Trigonometry Examples

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

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.

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

Step 1.1.2

Separate negation.

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

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

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

Step 1.1.7

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

$(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (30) - \sin (15) \sin (30)$

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

$(\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (30) - \sin (15) \sin (30)$

Step 1.1.8.1.1.2

Combine using the product rule for radicals.

$(\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (30) - \sin (15) \sin (30)$

Step 1.1.8.1.1.3

Multiply $2$ by $3$ .

$(\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (30) - \sin (15) \sin (30)$

Step 1.1.8.1.1.4

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (30) - \sin (15) \sin (30)$

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 $\frac{1}{2}$ .

$(\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) \cos (30) - \sin (15) \sin (30)$

Step 1.1.8.1.2.2

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) \cos (30) - \sin (15) \sin (30)$

Step 1.1.8.2

Combine the numerators over the common denominator.

$\frac{\sqrt{6} + \sqrt{2}}{4} \cos (30) - \sin (15) \sin (30)$

Step 1.2

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

$\frac{\sqrt{6} + \sqrt{2}}{4} \cdot \frac{\sqrt{3}}{2} - \sin (15) \sin (30)$

Step 1.3

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

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Step 1.3.1

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

$\frac{(\sqrt{6} + \sqrt{2}) \sqrt{3}}{4 \cdot 2} - \sin (15) \sin (30)$

Step 1.3.2

Multiply $4$ by $2$ .

$\frac{(\sqrt{6} + \sqrt{2}) \sqrt{3}}{8} - \sin (15) \sin (30)$

Step 1.4

Apply the distributive property.

$\frac{\sqrt{6} \sqrt{3} + \sqrt{2} \sqrt{3}}{8} - \sin (15) \sin (30)$

Step 1.5

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 3} + \sqrt{2} \sqrt{3}}{8} - \sin (15) \sin (30)$

Step 1.6

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 3} + \sqrt{2 \cdot 3}}{8} - \sin (15) \sin (30)$

Step 1.7

Simplify each term.

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Step 1.7.1

Multiply $6$ by $3$ .

$\frac{\sqrt{18} + \sqrt{2 \cdot 3}}{8} - \sin (15) \sin (30)$

Step 1.7.2

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

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Step 1.7.2.1

Factor $9$ out of $18$ .

$\frac{\sqrt{9 (2)} + \sqrt{2 \cdot 3}}{8} - \sin (15) \sin (30)$

Step 1.7.2.2

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

$\frac{\sqrt{3^{2} \cdot 2} + \sqrt{2 \cdot 3}}{8} - \sin (15) \sin (30)$

Step 1.7.3

Pull terms out from under the radical.

$\frac{3 \sqrt{2} + \sqrt{2 \cdot 3}}{8} - \sin (15) \sin (30)$

Step 1.7.4

Multiply $2$ by $3$ .

$\frac{3 \sqrt{2} + \sqrt{6}}{8} - \sin (15) \sin (30)$

Step 1.8

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

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Step 1.8.1

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

$\frac{3 \sqrt{2} + \sqrt{6}}{8} - \sin (45 - 30) \sin (30)$

Step 1.8.2

Separate negation.

$\frac{3 \sqrt{2} + \sqrt{6}}{8} - \sin (45 - (30)) \sin (30)$

Step 1.8.3

Apply the difference of angles identity.

$\frac{3 \sqrt{2} + \sqrt{6}}{8} - (\sin (45) \cos (30) - \cos (45) \sin (30)) \sin (30)$

Step 1.8.4

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

$\frac{3 \sqrt{2} + \sqrt{6}}{8} - (\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30)) \sin (30)$

Step 1.8.5

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

$\frac{3 \sqrt{2} + \sqrt{6}}{8} - (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)) \sin (30)$

Step 1.8.6

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

$\frac{3 \sqrt{2} + \sqrt{6}}{8} - (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30)) \sin (30)$

Step 1.8.7

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

$\frac{3 \sqrt{2} + \sqrt{6}}{8} - (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (30)$

Step 1.8.8

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

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Step 1.8.8.1

Simplify each term.

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Step 1.8.8.1.1

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

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Step 1.8.8.1.1.1

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

$\frac{3 \sqrt{2} + \sqrt{6}}{8} - (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (30)$

Step 1.8.8.1.1.2

Combine using the product rule for radicals.

$\frac{3 \sqrt{2} + \sqrt{6}}{8} - (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (30)$

Step 1.8.8.1.1.3

Multiply $2$ by $3$ .

$\frac{3 \sqrt{2} + \sqrt{6}}{8} - (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (30)$

Step 1.8.8.1.1.4

Multiply $2$ by $2$ .

$\frac{3 \sqrt{2} + \sqrt{6}}{8} - (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (30)$

Step 1.8.8.1.2

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

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Step 1.8.8.1.2.1

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

$\frac{3 \sqrt{2} + \sqrt{6}}{8} - (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) \sin (30)$

Step 1.8.8.1.2.2

Multiply $2$ by $2$ .

$\frac{3 \sqrt{2} + \sqrt{6}}{8} - (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) \sin (30)$

Step 1.8.8.2

Combine the numerators over the common denominator.

$\frac{3 \sqrt{2} + \sqrt{6}}{8} - \frac{\sqrt{6} - \sqrt{2}}{4} \sin (30)$

Step 1.9

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

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

Step 1.10

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

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Step 1.10.1

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

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

Step 1.10.2

Multiply $2$ by $4$ .

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $- - \sqrt{2}$ .

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Step 3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.2

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

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

Step 4

Simplify terms.

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Step 4.1

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

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

Step 4.2

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

$\frac{4 \sqrt{2} + 0}{8}$

Step 4.3

Add $4 \sqrt{2}$ and $0$ .

$\frac{4 \sqrt{2}}{8}$

Step 4.4

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

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Step 4.4.1

Factor $4$ out of $4 \sqrt{2}$ .

$\frac{4 (\sqrt{2})}{8}$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $4$ out of $8$ .

$\frac{4 \sqrt{2}}{4 \cdot 2}$

Step 4.4.2.2

Cancel the common factor.

$\frac{4 \sqrt{2}}{4 \cdot 2}$

Step 4.4.2.3

Rewrite the expression.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.70710678 \dots$