Trigonometry Examples

$\cos (25) \cos (15) - \sin (25) \sin (15)$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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Step 1.1.1.1

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

$\cos (25) \cos (45 - 30) - \sin (25) \sin (15)$

Step 1.1.1.2

Separate negation.

$\cos (25) \cos (45 - (30)) - \sin (25) \sin (15)$

Step 1.1.1.3

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

$\cos (25) (\cos (45) \cos (30) + \sin (45) \sin (30)) - \sin (25) \sin (15)$

Step 1.1.1.4

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

$\cos (25) (\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30)) - \sin (25) \sin (15)$

Step 1.1.1.5

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

$\cos (25) (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30)) - \sin (25) \sin (15)$

Step 1.1.1.6

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

$\cos (25) (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30)) - \sin (25) \sin (15)$

Step 1.1.1.7

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

$\cos (25) (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (25) \sin (15)$

Step 1.1.1.8

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

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Step 1.1.1.8.1

Simplify each term.

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Step 1.1.1.8.1.1

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

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Step 1.1.1.8.1.1.1

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

$\cos (25) (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (25) \sin (15)$

Step 1.1.1.8.1.1.2

Combine using the product rule for radicals.

$\cos (25) (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (25) \sin (15)$

Step 1.1.1.8.1.1.3

Multiply $2$ by $3$ .

$\cos (25) (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (25) \sin (15)$

Step 1.1.1.8.1.1.4

Multiply $2$ by $2$ .

$\cos (25) (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (25) \sin (15)$

Step 1.1.1.8.1.2

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

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Step 1.1.1.8.1.2.1

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

$\cos (25) (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) - \sin (25) \sin (15)$

Step 1.1.1.8.1.2.2

Multiply $2$ by $2$ .

$\cos (25) (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) - \sin (25) \sin (15)$

Step 1.1.1.8.2

Combine the numerators over the common denominator.

$\cos (25) \frac{\sqrt{6} + \sqrt{2}}{4} - \sin (25) \sin (15)$

Step 1.1.2

Combine $\cos (25)$ and $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

$\frac{\cos (25) (\sqrt{6} + \sqrt{2})}{4} - \sin (25) \sin (15)$

Step 1.1.3

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

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Step 1.1.3.1

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

$\frac{\cos (25) (\sqrt{6} + \sqrt{2})}{4} - \sin (25) \sin (45 - 30)$

Step 1.1.3.2

Separate negation.

$\frac{\cos (25) (\sqrt{6} + \sqrt{2})}{4} - \sin (25) \sin (45 - (30))$

Step 1.1.3.3

Apply the difference of angles identity.

$\frac{\cos (25) (\sqrt{6} + \sqrt{2})}{4} - \sin (25) (\sin (45) \cos (30) - \cos (45) \sin (30))$

Step 1.1.3.4

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

$\frac{\cos (25) (\sqrt{6} + \sqrt{2})}{4} - \sin (25) (\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30))$

Step 1.1.3.5

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

$\frac{\cos (25) (\sqrt{6} + \sqrt{2})}{4} - \sin (25) (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30))$

Step 1.1.3.6

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

$\frac{\cos (25) (\sqrt{6} + \sqrt{2})}{4} - \sin (25) (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30))$

Step 1.1.3.7

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

$\frac{\cos (25) (\sqrt{6} + \sqrt{2})}{4} - \sin (25) (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 1.1.3.8

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

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Step 1.1.3.8.1

Simplify each term.

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Step 1.1.3.8.1.1

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

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Step 1.1.3.8.1.1.1

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

$\frac{\cos (25) (\sqrt{6} + \sqrt{2})}{4} - \sin (25) (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 1.1.3.8.1.1.2

Combine using the product rule for radicals.

$\frac{\cos (25) (\sqrt{6} + \sqrt{2})}{4} - \sin (25) (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 1.1.3.8.1.1.3

Multiply $2$ by $3$ .

$\frac{\cos (25) (\sqrt{6} + \sqrt{2})}{4} - \sin (25) (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 1.1.3.8.1.1.4

Multiply $2$ by $2$ .

$\frac{\cos (25) (\sqrt{6} + \sqrt{2})}{4} - \sin (25) (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 1.1.3.8.1.2

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

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Step 1.1.3.8.1.2.1

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

$\frac{\cos (25) (\sqrt{6} + \sqrt{2})}{4} - \sin (25) (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})$

Step 1.1.3.8.1.2.2

Multiply $2$ by $2$ .

$\frac{\cos (25) (\sqrt{6} + \sqrt{2})}{4} - \sin (25) (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})$

Step 1.1.3.8.2

Combine the numerators over the common denominator.

$\frac{\cos (25) (\sqrt{6} + \sqrt{2})}{4} - \sin (25) \frac{\sqrt{6} - \sqrt{2}}{4}$

Step 1.1.4

Combine $\frac{\sqrt{6} - \sqrt{2}}{4}$ and $\sin (25)$ .

$\frac{\cos (25) (\sqrt{6} + \sqrt{2})}{4} - \frac{(\sqrt{6} - \sqrt{2}) \sin (25)}{4}$

Step 1.2

Combine the numerators over the common denominator.

$\frac{\cos (25) (\sqrt{6} + \sqrt{2}) - (\sqrt{6} - \sqrt{2}) \sin (25)}{4}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the distributive property.

$\frac{\cos (25) \sqrt{6} + \cos (25) \sqrt{2} - (\sqrt{6} - \sqrt{2}) \sin (25)}{4}$

Step 2.2

Apply the distributive property.

$\frac{\cos (25) \sqrt{6} + \cos (25) \sqrt{2} + (- \sqrt{6} - - \sqrt{2}) \sin (25)}{4}$

Step 2.3

Multiply $- - \sqrt{2}$ .

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Step 2.3.1

Multiply $- 1$ by $- 1$ .

$\frac{\cos (25) \sqrt{6} + \cos (25) \sqrt{2} + (- \sqrt{6} + 1 \sqrt{2}) \sin (25)}{4}$

Step 2.3.2

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

$\frac{\cos (25) \sqrt{6} + \cos (25) \sqrt{2} + (- \sqrt{6} + \sqrt{2}) \sin (25)}{4}$

Step 2.4

Apply the distributive property.

$\frac{\cos (25) \sqrt{6} + \cos (25) \sqrt{2} - \sqrt{6} \sin (25) + \sqrt{2} \sin (25)}{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\cos (25) \sqrt{6} + \cos (25) \sqrt{2} - \sqrt{6} \sin (25) + \sqrt{2} \sin (25)}{4}$

Decimal Form:

$0.76604444 \dots$