Trigonometry Examples

cos (165)

$\cos (165)$

Step 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)

$- \cos (15)$

Step 2

Split

15

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

- cos (45 - 30)

$- \cos (45 - 30)$

Step 3

Separate negation.

- cos (45 - (30))

$- \cos (45 - (30))$

Step 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))

$- (\cos (45) \cos (30) + \sin (45) \sin (30))$

Step 5

The exact value of

cos (45)

$\cos (45)$ is

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

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

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

$- (\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30))$

Step 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))

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

Step 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} sin (30))

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

Step 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})

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

Step 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 9.1

Simplify each term.

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Step 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 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})

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

Step 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})

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

Step 9.1.1.3

Multiply

2

$2$ by

3

$3$ .

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

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

Step 9.1.1.4

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

Step 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 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})

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

Step 9.1.2.2

Multiply

2

$2$ by

2

$2$ .

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

$- (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4})$

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

$- (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4})$

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

$- (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4})$

Step 9.2

Combine the numerators over the common denominator.

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.96592582 \dots$