Trigonometry Examples

cos (165)

$\cos (165)$

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

cos (165)

$\cos (165)$ is

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

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

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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 (15) \cdot \frac{π}{180}

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

Step 2.2

Split

15

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

- cos (45 - 30) \cdot \frac{π}{180}

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

Step 2.3

Separate negation.

- cos (45 - (30)) \cdot \frac{π}{180}

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

Step 2.4

Apply the difference of angles identity

cos (x - y) = cos (x) cos (y) + sin (x) sin (y)

$\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

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

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

Step 2.5

The exact value of

cos (45)

$\cos (45)$ is

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

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

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

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

Step 2.6

The exact value of

cos (30)

$\cos (30)$ is

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

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

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

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

Step 2.7

The exact value of

sin (45)

$\sin (45)$ is

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

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

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

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

Step 2.8

The exact value of

sin (30)

$\sin (30)$ is

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 2.9

Simplify

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

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

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Step 2.9.1

Simplify each term.

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Step 2.9.1.1

Multiply

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

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

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Step 2.9.1.1.1

Multiply

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

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

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

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

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

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

Step 2.9.1.1.2

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}

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

Step 2.9.1.1.3

Multiply

2

$2$ by

3

$3$ .

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

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

Step 2.9.1.1.4

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

Step 2.9.1.2

Multiply

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

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

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Step 2.9.1.2.1

Multiply

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

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

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 2.9.1.2.2

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

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

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

Step 2.9.2

Combine the numerators over the common denominator.

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

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

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

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

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

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

Step 3

Multiply

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

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

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Step 3.1

Multiply

\frac{π}{180}

$\frac{π}{180}$ by

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

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

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

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

Step 3.2

Multiply

180

$180$ by

4

$4$ .

- \frac{π (\sqrt{6} + \sqrt{2})}{720}

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

- \frac{π (\sqrt{6} + \sqrt{2})}{720}

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