Trigonometry Examples

\cos (165)

$\cos (165)$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case,

165

$165$ can be split into

120 + 45

$120 + 45$ .

\cos (120 + 45)

$\cos (120 + 45)$

Step 2

Use the sum formula for cosine to simplify the expression. The formula states that

\cos (A + B) = - (\cos (A) \cos (B) + \sin (A) \sin (B))

$\cos (A + B) = - (\cos (A) \cos (B) + \sin (A) \sin (B))$ .

\cos (45) \cdot \cos (120) - \sin (45) \cdot \sin (120)

$\cos (45) \cdot \cos (120) - \sin (45) \cdot \sin (120)$

Step 3

Remove parentheses.

\cos (45) \cdot \cos (120) - \sin (45) \cdot \sin (120)

$\cos (45) \cdot \cos (120) - \sin (45) \cdot \sin (120)$

Step 4

Simplify each term.

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Step 4.1

The exact value of

\cos (45)

$\cos (45)$ is

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

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

\frac{\sqrt{2}}{2} \cdot \cos (120) - \sin (45) \cdot \sin (120)

$\frac{\sqrt{2}}{2} \cdot \cos (120) - \sin (45) \cdot \sin (120)$

Step 4.2

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.

\frac{\sqrt{2}}{2} \cdot (- \cos (60)) - \sin (45) \cdot \sin (120)

$\frac{\sqrt{2}}{2} \cdot (- \cos (60)) - \sin (45) \cdot \sin (120)$

Step 4.3

The exact value of

\cos (60)

$\cos (60)$ is

\frac{1}{2}

$\frac{1}{2}$ .

\frac{\sqrt{2}}{2} \cdot (- \frac{1}{2}) - \sin (45) \cdot \sin (120)

$\frac{\sqrt{2}}{2} \cdot (- \frac{1}{2}) - \sin (45) \cdot \sin (120)$

Step 4.4

Multiply

\frac{\sqrt{2}}{2} (- \frac{1}{2})

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

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Step 4.4.1

Multiply

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

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

\frac{1}{2}

$\frac{1}{2}$ .

- \frac{\sqrt{2}}{2 \cdot 2} - \sin (45) \cdot \sin (120)

$- \frac{\sqrt{2}}{2 \cdot 2} - \sin (45) \cdot \sin (120)$

Step 4.4.2

Multiply

2

$2$ by

2

$2$ .

- \frac{\sqrt{2}}{4} - \sin (45) \cdot \sin (120)

$- \frac{\sqrt{2}}{4} - \sin (45) \cdot \sin (120)$

- \frac{\sqrt{2}}{4} - \sin (45) \cdot \sin (120)

$- \frac{\sqrt{2}}{4} - \sin (45) \cdot \sin (120)$

Step 4.5

The exact value of

\sin (45)

$\sin (45)$ is

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

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

- \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{2} \cdot \sin (120)

$- \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{2} \cdot \sin (120)$

Step 4.6

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

- \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{2} \cdot \sin (60)

$- \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{2} \cdot \sin (60)$

Step 4.7

The exact value of

\sin (60)

$\sin (60)$ is

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

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

- \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}

$- \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$

Step 4.8

Multiply

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

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

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Step 4.8.1

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}

$- \frac{\sqrt{2}}{4} - \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}$

Step 4.8.2

Combine using the product rule for radicals.

- \frac{\sqrt{2}}{4} - \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}

$- \frac{\sqrt{2}}{4} - \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}$

Step 4.8.3

Multiply

3

$3$ by

2

$2$ .

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

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

Step 4.8.4

Multiply

2

$2$ by

2

$2$ .

- \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}

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

Step 5

Simplify.

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Step 5.1

Combine the numerators over the common denominator.

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

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

Step 5.2

Factor

- 1

$- 1$ out of

- \sqrt{2}

$- \sqrt{2}$ .

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

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

Step 5.3

Factor

- 1

$- 1$ out of

- \sqrt{6}

$- \sqrt{6}$ .

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

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

Step 5.4

Factor

- 1

$- 1$ out of

- (\sqrt{2}) - (\sqrt{6})

$- (\sqrt{2}) - (\sqrt{6})$ .

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

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

Step 5.5

Simplify the expression.

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Step 5.5.1

Rewrite

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

$- (\sqrt{2} + \sqrt{6})$ as

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

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

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

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

Step 5.5.2

Move the negative in front of the fraction.

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

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

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

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

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

- 0.96592582 \dots

$- 0.96592582 \dots$