Trigonometry Examples

$\cos (105)$

Step 1

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

Step 2

The exact value of $\cos (105)$ is $- \frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

$- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \sin (45)) \cdot \frac{π}{180}$ radians

Step 2.7

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

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

Step 2.8

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

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

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

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Step 2.8.1.1.1

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

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

Step 2.8.1.1.2

Combine using the product rule for radicals.

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

Step 2.8.1.1.3

Multiply $3$ by $2$ .

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

Step 2.8.1.1.4

Multiply $2$ by $2$ .

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

Step 2.8.1.2

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

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Step 2.8.1.2.1

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 2.8.1.2.2

Multiply $2$ by $2$ .

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

Step 2.8.2

Combine the numerators over the common denominator.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Multiply $180$ by $4$ .

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