Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Apply the difference of angles identity.

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

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

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

Step 1.1.7

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

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Step 1.1.7.1

Simplify each term.

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Step 1.1.7.1.1

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

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Step 1.1.7.1.1.1

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

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

Step 1.1.7.1.1.2

Combine using the product rule for radicals.

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

Step 1.1.7.1.1.3

Multiply $2$ by $3$ .

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

Step 1.1.7.1.1.4

Multiply $2$ by $2$ .

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

Step 1.1.7.1.2

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

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Step 1.1.7.1.2.1

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

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

Step 1.1.7.1.2.2

Multiply $2$ by $2$ .

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

Step 1.1.7.2

Combine the numerators over the common denominator.

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

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.

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Multiply $4$ by $2$ .

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

Step 1.5

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

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

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

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

Step 1.5.4

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

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

Step 1.5.5

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

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

Step 1.5.6

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

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

Step 1.5.7

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

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Step 1.5.7.1

Simplify each term.

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Step 1.5.7.1.1

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

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Step 1.5.7.1.1.1

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

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

Step 1.5.7.1.1.2

Combine using the product rule for radicals.

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

Step 1.5.7.1.1.3

Multiply $2$ by $3$ .

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

Step 1.5.7.1.1.4

Multiply $2$ by $2$ .

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

Step 1.5.7.1.2

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

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Step 1.5.7.1.2.1

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

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

Step 1.5.7.1.2.2

Multiply $2$ by $2$ .

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

Step 1.5.7.2

Combine the numerators over the common denominator.

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

Step 1.6

Evaluate $\sin (\frac{2}{3})$ .

$- \frac{\sqrt{6} - \sqrt{2}}{8} + \frac{\sqrt{6} + \sqrt{2}}{4} \cdot 0.6183698$

Step 1.7

Multiply $\frac{\sqrt{6} + \sqrt{2}}{4} \cdot 0.6183698$ .

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Step 1.7.1

Combine $\frac{\sqrt{6} + \sqrt{2}}{4}$ and $0.6183698$ .

$- \frac{\sqrt{6} - \sqrt{2}}{8} + \frac{(\sqrt{6} + \sqrt{2}) \cdot 0.6183698}{4}$

Step 1.7.2

Multiply $\sqrt{6} + \sqrt{2}$ by $0.6183698$ .

$- \frac{\sqrt{6} - \sqrt{2}}{8} + \frac{2.38919745}{4}$

Step 1.8

Divide $2.38919745$ by $4$ .

$- \frac{\sqrt{6} - \sqrt{2}}{8} + 0.59729936$

Step 2

To write $0.59729936$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- \frac{\sqrt{6} - \sqrt{2}}{8} + 0.59729936 \cdot \frac{8}{8}$

Step 3

Combine fractions.

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Step 3.1

Combine $0.59729936$ and $\frac{8}{8}$ .

$- \frac{\sqrt{6} - \sqrt{2}}{8} + \frac{0.59729936 \cdot 8}{8}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{- (\sqrt{6} - \sqrt{2}) + 0.59729936 \cdot 8}{8}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{- \sqrt{6} - - \sqrt{2} + 0.59729936 \cdot 8}{8}$

Step 4.2

Multiply $- - \sqrt{2}$ .

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- \sqrt{6} + 1 \sqrt{2} + 0.59729936 \cdot 8}{8}$

Step 4.2.2

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

$\frac{- \sqrt{6} + \sqrt{2} + 0.59729936 \cdot 8}{8}$

Step 4.3

Multiply $0.59729936$ by $8$ .

$\frac{- \sqrt{6} + \sqrt{2} + 4.7783949}{8}$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $- 1$ out of $- \sqrt{6}$ .

$\frac{- (\sqrt{6}) + \sqrt{2} + 4.7783949}{8}$

Step 5.2

Factor $- 1$ out of $\sqrt{2}$ .

$\frac{- (\sqrt{6}) - 1 (- \sqrt{2}) + 4.7783949}{8}$

Step 5.3

Factor $- 1$ out of $- (\sqrt{6}) - 1 (- \sqrt{2})$ .

$\frac{- (\sqrt{6} - \sqrt{2}) + 4.7783949}{8}$

Step 5.4

Rewrite $4.7783949$ as $- 1 (- 4.7783949)$ .

$\frac{- (\sqrt{6} - \sqrt{2}) - 1 (- 4.7783949)}{8}$

Step 5.5

Factor $- 1$ out of $- (\sqrt{6} - \sqrt{2}) - 1 (- 4.7783949)$ .

$\frac{- (\sqrt{6} - \sqrt{2} - 4.7783949)}{8}$

Step 5.6

Simplify the expression.

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Step 5.6.1

Rewrite $- (\sqrt{6} - \sqrt{2} - 4.7783949)$ as $- 1 (\sqrt{6} - \sqrt{2} - 4.7783949)$ .

$\frac{- 1 (\sqrt{6} - \sqrt{2} - 4.7783949)}{8}$

Step 5.6.2

Move the negative in front of the fraction.

$- \frac{\sqrt{6} - \sqrt{2} - 4.7783949}{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{6} - \sqrt{2} - 4.7783949}{8}$

Decimal Form:

$0.46788984 \dots$