Precalculus Examples

$\cos (75) \cdot 20$

Step 1

To convert degrees to radians, multiply by $\frac{π}{180 °}$ , since a full circle is $360 °$ or $2 π$ radians.

$\cos (75) \cdot 20 ° \cdot \frac{π}{180 °}$ radians

Step 2

Cancel the common factor of $20$ .

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Step 2.1

Factor $20$ out of $\cos (75) \cdot 20$ .

$20 \cdot \cos (75) \cdot \frac{π}{180}$ radians

Step 2.2

Factor $20$ out of $180$ .

$20 \cdot \cos (75) \cdot \frac{π}{20 (9)}$ radians

Step 2.3

Cancel the common factor.

$20 \cdot \cos (75) \cdot \frac{π}{20 \cdot 9}$ radians

Step 2.4

Rewrite the expression.

$\cos (75) \cdot \frac{π}{9}$ radians

Step 3

Combine $\cos (75)$ and $\frac{π}{9}$ .

$\frac{\cos (75) π}{9}$ radians

Step 4

The exact value of $\cos (75)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 4.1

Split $75$ into two angles where the values of the six trigonometric functions are known.

$\frac{\cos (30 + 45) π}{9}$ radians

Step 4.2

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

$\frac{(\cos (30) \cos (45) - \sin (30) \sin (45)) π}{9}$ radians

Step 4.3

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

$\frac{(\frac{\sqrt{3}}{2} \cdot \cos (45) - \sin (30) \sin (45)) π}{9}$ radians

Step 4.4

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

$\frac{(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (30) \sin (45)) π}{9}$ radians

Step 4.5

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

$\frac{(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \sin (45)) π}{9}$ radians

Step 4.6

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

$\frac{(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) π}{9}$ radians

Step 4.7

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

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Step 4.7.1

Simplify each term.

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Step 4.7.1.1

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

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Step 4.7.1.1.1

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

$\frac{(\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) π}{9}$ radians

Step 4.7.1.1.2

Combine using the product rule for radicals.

$\frac{(\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) π}{9}$ radians

Step 4.7.1.1.3

Multiply $3$ by $2$ .

$\frac{(\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) π}{9}$ radians

Step 4.7.1.1.4

Multiply $2$ by $2$ .

$\frac{(\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) π}{9}$ radians

Step 4.7.1.2

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

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Step 4.7.1.2.1

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

$\frac{(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) π}{9}$ radians

Step 4.7.1.2.2

Multiply $2$ by $2$ .

$\frac{(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) π}{9}$ radians

Step 4.7.2

Combine the numerators over the common denominator.

$\frac{\frac{\sqrt{6} - \sqrt{2}}{4} \cdot π}{9}$ radians

Step 5

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

$\frac{\frac{(\sqrt{6} - \sqrt{2}) π}{4}}{9}$ radians

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\frac{(\sqrt{6} - \sqrt{2}) π}{4} \cdot \frac{1}{9}$ radians

Step 7

Multiply $\frac{(\sqrt{6} - \sqrt{2}) π}{4} \cdot \frac{1}{9}$ .

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Step 7.1

Multiply $\frac{(\sqrt{6} - \sqrt{2}) π}{4}$ by $\frac{1}{9}$ .

$\frac{(\sqrt{6} - \sqrt{2}) π}{4 \cdot 9}$ radians

Step 7.2

Multiply $4$ by $9$ .

$\frac{(\sqrt{6} - \sqrt{2}) π}{36}$ radians