Trigonometry Examples

cos (25) cos (15) - sin (25) sin (15)

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

Step 1

The polynomial cannot be factored using the grouping method. Try a different method, or if you aren't sure, choose Factor.

The polynomial cannot be factored using the grouping method.

Step 2

Evaluate

$\cos (25)$ .

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

Step 3

The exact value of

$\cos (15)$ is

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

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Step 3.1

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 3.2

Separate negation.

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

Step 3.3

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 3.4

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 3.5

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 3.6

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 3.7

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 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 3.8.1

Simplify each term.

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Step 3.8.1.1

Multiply

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

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Step 3.8.1.1.1

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 3.8.1.1.2

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 3.8.1.1.3

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 3.8.1.1.4

Multiply

$2$ by

$2$ .

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

Step 3.8.1.2

Multiply

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

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Step 3.8.1.2.1

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 3.8.1.2.2

Multiply

$2$ by

$2$ .

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

Step 3.8.2

Combine the numerators over the common denominator.

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

Step 4

Multiply

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

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Step 4.1

Combine

$0.90630778$ and

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

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

Step 4.2

Multiply

$0.90630778$ by

$\sqrt{6} + \sqrt{2}$ .

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

Step 5

Divide

$3.50170439$ by

$4$ .

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

Step 6

Evaluate

$\sin (25)$ .

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

Step 7

Multiply

$- 1$ by

$0.42261826$ .

$0.87542609 - 0.42261826 \sin (15)$

Step 8

The exact value of

$\sin (15)$ is

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

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Step 8.1

Split

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

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

Step 8.2

Separate negation.

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

Step 8.3

Apply the difference of angles identity.

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

Step 8.4

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 8.5

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 8.6

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 8.7

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 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 8.8.1

Simplify each term.

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Step 8.8.1.1

Multiply

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

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Step 8.8.1.1.1

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 8.8.1.1.2

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 8.8.1.1.3

Multiply

$2$ by

$3$ .

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

Step 8.8.1.1.4

Multiply

$2$ by

$2$ .

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

Step 8.8.1.2

Multiply

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

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Step 8.8.1.2.1

Multiply

$\frac{1}{2}$ by

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

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

Step 8.8.1.2.2

Multiply

$2$ by

$2$ .

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

Step 8.8.2

Combine the numerators over the common denominator.

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

Step 9

Multiply

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

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Step 9.1

Combine

$- 0.42261826$ and

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

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

Step 9.2

Multiply

$- 0.42261826$ by

$\sqrt{6} - \sqrt{2}$ .

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

Step 10

Divide

$- 0.43752661$ by

$4$ .

$0.87542609 - 0.10938165$

Step 11

Subtract

$0.10938165$ from

$0.87542609$ .

$0.76604444$