Trigonometry Examples

$\sin (\frac{π}{4}) \cos (\frac{11 π}{12})$

Step 1

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

$\frac{\sqrt{2}}{2} \cos (\frac{11 π}{12})$

Step 2

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

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Step 2.1

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.

$\frac{\sqrt{2}}{2} (- \cos (\frac{π}{12}))$

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

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

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Step 2.8.1.1.1

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

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

Step 2.8.1.1.2

Combine using the product rule for radicals.

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

Step 2.8.1.1.3

Multiply $2$ by $3$ .

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

Step 2.8.1.1.4

Multiply $2$ by $2$ .

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

Step 2.8.1.2

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

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Step 2.8.1.2.1

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

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

Step 2.8.1.2.2

Multiply $2$ by $2$ .

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

Step 2.8.2

Combine the numerators over the common denominator.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Multiply $2$ by $4$ .

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

Step 4

Simplify terms.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Combine using the product rule for radicals.

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

Step 4.3

Combine using the product rule for radicals.

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

Step 5

Simplify each term.

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Step 5.1

Multiply $2$ by $6$ .

$- \frac{\sqrt{12} + \sqrt{2 \cdot 2}}{8}$

Step 5.2

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.2.1

Factor $4$ out of $12$ .

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

Step 5.2.2

Rewrite $4$ as $2^{2}$ .

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

Step 5.3

Pull terms out from under the radical.

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

Step 5.4

Multiply $2$ by $2$ .

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

Step 5.5

Rewrite $4$ as $2^{2}$ .

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

Step 5.6

Pull terms out from under the radical, assuming positive real numbers.

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

Step 6

Cancel the common factor of $2 \sqrt{3} + 2$ and $8$ .

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Step 6.1

Factor $2$ out of $2 \sqrt{3}$ .

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

Step 6.2

Factor $2$ out of $2$ .

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

Step 6.3

Factor $2$ out of $2 (\sqrt{3}) + 2 (1)$ .

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

Step 6.4

Cancel the common factors.

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Step 6.4.1

Factor $2$ out of $8$ .

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

Step 6.4.2

Cancel the common factor.

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

Step 6.4.3

Rewrite the expression.

$- \frac{\sqrt{3} + 1}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{3} + 1}{4}$

Decimal Form:

$- 0.68301270 \dots$