1.11.1

Combine using the product rule for radicals.

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

Step 1.11.2

Multiply $3$ by $2$ .

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

Step 1.12

Simplify each term.

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Step 1.12.1

Multiply $6$ by $3$ .

$\frac{\sqrt{6} + \sqrt{2}}{8} + \frac{\sqrt{18} - \sqrt{6}}{8}$

Step 1.12.2

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

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Step 1.12.2.1

Factor $9$ out of $18$ .

$\frac{\sqrt{6} + \sqrt{2}}{8} + \frac{\sqrt{9 (2)} - \sqrt{6}}{8}$

Step 1.12.2.2

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

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

Step 1.12.3

Pull terms out from under the radical.

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

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

$\frac{0 + \sqrt{2} + 3 \sqrt{2}}{8}$

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Cancel the common factor of $4$ and $8$ .

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Step 2.5.1

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

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

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $4$ out of $8$ .

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

Step 2.5.2.2

Cancel the common factor.

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

Step 2.5.2.3

Rewrite the expression.

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.70710678 \dots$