Trigonometry Examples

$\cos (60) \cos (15) + \sin (60) \sin (15)$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$\frac{1}{2} \cos (15) + \sin (60) \sin (15)$

Step 1.2

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

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Step 1.2.1

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

$\frac{1}{2} \cos (45 - 30) + \sin (60) \sin (15)$

Step 1.2.2

Separate negation.

$\frac{1}{2} \cos (45 - (30)) + \sin (60) \sin (15)$

Step 1.2.3

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

$\frac{1}{2} (\cos (45) \cos (30) + \sin (45) \sin (30)) + \sin (60) \sin (15)$

Step 1.2.4

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

$\frac{1}{2} (\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30)) + \sin (60) \sin (15)$

Step 1.2.5

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

$\frac{1}{2} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30)) + \sin (60) \sin (15)$

Step 1.2.6

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

$\frac{1}{2} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30)) + \sin (60) \sin (15)$

Step 1.2.7

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

$\frac{1}{2} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + \sin (60) \sin (15)$

Step 1.2.8

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

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Step 1.2.8.1

Simplify each term.

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Step 1.2.8.1.1

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

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Step 1.2.8.1.1.1

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

$\frac{1}{2} (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + \sin (60) \sin (15)$

Step 1.2.8.1.1.2

Combine using the product rule for radicals.

$\frac{1}{2} (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + \sin (60) \sin (15)$

Step 1.2.8.1.1.3

Multiply $2$ by $3$ .

$\frac{1}{2} (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + \sin (60) \sin (15)$

Step 1.2.8.1.1.4

Multiply $2$ by $2$ .

$\frac{1}{2} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + \sin (60) \sin (15)$

Step 1.2.8.1.2

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

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Step 1.2.8.1.2.1

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

$\frac{1}{2} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) + \sin (60) \sin (15)$

Step 1.2.8.1.2.2

Multiply $2$ by $2$ .

$\frac{1}{2} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) + \sin (60) \sin (15)$

Step 1.2.8.2

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{\sqrt{6} + \sqrt{2}}{4} + \sin (60) \sin (15)$

Step 1.3

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

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Step 1.3.1

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

$\frac{\sqrt{6} + \sqrt{2}}{2 \cdot 4} + \sin (60) \sin (15)$

Step 1.3.2

Multiply $2$ by $4$ .

$\frac{\sqrt{6} + \sqrt{2}}{8} + \sin (60) \sin (15)$

Step 1.4

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

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

Step 1.5

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

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Step 1.5.1

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

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

Step 1.5.2

Separate negation.

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

Step 1.5.3

Apply the difference of angles identity.

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

Step 1.5.4

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

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

Step 1.5.5

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

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

Step 1.5.6

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

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

Step 1.5.7

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

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

Step 1.5.8

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

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Step 1.5.8.1

Simplify each term.

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Step 1.5.8.1.1

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

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Step 1.5.8.1.1.1

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

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

Step 1.5.8.1.1.2

Combine using the product rule for radicals.

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

Step 1.5.8.1.1.3

Multiply $2$ by $3$ .

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

Step 1.5.8.1.1.4

Multiply $2$ by $2$ .

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

Step 1.5.8.1.2

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

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Step 1.5.8.1.2.1

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

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

Step 1.5.8.1.2.2

Multiply $2$ by $2$ .

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

Step 1.5.8.2

Combine the numerators over the common denominator.

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

Step 1.6

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

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Step 1.6.1

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

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

Step 1.6.2

Multiply $2$ by $4$ .

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

Step 1.7

Apply the distributive property.

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

Step 1.8

Combine using the product rule for radicals.

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

Step 1.9

Multiply $\sqrt{3} (- \sqrt{2})$ .

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Step 1.9.1

Combine using the product rule for radicals.

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

Step 1.9.2

Multiply $3$ by $2$ .

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

Step 1.10

Simplify each term.

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Step 1.10.1

Multiply $3$ by $6$ .

$\frac{\sqrt{6} + \sqrt{2}}{8} + \frac{\sqrt{18} - \sqrt{6}}{8}$

Step 1.10.2

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

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Step 1.10.2.1

Factor $9$ out of $18$ .

$\frac{\sqrt{6} + \sqrt{2}}{8} + \frac{\sqrt{9 (2)} - \sqrt{6}}{8}$

Step 1.10.2.2

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

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

Step 1.10.3

Pull terms out from under the radical.

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

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

$\frac{0 + \sqrt{2} + 3 \sqrt{2}}{8}$

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $4$ out of $8$ .

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

Step 2.5.2.2

Cancel the common factor.

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

Step 2.5.2.3

Rewrite the expression.

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.70710678 \dots$