Trigonometry Examples

$\sin (48) \cos (15) - \cos (48) \sin (15)$

Step 1

Evaluate $\sin (48)$ .

$0.74314482 \cos (15) - \cos (48) \sin (15)$

Step 2

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

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Step 2.1

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

$0.74314482 \cos (45 - 30) - \cos (48) \sin (15)$

Step 2.2

Separate negation.

$0.74314482 \cos (45 - (30)) - \cos (48) \sin (15)$

Step 2.3

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

$0.74314482 (\cos (45) \cos (30) + \sin (45) \sin (30)) - \cos (48) \sin (15)$

Step 2.4

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

$0.74314482 (\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30)) - \cos (48) \sin (15)$

Step 2.5

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

$0.74314482 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30)) - \cos (48) \sin (15)$

Step 2.6

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

$0.74314482 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30)) - \cos (48) \sin (15)$

Step 2.7

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

$0.74314482 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \cos (48) \sin (15)$

Step 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 2.8.1

Simplify each term.

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Step 2.8.1.1

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

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Step 2.8.1.1.1

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

$0.74314482 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \cos (48) \sin (15)$

Step 2.8.1.1.2

Combine using the product rule for radicals.

$0.74314482 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \cos (48) \sin (15)$

Step 2.8.1.1.3

Multiply $2$ by $3$ .

$0.74314482 (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \cos (48) \sin (15)$

Step 2.8.1.1.4

Multiply $2$ by $2$ .

$0.74314482 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \cos (48) \sin (15)$

Step 2.8.1.2

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

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Step 2.8.1.2.1

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

$0.74314482 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) - \cos (48) \sin (15)$

Step 2.8.1.2.2

Multiply $2$ by $2$ .

$0.74314482 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) - \cos (48) \sin (15)$

Step 2.8.2

Combine the numerators over the common denominator.

$0.74314482 \frac{\sqrt{6} + \sqrt{2}}{4} - \cos (48) \sin (15)$

Step 3

Multiply $0.74314482 \frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 3.1

Combine $0.74314482$ and $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

$\frac{0.74314482 (\sqrt{6} + \sqrt{2})}{4} - \cos (48) \sin (15)$

Step 3.2

Multiply $0.74314482$ by $\sqrt{6} + \sqrt{2}$ .

$\frac{2.87129111}{4} - \cos (48) \sin (15)$

Step 4

Divide $2.87129111$ by $4$ .

$0.71782277 - \cos (48) \sin (15)$

Step 5

Evaluate $\cos (48)$ .

$0.71782277 - 1 \cdot 0.6691306 \sin (15)$

Step 6

Multiply $- 1$ by $0.6691306$ .

$0.71782277 - 0.6691306 \sin (15)$

Step 7

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

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Step 7.1

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

$0.71782277 - 0.6691306 \sin (45 - 30)$

Step 7.2

Separate negation.

$0.71782277 - 0.6691306 \sin (45 - (30))$

Step 7.3

Apply the difference of angles identity.

$0.71782277 - 0.6691306 (\sin (45) \cos (30) - \cos (45) \sin (30))$

Step 7.4

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

$0.71782277 - 0.6691306 (\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30))$

Step 7.5

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

$0.71782277 - 0.6691306 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30))$

Step 7.6

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

$0.71782277 - 0.6691306 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30))$

Step 7.7

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

$0.71782277 - 0.6691306 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 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 7.8.1

Simplify each term.

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Step 7.8.1.1

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

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Step 7.8.1.1.1

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

$0.71782277 - 0.6691306 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 7.8.1.1.2

Combine using the product rule for radicals.

$0.71782277 - 0.6691306 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 7.8.1.1.3

Multiply $2$ by $3$ .

$0.71782277 - 0.6691306 (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 7.8.1.1.4

Multiply $2$ by $2$ .

$0.71782277 - 0.6691306 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 7.8.1.2

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

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Step 7.8.1.2.1

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

$0.71782277 - 0.6691306 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})$

Step 7.8.1.2.2

Multiply $2$ by $2$ .

$0.71782277 - 0.6691306 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})$

Step 7.8.2

Combine the numerators over the common denominator.

$0.71782277 - 0.6691306 \frac{\sqrt{6} - \sqrt{2}}{4}$

Step 8

Multiply $- 0.6691306 \frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 8.1

Combine $- 0.6691306$ and $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

$0.71782277 + \frac{- 0.6691306 (\sqrt{6} - \sqrt{2})}{4}$

Step 8.2

Multiply $- 0.6691306$ by $\sqrt{6} - \sqrt{2}$ .

$0.71782277 + \frac{- 0.69273497}{4}$

Step 9

Divide $- 0.69273497$ by $4$ .

$0.71782277 - 0.17318374$

Step 10

Subtract $0.17318374$ from $0.71782277$ .

$0.54463903$