Trigonometry Examples

$2 \cos^{2} (15) - 1$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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

$2 \cos^{2} (45 - 30) - 1$

Step 1.1.2

Separate negation.

$2 \cos^{2} (45 - (30)) - 1$

Step 1.1.3

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

$2 {(\cos (45) \cos (30) + \sin (45) \sin (30))}^{2} - 1$

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

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

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

Step 1.1.7

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

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

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

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

Step 1.1.8.1.1.2

Combine using the product rule for radicals.

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

Step 1.1.8.1.1.3

Multiply $2$ by $3$ .

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

Step 1.1.8.1.1.4

Multiply $2$ by $2$ .

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

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 $\frac{1}{2}$ .

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

Step 1.1.8.1.2.2

Multiply $2$ by $2$ .

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

Step 1.1.8.2

Combine the numerators over the common denominator.

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

Step 1.2

Apply the product rule to $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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

Step 1.3

Raise $4$ to the power of $2$ .

$2 \frac{{(\sqrt{6} + \sqrt{2})}^{2}}{16} - 1$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Factor $2$ out of $16$ .

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

Step 1.4.2

Cancel the common factor.

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

Step 1.4.3

Rewrite the expression.

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

Step 1.5

Rewrite ${(\sqrt{6} + \sqrt{2})}^{2}$ as $(\sqrt{6} + \sqrt{2}) (\sqrt{6} + \sqrt{2})$ .

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

Step 1.6

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

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Step 1.6.1

Apply the distributive property.

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

Step 1.6.2

Apply the distributive property.

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

Step 1.6.3

Apply the distributive property.

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

Step 1.7

Simplify and combine like terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Combine using the product rule for radicals.

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

Step 1.7.1.2

Multiply $6$ by $6$ .

$\frac{\sqrt{36} + \sqrt{6} \sqrt{2} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{8} - 1$

Step 1.7.1.3

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

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

Step 1.7.1.4

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

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

Step 1.7.1.5

Combine using the product rule for radicals.

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

Step 1.7.1.6

Multiply $6$ by $2$ .

$\frac{6 + \sqrt{12} + \sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{2}}{8} - 1$

Step 1.7.1.7

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

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Step 1.7.1.7.1

Factor $4$ out of $12$ .

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

Step 1.7.1.7.2

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

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

Step 1.7.1.8

Pull terms out from under the radical.

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

Step 1.7.1.9

Combine using the product rule for radicals.

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

Step 1.7.1.10

Multiply $2$ by $6$ .

$\frac{6 + 2 \sqrt{3} + \sqrt{12} + \sqrt{2} \sqrt{2}}{8} - 1$

Step 1.7.1.11

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

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Step 1.7.1.11.1

Factor $4$ out of $12$ .

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

Step 1.7.1.11.2

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

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

Step 1.7.1.12

Pull terms out from under the radical.

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

Step 1.7.1.13

Combine using the product rule for radicals.

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

Step 1.7.1.14

Multiply $2$ by $2$ .

$\frac{6 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{4}}{8} - 1$

Step 1.7.1.15

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

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

Step 1.7.1.16

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

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

Step 1.7.2

Add $6$ and $2$ .

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

Step 1.7.3

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

$\frac{8 + 4 \sqrt{3}}{8} - 1$

Step 1.8

Cancel the common factor of $8 + 4 \sqrt{3}$ and $8$ .

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Step 1.8.1

Factor $4$ out of $8$ .

$\frac{4 \cdot 2 + 4 \sqrt{3}}{8} - 1$

Step 1.8.2

Factor $4$ out of $4 \sqrt{3}$ .

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

Step 1.8.3

Factor $4$ out of $4 \cdot 2 + 4 (\sqrt{3})$ .

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

Step 1.8.4

Cancel the common factors.

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Step 1.8.4.1

Factor $4$ out of $8$ .

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

Step 1.8.4.2

Cancel the common factor.

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

Step 1.8.4.3

Rewrite the expression.

$\frac{2 + \sqrt{3}}{2} - 1$

Step 2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3

Combine fractions.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Multiply $- 1$ by $2$ .

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

Step 4.2

Subtract $2$ from $2$ .

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

Step 4.3

Add $0$ and $\sqrt{3}$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.86602540 \dots$