Trigonometry Examples

$\sin (15) \cos (45) \cos (15) \sin (45)$

Step 1

The exact value of $\sin (15)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 1.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\sin (45 - 30) \cos (45) \cos (15) \sin (45)$

Step 1.2

Separate negation.

$\sin (45 - (30)) \cos (45) \cos (15) \sin (45)$

Step 1.3

Apply the difference of angles identity.

$(\sin (45) \cos (30) - \cos (45) \sin (30)) \cos (45) \cos (15) \sin (45)$

Step 1.4

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$(\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30)) \cos (45) \cos (15) \sin (45)$

Step 1.5

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)) \cos (45) \cos (15) \sin (45)$

Step 1.6

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30)) \cos (45) \cos (15) \sin (45)$

Step 1.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (45) \cos (15) \sin (45)$

Step 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 1.8.1

Simplify each term.

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Step 1.8.1.1

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

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Step 1.8.1.1.1

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

$(\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (45) \cos (15) \sin (45)$

Step 1.8.1.1.2

Combine using the product rule for radicals.

$(\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (45) \cos (15) \sin (45)$

Step 1.8.1.1.3

Multiply $2$ by $3$ .

$(\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (45) \cos (15) \sin (45)$

Step 1.8.1.1.4

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (45) \cos (15) \sin (45)$

Step 1.8.1.2

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

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Step 1.8.1.2.1

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

$(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) \cos (45) \cos (15) \sin (45)$

Step 1.8.1.2.2

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) \cos (45) \cos (15) \sin (45)$

Step 1.8.2

Combine the numerators over the common denominator.

$\frac{\sqrt{6} - \sqrt{2}}{4} \cos (45) \cos (15) \sin (45)$

Step 2

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{\sqrt{2}}{2} \cos (15) \sin (45)$

Step 3

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

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Step 3.1

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

$\frac{(\sqrt{6} - \sqrt{2}) \sqrt{2}}{4 \cdot 2} \cos (15) \sin (45)$

Step 3.2

Multiply $4$ by $2$ .

$\frac{(\sqrt{6} - \sqrt{2}) \sqrt{2}}{8} \cos (15) \sin (45)$

Step 4

Apply the distributive property.

$\frac{\sqrt{6} \sqrt{2} - \sqrt{2} \sqrt{2}}{8} \cos (15) \sin (45)$

Step 5

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 2} - \sqrt{2} \sqrt{2}}{8} \cos (15) \sin (45)$

Step 6

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

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Step 6.1

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{6 \cdot 2} - ({\sqrt{2}}^{1} \sqrt{2})}{8} \cos (15) \sin (45)$

Step 6.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{6 \cdot 2} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{8} \cos (15) \sin (45)$

Step 6.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{6 \cdot 2} - {\sqrt{2}}^{1 + 1}}{8} \cos (15) \sin (45)$

Step 6.4

Add $1$ and $1$ .

$\frac{\sqrt{6 \cdot 2} - {\sqrt{2}}^{2}}{8} \cos (15) \sin (45)$

Step 7

Simplify each term.

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Step 7.1

Multiply $6$ by $2$ .

$\frac{\sqrt{12} - {\sqrt{2}}^{2}}{8} \cos (15) \sin (45)$

Step 7.2

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

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Step 7.2.1

Factor $4$ out of $12$ .

$\frac{\sqrt{4 (3)} - {\sqrt{2}}^{2}}{8} \cos (15) \sin (45)$

Step 7.2.2

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

$\frac{\sqrt{2^{2} \cdot 3} - {\sqrt{2}}^{2}}{8} \cos (15) \sin (45)$

Step 7.3

Pull terms out from under the radical.

$\frac{2 \sqrt{3} - {\sqrt{2}}^{2}}{8} \cos (15) \sin (45)$

Step 7.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 7.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{2 \sqrt{3} - {(2^{\frac{1}{2}})}^{2}}{8} \cos (15) \sin (45)$

Step 7.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \sqrt{3} - 2^{\frac{1}{2} \cdot 2}}{8} \cos (15) \sin (45)$

Step 7.4.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{2 \sqrt{3} - 2^{\frac{2}{2}}}{8} \cos (15) \sin (45)$

Step 7.4.4

Cancel the common factor of $2$ .

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Step 7.4.4.1

Cancel the common factor.

$\frac{2 \sqrt{3} - 2^{\frac{2}{2}}}{8} \cos (15) \sin (45)$

Step 7.4.4.2

Rewrite the expression.

$\frac{2 \sqrt{3} - 2^{1}}{8} \cos (15) \sin (45)$

Step 7.4.5

Evaluate the exponent.

$\frac{2 \sqrt{3} - 1 \cdot 2}{8} \cos (15) \sin (45)$

Step 7.5

Multiply $- 1$ by $2$ .

$\frac{2 \sqrt{3} - 2}{8} \cos (15) \sin (45)$

Step 8

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

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Step 8.1

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

$\frac{2 (\sqrt{3}) - 2}{8} \cos (15) \sin (45)$

Step 8.2

Factor $2$ out of $- 2$ .

$\frac{2 (\sqrt{3}) + 2 (- 1)}{8} \cos (15) \sin (45)$

Step 8.3

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

$\frac{2 (\sqrt{3} - 1)}{8} \cos (15) \sin (45)$

Step 8.4

Cancel the common factors.

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Step 8.4.1

Factor $2$ out of $8$ .

$\frac{2 (\sqrt{3} - 1)}{2 (4)} \cos (15) \sin (45)$

Step 8.4.2

Cancel the common factor.

$\frac{2 (\sqrt{3} - 1)}{2 \cdot 4} \cos (15) \sin (45)$

Step 8.4.3

Rewrite the expression.

$\frac{\sqrt{3} - 1}{4} \cos (15) \sin (45)$

Step 9

The exact value of $\cos (15)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 9.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{\sqrt{3} - 1}{4} \cos (45 - 30) \sin (45)$

Step 9.2

Separate negation.

$\frac{\sqrt{3} - 1}{4} \cos (45 - (30)) \sin (45)$

Step 9.3

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

$\frac{\sqrt{3} - 1}{4} (\cos (45) \cos (30) + \sin (45) \sin (30)) \sin (45)$

Step 9.4

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{3} - 1}{4} (\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30)) \sin (45)$

Step 9.5

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{3} - 1}{4} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30)) \sin (45)$

Step 9.6

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{3} - 1}{4} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30)) \sin (45)$

Step 9.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\frac{\sqrt{3} - 1}{4} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (45)$

Step 9.8

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

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Step 9.8.1

Simplify each term.

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Step 9.8.1.1

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

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Step 9.8.1.1.1

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

$\frac{\sqrt{3} - 1}{4} (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (45)$

Step 9.8.1.1.2

Combine using the product rule for radicals.

$\frac{\sqrt{3} - 1}{4} (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (45)$

Step 9.8.1.1.3

Multiply $2$ by $3$ .

$\frac{\sqrt{3} - 1}{4} (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (45)$

Step 9.8.1.1.4

Multiply $2$ by $2$ .

$\frac{\sqrt{3} - 1}{4} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (45)$

Step 9.8.1.2

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

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Step 9.8.1.2.1

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

$\frac{\sqrt{3} - 1}{4} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) \sin (45)$

Step 9.8.1.2.2

Multiply $2$ by $2$ .

$\frac{\sqrt{3} - 1}{4} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) \sin (45)$

Step 9.8.2

Combine the numerators over the common denominator.

$\frac{\sqrt{3} - 1}{4} \cdot \frac{\sqrt{6} + \sqrt{2}}{4} \sin (45)$

Step 10

Multiply $\frac{\sqrt{3} - 1}{4} \cdot \frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 10.1

Multiply $\frac{\sqrt{3} - 1}{4}$ by $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

$\frac{(\sqrt{3} - 1) (\sqrt{6} + \sqrt{2})}{4 \cdot 4} \sin (45)$

Step 10.2

Multiply $4$ by $4$ .

$\frac{(\sqrt{3} - 1) (\sqrt{6} + \sqrt{2})}{16} \sin (45)$

Step 11

Simplify the numerator.

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Step 11.1

Expand $(\sqrt{3} - 1) (\sqrt{6} + \sqrt{2})$ using the FOIL Method.

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Step 11.1.1

Apply the distributive property.

$\frac{\sqrt{3} (\sqrt{6} + \sqrt{2}) - 1 (\sqrt{6} + \sqrt{2})}{16} \sin (45)$

Step 11.1.2

Apply the distributive property.

$\frac{\sqrt{3} \sqrt{6} + \sqrt{3} \sqrt{2} - 1 (\sqrt{6} + \sqrt{2})}{16} \sin (45)$

Step 11.1.3

Apply the distributive property.

$\frac{\sqrt{3} \sqrt{6} + \sqrt{3} \sqrt{2} - 1 \sqrt{6} - 1 \sqrt{2}}{16} \sin (45)$

Step 11.2

Simplify and combine like terms.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 6} + \sqrt{3} \sqrt{2} - 1 \sqrt{6} - 1 \sqrt{2}}{16} \sin (45)$

Step 11.2.1.2

Multiply $3$ by $6$ .

$\frac{\sqrt{18} + \sqrt{3} \sqrt{2} - 1 \sqrt{6} - 1 \sqrt{2}}{16} \sin (45)$

Step 11.2.1.3

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

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Step 11.2.1.3.1

Factor $9$ out of $18$ .

$\frac{\sqrt{9 (2)} + \sqrt{3} \sqrt{2} - 1 \sqrt{6} - 1 \sqrt{2}}{16} \sin (45)$

Step 11.2.1.3.2

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

$\frac{\sqrt{3^{2} \cdot 2} + \sqrt{3} \sqrt{2} - 1 \sqrt{6} - 1 \sqrt{2}}{16} \sin (45)$

Step 11.2.1.4

Pull terms out from under the radical.

$\frac{3 \sqrt{2} + \sqrt{3} \sqrt{2} - 1 \sqrt{6} - 1 \sqrt{2}}{16} \sin (45)$

Step 11.2.1.5

Combine using the product rule for radicals.

$\frac{3 \sqrt{2} + \sqrt{3 \cdot 2} - 1 \sqrt{6} - 1 \sqrt{2}}{16} \sin (45)$

Step 11.2.1.6

Multiply $3$ by $2$ .

$\frac{3 \sqrt{2} + \sqrt{6} - 1 \sqrt{6} - 1 \sqrt{2}}{16} \sin (45)$

Step 11.2.1.7

Rewrite $- 1 \sqrt{6}$ as $- \sqrt{6}$ .

$\frac{3 \sqrt{2} + \sqrt{6} - \sqrt{6} - 1 \sqrt{2}}{16} \sin (45)$

Step 11.2.1.8

Rewrite $- 1 \sqrt{2}$ as $- \sqrt{2}$ .

$\frac{3 \sqrt{2} + \sqrt{6} - \sqrt{6} - \sqrt{2}}{16} \sin (45)$

Step 11.2.2

Subtract $\sqrt{2}$ from $3 \sqrt{2}$ .

$\frac{2 \sqrt{2} + \sqrt{6} - \sqrt{6}}{16} \sin (45)$

Step 11.2.3

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

$\frac{2 \sqrt{2} + 0}{16} \sin (45)$

Step 11.2.4

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

$\frac{2 \sqrt{2}}{16} \sin (45)$

Step 12

Cancel the common factor of $2$ and $16$ .

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Step 12.1

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

$\frac{2 (\sqrt{2})}{16} \sin (45)$

Step 12.2

Cancel the common factors.

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Step 12.2.1

Factor $2$ out of $16$ .

$\frac{2 \sqrt{2}}{2 \cdot 8} \sin (45)$

Step 12.2.2

Cancel the common factor.

$\frac{2 \sqrt{2}}{2 \cdot 8} \sin (45)$

Step 12.2.3

Rewrite the expression.

$\frac{\sqrt{2}}{8} \sin (45)$

Step 13

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

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

Step 14

Multiply $\frac{\sqrt{2}}{8} \cdot \frac{\sqrt{2}}{2}$ .

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Step 14.1

Multiply $\frac{\sqrt{2}}{8}$ by $\frac{\sqrt{2}}{2}$ .

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

Step 14.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{{\sqrt{2}}^{1} \sqrt{2}}{8 \cdot 2}$

Step 14.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}{8 \cdot 2}$

Step 14.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{\sqrt{2}}^{1 + 1}}{8 \cdot 2}$

Step 14.5

Add $1$ and $1$ .

$\frac{{\sqrt{2}}^{2}}{8 \cdot 2}$

Step 14.6

Multiply $8$ by $2$ .

$\frac{{\sqrt{2}}^{2}}{16}$

Step 15

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 15.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{{(2^{\frac{1}{2}})}^{2}}{16}$

Step 15.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2^{\frac{1}{2} \cdot 2}}{16}$

Step 15.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{2^{\frac{2}{2}}}{16}$

Step 15.4

Cancel the common factor of $2$ .

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Step 15.4.1

Cancel the common factor.

$\frac{2^{\frac{2}{2}}}{16}$

Step 15.4.2

Rewrite the expression.

$\frac{2^{1}}{16}$

Step 15.5

Evaluate the exponent.

$\frac{2}{16}$

Step 16

Cancel the common factor of $2$ and $16$ .

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Step 16.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{16}$

Step 16.2

Cancel the common factors.

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Step 16.2.1

Factor $2$ out of $16$ .

$\frac{2 \cdot 1}{2 \cdot 8}$

Step 16.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 8}$

Step 16.2.3

Rewrite the expression.

$\frac{1}{8}$

Step 17

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{8}$

Decimal Form:

$0.125$