Trigonometry Examples

$2 \sin (15) \cos (15)$

Step 1

The complement of $2 \sin (15) \cos (15)$ is the angle that when added to $2 \sin (15) \cos (15)$ forms a right angle ( $90 °$ ).

$90 ° - 2 \sin (15) \cos (15)$

Step 2

Simplify each term.

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Step 2.1

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

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Step 2.1.1

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

Step 2.1.2

Separate negation.

Step 2.1.3

Apply the difference of angles identity.

Step 2.1.4

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

Step 2.1.5

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

Step 2.1.6

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

Step 2.1.7

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

Step 2.1.8

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

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Step 2.1.8.1

Simplify each term.

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Step 2.1.8.1.1

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

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Step 2.1.8.1.1.1

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

Step 2.1.8.1.1.2

Combine using the product rule for radicals.

Step 2.1.8.1.1.3

Multiply $2$ by $3$ .

Step 2.1.8.1.1.4

Multiply $2$ by $2$ .

Step 2.1.8.1.2

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

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Step 2.1.8.1.2.1

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

Step 2.1.8.1.2.2

Multiply $2$ by $2$ .

Step 2.1.8.2

Combine the numerators over the common denominator.

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Factor $2$ out of $- 2$ .

Step 2.2.2

Factor $2$ out of $4$ .

Step 2.2.3

Cancel the common factor.

Step 2.2.4

Rewrite the expression.

Step 2.3

Rewrite $- 1 \frac{\sqrt{6} - \sqrt{2}}{2}$ as $- \frac{\sqrt{6} - \sqrt{2}}{2}$ .

Step 2.4

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

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Step 2.4.1

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

Step 2.4.2

Separate negation.

Step 2.4.3

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

Step 2.4.4

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

Step 2.4.5

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

Step 2.4.6

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

Step 2.4.7

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

Step 2.4.8

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

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Step 2.4.8.1

Simplify each term.

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Step 2.4.8.1.1

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

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Step 2.4.8.1.1.1

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

Step 2.4.8.1.1.2

Combine using the product rule for radicals.

Step 2.4.8.1.1.3

Multiply $2$ by $3$ .

Step 2.4.8.1.1.4

Multiply $2$ by $2$ .

Step 2.4.8.1.2

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

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Step 2.4.8.1.2.1

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

Step 2.4.8.1.2.2

Multiply $2$ by $2$ .

Step 2.4.8.2

Combine the numerators over the common denominator.

$90 - \frac{\sqrt{6} - \sqrt{2}}{2} \cdot \frac{\sqrt{6} + \sqrt{2}}{4 °}$

Step 2.5

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

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Step 2.5.1

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

Step 2.5.2

Multiply $4$ by $2$ .

$90 - \frac{(\sqrt{6} + \sqrt{2}) (\sqrt{6} - \sqrt{2})}{8 °}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

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

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Step 2.6.1.1

Apply the distributive property.

$90 - \frac{\sqrt{6} (\sqrt{6} - \sqrt{2}) + \sqrt{2} (\sqrt{6} - \sqrt{2})}{8 °}$

Step 2.6.1.2

Apply the distributive property.

$90 - \frac{\sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{2} (\sqrt{6} - \sqrt{2})}{8 °}$

Step 2.6.1.3

Apply the distributive property.

$90 - \frac{\sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{8 °}$

Step 2.6.2

Simplify and combine like terms.

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Step 2.6.2.1

Simplify each term.

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Step 2.6.2.1.1

Combine using the product rule for radicals.

$90 - \frac{\sqrt{6 \cdot 6} + \sqrt{6} (- \sqrt{2}) + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{8 °}$

Step 2.6.2.1.2

Multiply $6$ by $6$ .

$90 - \frac{\sqrt{36} + \sqrt{6} (- \sqrt{2}) + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{8 °}$

Step 2.6.2.1.3

Rewrite $36$ as $6^{2}$ .

$90 - \frac{\sqrt{6^{2}} + \sqrt{6} (- \sqrt{2}) + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{8 °}$

Step 2.6.2.1.4

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

$90 - \frac{6 + \sqrt{6} (- \sqrt{2}) + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{8 °}$

Step 2.6.2.1.5

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

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Step 2.6.2.1.5.1

Combine using the product rule for radicals.

$90 - \frac{6 - \sqrt{6 \cdot 2} + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{8 °}$

Step 2.6.2.1.5.2

Multiply $6$ by $2$ .

$90 - \frac{6 - \sqrt{12} + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{8 °}$

Step 2.6.2.1.6

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

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Step 2.6.2.1.6.1

Factor $4$ out of $12$ .

$90 - \frac{6 - \sqrt{4 (3)} + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{8 °}$

Step 2.6.2.1.6.2

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

$90 - \frac{6 - \sqrt{2^{2} \cdot 3} + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{8 °}$

Step 2.6.2.1.7

Pull terms out from under the radical.

$90 - \frac{6 - (2 \sqrt{3}) + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{8 °}$

Step 2.6.2.1.8

Multiply $2$ by $- 1$ .

$90 - \frac{6 - 2 \sqrt{3} + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{8 °}$

Step 2.6.2.1.9

Combine using the product rule for radicals.

$90 - \frac{6 - 2 \sqrt{3} + \sqrt{2 \cdot 6} + \sqrt{2} (- \sqrt{2})}{8 °}$

Step 2.6.2.1.10

Multiply $2$ by $6$ .

$90 - \frac{6 - 2 \sqrt{3} + \sqrt{12} + \sqrt{2} (- \sqrt{2})}{8 °}$

Step 2.6.2.1.11

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

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Step 2.6.2.1.11.1

Factor $4$ out of $12$ .

$90 - \frac{6 - 2 \sqrt{3} + \sqrt{4 (3)} + \sqrt{2} (- \sqrt{2})}{8 °}$

Step 2.6.2.1.11.2

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

$90 - \frac{6 - 2 \sqrt{3} + \sqrt{2^{2} \cdot 3} + \sqrt{2} (- \sqrt{2})}{8 °}$

Step 2.6.2.1.12

Pull terms out from under the radical.

$90 - \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{2} (- \sqrt{2})}{8 °}$

Step 2.6.2.1.13

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

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Step 2.6.2.1.13.1

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

$90 - \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - (\sqrt{2} \sqrt{2})}{8 °}$

Step 2.6.2.1.13.2

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

$90 - \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - (\sqrt{2} \sqrt{2})}{8 °}$

Step 2.6.2.1.13.3

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

$90 - \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - {\sqrt{2}}^{1 + 1}}{8 °}$

Step 2.6.2.1.13.4

Add $1$ and $1$ .

$90 - \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - {\sqrt{2}}^{2}}{8 °}$

Step 2.6.2.1.14

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

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Step 2.6.2.1.14.1

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

$90 - \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - {(2^{\frac{1}{2}})}^{2}}{8 °}$

Step 2.6.2.1.14.2

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

$90 - \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - 2^{\frac{1}{2} \cdot 2}}{8 °}$

Step 2.6.2.1.14.3

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

$90 - \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - 2^{\frac{2}{2}}}{8 °}$

Step 2.6.2.1.14.4

Cancel the common factor of $2$ .

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Step 2.6.2.1.14.4.1

Cancel the common factor.

$90 - \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - 2^{\frac{2}{2}}}{8 °}$

Step 2.6.2.1.14.4.2

Rewrite the expression.

$90 - \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - 2}{8 °}$

Step 2.6.2.1.14.5

Evaluate the exponent.

$90 - \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - 1 \cdot 2}{8 °}$

Step 2.6.2.1.15

Multiply $- 1$ by $2$ .

$90 - \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - 2}{8 °}$

Step 2.6.2.2

Subtract $2$ from $6$ .

$90 - \frac{4 - 2 \sqrt{3} + 2 \sqrt{3}}{8 °}$

Step 2.6.2.3

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

$90 - \frac{4 + 0}{8 °}$

Step 2.6.2.4

Add $4$ and $0$ .

$90 - \frac{4}{8 °}$

Step 2.7

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

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Step 2.7.1

Factor $4$ out of $4$ .

$90 - \frac{4 (1)}{8 °}$

Step 2.7.2

Cancel the common factors.

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Step 2.7.2.1

Factor $4$ out of $8$ .

Step 2.7.2.2

Cancel the common factor.

Step 2.7.2.3

Rewrite the expression.

$90 - \frac{1}{2 °}$

Step 3

To write $90$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$90 \cdot \frac{2}{2} - \frac{1}{2 °}$

Step 4

Combine $90$ and $\frac{2}{2}$ .

$\frac{90 \cdot 2}{2} - \frac{1}{2 °}$

Step 5

Combine the numerators over the common denominator.

$\frac{90 \cdot 2 - 1}{2 °}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $90$ by $2$ .

$\frac{180 - 1}{2 °}$

Step 6.2

Subtract $1$ from $180$ .

$\frac{179}{2 °}$