Precalculus Examples

$\cos (45 - 60)$

Step 1

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

Step 2

Subtract $60$ from $45$ .

$\cos (- 15) \cdot \frac{π}{180}$ radians

Step 3

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

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Step 3.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\cos (15) \cdot \frac{π}{180}$ radians

Step 3.2

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

$\cos (45 - 30) \cdot \frac{π}{180}$ radians

Step 3.3

Separate negation.

$\cos (45 - (30)) \cdot \frac{π}{180}$ radians

Step 3.4

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

$(\cos (45) \cos (30) + \sin (45) \sin (30)) \cdot \frac{π}{180}$ radians

Step 3.5

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

$(\frac{\sqrt{2}}{2} \cdot \cos (30) + \sin (45) \sin (30)) \cdot \frac{π}{180}$ radians

Step 3.6

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

$(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30)) \cdot \frac{π}{180}$ radians

Step 3.7

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

$(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \sin (30)) \cdot \frac{π}{180}$ radians

Step 3.8

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

$(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot \frac{π}{180}$ radians

Step 3.9

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

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Step 3.9.1

Simplify each term.

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Step 3.9.1.1

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

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Step 3.9.1.1.1

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

$(\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot \frac{π}{180}$ radians

Step 3.9.1.1.2

Combine using the product rule for radicals.

$(\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot \frac{π}{180}$ radians

Step 3.9.1.1.3

Multiply $2$ by $3$ .

$(\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot \frac{π}{180}$ radians

Step 3.9.1.1.4

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot \frac{π}{180}$ radians

Step 3.9.1.2

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

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Step 3.9.1.2.1

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

$(\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) \cdot \frac{π}{180}$ radians

Step 3.9.1.2.2

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) \cdot \frac{π}{180}$ radians

Step 3.9.2

Combine the numerators over the common denominator.

$\frac{\sqrt{6} + \sqrt{2}}{4} \cdot \frac{π}{180}$ radians

Step 4

Multiply $\frac{\sqrt{6} + \sqrt{2}}{4} \cdot \frac{π}{180}$ .

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Step 4.1

Multiply $\frac{\sqrt{6} + \sqrt{2}}{4}$ by $\frac{π}{180}$ .

$\frac{(\sqrt{6} + \sqrt{2}) π}{4 \cdot 180}$ radians

Step 4.2

Multiply $4$ by $180$ .

$\frac{(\sqrt{6} + \sqrt{2}) π}{720}$ radians