Trigonometry Examples

\sin (105)

$\sin (105)$

Step 1

To convert degrees to radians, multiply by

\frac{π}{180 °}

$\frac{π}{180 °}$ , since a full circle is

360 °

$360 °$ or

2 π

$2 π$ radians.

Step 2

The exact value of

\sin (105)

$\sin (105)$ is

\frac{\sqrt{2} + \sqrt{6}}{4}

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

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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}

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

Step 2.2

Split

75

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

\sin (30 + 45) \cdot \frac{π}{180}

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

Step 2.3

Apply the sum of angles identity.

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

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

Step 2.4

The exact value of

\sin (30)

$\sin (30)$ is

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 2.5

The exact value of

\cos (45)

$\cos (45)$ is

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

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

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

Step 2.6

The exact value of

\cos (30)

$\cos (30)$ is

\frac{\sqrt{3}}{2}

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

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

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

Step 2.7

The exact value of

\sin (45)

$\sin (45)$ is

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

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

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

Step 2.8

Simplify

\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}

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

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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}

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

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Step 2.8.1.1.1

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

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

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

Step 2.8.1.1.2

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

Step 2.8.1.2

Multiply

\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}

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

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Step 2.8.1.2.1

Multiply

\frac{\sqrt{3}}{2}

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

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

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

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

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

Step 2.8.1.2.3

Multiply

3

$3$ by

2

$2$ .

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

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

Step 2.8.1.2.4

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

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

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

Step 2.8.2

Combine the numerators over the common denominator.

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

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

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

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

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

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

Step 3

Multiply

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

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

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Step 3.1

Multiply

\frac{\sqrt{2} + \sqrt{6}}{4}

$\frac{\sqrt{2} + \sqrt{6}}{4}$ by

\frac{π}{180}

$\frac{π}{180}$ .

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

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

Step 3.2

Multiply

4

$4$ by

180

$180$ .

\frac{(\sqrt{2} + \sqrt{6}) π}{720}

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

\frac{(\sqrt{2} + \sqrt{6}) π}{720}

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