Trigonometry Examples

$\sin (105)$

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 $\sin (105)$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Apply the sum of angles identity.

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

Step 2.4

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

$(\frac{1}{2} \cdot \cos (45) + \cos (30) \sin (45)) \cdot \frac{π}{180}$ radians

Step 2.5

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)) \cdot \frac{π}{180}$ radians

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

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

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Step 2.8.1.1.1

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

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

Step 2.8.1.1.2

Multiply $2$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \cdot \frac{π}{180}$ radians

Step 2.8.1.2

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

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Step 2.8.1.2.1

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

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}) \cdot \frac{π}{180}$ radians

Step 2.8.1.2.2

Combine using the product rule for radicals.

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}) \cdot \frac{π}{180}$ radians

Step 2.8.1.2.3

Multiply $3$ by $2$ .

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

Step 2.8.1.2.4

Multiply $2$ by $2$ .

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

Step 2.8.2

Combine the numerators over the common denominator.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Multiply $4$ by $180$ .

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