Expand Using Sum/Difference Formulas cos((19pi)/12)

$\cos (\frac{19 π}{12})$

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $\frac{19 π}{12}$ can be split into $\frac{5 π}{4} + \frac{π}{3}$ .

$\cos (\frac{5 π}{4} + \frac{π}{3})$

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

$\cos (\frac{π}{3}) \cdot \cos (\frac{5 π}{4}) - \sin (\frac{π}{3}) \cdot \sin (\frac{5 π}{4})$

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$(\frac{1}{2}) \cdot \cos (\frac{5 π}{4}) - \sin (\frac{π}{3}) \cdot \sin (\frac{5 π}{4})$

Simplify $\cos (\frac{5 π}{4})$ .

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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 third quadrant.

$(\frac{1}{2}) \cdot (- \cos (\frac{π}{4})) - \sin (\frac{π}{3}) \cdot \sin (\frac{5 π}{4})$

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$(\frac{1}{2}) \cdot (- \frac{\sqrt{2}}{2}) - \sin (\frac{π}{3}) \cdot \sin (\frac{5 π}{4})$

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

$(\frac{1}{2}) \cdot (- \frac{\sqrt{2}}{2}) - \frac{\sqrt{3}}{2} \cdot \sin (\frac{5 π}{4})$

Simplify $\sin (\frac{5 π}{4})$ .

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Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

$(\frac{1}{2}) \cdot (- \frac{\sqrt{2}}{2}) - \frac{\sqrt{3}}{2} \cdot (- \sin (\frac{π}{4}))$

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Simplify each term.

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

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

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

Multiply $2$ by $2$ .

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

Multiply $- \frac{\sqrt{3}}{2} (- \frac{\sqrt{2}}{2})$ .

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Multiply $- 1$ by $- 1$ .

$- \frac{\sqrt{2}}{4} + 1 \frac{\sqrt{3}}{2} \frac{\sqrt{2}}{2}$

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

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

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

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

Combine using the product rule for radicals.

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

Multiply $3$ by $2$ .

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

Multiply $2$ by $2$ .

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

Simplify.

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Combine the numerators over the common denominator.

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

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

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

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

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

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

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

Simplify the expression.

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

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

Move the negative in front of the fraction.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.25881904 \dots$

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