Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Evaluate $\cos (25)$ .

$0.90630778 \cos (15) - \sin (25) \sin (15)$

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.

$0.90630778 \cos (45 - 30) - \sin (25) \sin (15)$

Step 1.2.2

Separate negation.

$0.90630778 \cos (45 - (30)) - \sin (25) \sin (15)$

Step 1.2.3

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

$0.90630778 (\cos (45) \cos (30) + \sin (45) \sin (30)) - \sin (25) \sin (15)$

Step 1.2.4

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

$0.90630778 (\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30)) - \sin (25) \sin (15)$

Step 1.2.5

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

$0.90630778 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30)) - \sin (25) \sin (15)$

Step 1.2.6

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

$0.90630778 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30)) - \sin (25) \sin (15)$

Step 1.2.7

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

$0.90630778 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (25) \sin (15)$

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

$0.90630778 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (25) \sin (15)$

Step 1.2.8.1.1.2

Combine using the product rule for radicals.

$0.90630778 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (25) \sin (15)$

Step 1.2.8.1.1.3

Multiply $2$ by $3$ .

$0.90630778 (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (25) \sin (15)$

Step 1.2.8.1.1.4

Multiply $2$ by $2$ .

$0.90630778 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (25) \sin (15)$

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 $\frac{1}{2}$ .

$0.90630778 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) - \sin (25) \sin (15)$

Step 1.2.8.1.2.2

Multiply $2$ by $2$ .

$0.90630778 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) - \sin (25) \sin (15)$

Step 1.2.8.2

Combine the numerators over the common denominator.

$0.90630778 \frac{\sqrt{6} + \sqrt{2}}{4} - \sin (25) \sin (15)$

Step 1.3

Multiply $0.90630778 \frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 1.3.1

Combine $0.90630778$ and $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

$\frac{0.90630778 (\sqrt{6} + \sqrt{2})}{4} - \sin (25) \sin (15)$

Step 1.3.2

Multiply $0.90630778$ by $\sqrt{6} + \sqrt{2}$ .

$\frac{3.50170439}{4} - \sin (25) \sin (15)$

Step 1.4

Divide $3.50170439$ by $4$ .

$0.87542609 - \sin (25) \sin (15)$

Step 1.5

Evaluate $\sin (25)$ .

$0.87542609 - 1 \cdot 0.42261826 \sin (15)$

Step 1.6

Multiply $- 1$ by $0.42261826$ .

$0.87542609 - 0.42261826 \sin (15)$

Step 1.7

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

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Step 1.7.1

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

$0.87542609 - 0.42261826 \sin (45 - 30)$

Step 1.7.2

Separate negation.

$0.87542609 - 0.42261826 \sin (45 - (30))$

Step 1.7.3

Apply the difference of angles identity.

$0.87542609 - 0.42261826 (\sin (45) \cos (30) - \cos (45) \sin (30))$

Step 1.7.4

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

$0.87542609 - 0.42261826 (\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30))$

Step 1.7.5

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

$0.87542609 - 0.42261826 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30))$

Step 1.7.6

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

$0.87542609 - 0.42261826 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30))$

Step 1.7.7

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

$0.87542609 - 0.42261826 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 1.7.8

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

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Step 1.7.8.1

Simplify each term.

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Step 1.7.8.1.1

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

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Step 1.7.8.1.1.1

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

$0.87542609 - 0.42261826 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 1.7.8.1.1.2

Combine using the product rule for radicals.

$0.87542609 - 0.42261826 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 1.7.8.1.1.3

Multiply $2$ by $3$ .

$0.87542609 - 0.42261826 (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 1.7.8.1.1.4

Multiply $2$ by $2$ .

$0.87542609 - 0.42261826 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 1.7.8.1.2

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

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Step 1.7.8.1.2.1

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

$0.87542609 - 0.42261826 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})$

Step 1.7.8.1.2.2

Multiply $2$ by $2$ .

$0.87542609 - 0.42261826 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})$

Step 1.7.8.2

Combine the numerators over the common denominator.

$0.87542609 - 0.42261826 \frac{\sqrt{6} - \sqrt{2}}{4}$

Step 1.8

Multiply $- 0.42261826 \frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 1.8.1

Combine $- 0.42261826$ and $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

$0.87542609 + \frac{- 0.42261826 (\sqrt{6} - \sqrt{2})}{4}$

Step 1.8.2

Multiply $- 0.42261826$ by $\sqrt{6} - \sqrt{2}$ .

$0.87542609 + \frac{- 0.43752661}{4}$

Step 1.9

Divide $- 0.43752661$ by $4$ .

$0.87542609 - 0.10938165$

Step 2

Subtract $0.10938165$ from $0.87542609$ .

$0.76604444$