Precalculus Examples

$\cos (165 °)$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $165 °$ can be split into $120 + 45$ .

$\cos (120 + 45)$

Step 2

Use the sum formula for cosine to simplify the expression. The formula states that $\cos (A + B) = - (\cos (A) \cos (B) + \sin (A) \sin (B))$ .

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

Step 3

Remove parentheses.

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

Step 4

Simplify each term.

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Step 4.1

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

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

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.

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

Step 4.3

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

Multiply $2$ by $2$ .

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

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

Step 4.8

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

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Step 4.8.1

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

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

Step 4.8.2

Combine using the product rule for radicals.

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

Step 4.8.3

Multiply $3$ by $2$ .

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

Step 4.8.4

Multiply $2$ by $2$ .

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

Step 5

Simplify.

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Step 5.1

Combine the numerators over the common denominator.

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

Step 5.2

Factor $- 1$ out of $- \sqrt{2}$ .

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

Step 5.3

Factor $- 1$ out of $- \sqrt{6}$ .

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

Step 5.4

Factor $- 1$ out of $- (\sqrt{2}) - (\sqrt{6})$ .

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

Step 5.5

Simplify the expression.

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Step 5.5.1

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

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

Step 5.5.2

Move the negative in front of the fraction.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.96592582 \dots$