Expand Using Sum/Difference Formulas sin((11pi)/12)

$\sin (\frac{11 π}{12})$

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $\frac{11 π}{12}$ can be split into $\frac{2 π}{3} + \frac{π}{4}$ .

$\sin (\frac{2 π}{3} + \frac{π}{4})$

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

$\sin (\frac{2 π}{3}) \cos (\frac{π}{4}) + \cos (\frac{2 π}{3}) \sin (\frac{π}{4})$

Simplify $\sin (\frac{2 π}{3})$ .

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

$(\sin (\frac{π}{3})) \cos (\frac{π}{4}) + \cos (\frac{2 π}{3}) \sin (\frac{π}{4})$

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

$(\frac{\sqrt{3}}{2}) \cos (\frac{π}{4}) + \cos (\frac{2 π}{3}) \sin (\frac{π}{4})$

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

$(\frac{\sqrt{3}}{2}) (\frac{\sqrt{2}}{2}) + \cos (\frac{2 π}{3}) \sin (\frac{π}{4})$

Simplify $\cos (\frac{2 π}{3})$ .

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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 second quadrant.

$(\frac{\sqrt{3}}{2}) (\frac{\sqrt{2}}{2}) + (- \cos (\frac{π}{3})) \sin (\frac{π}{4})$

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

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

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

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

Simplify each term.

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

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

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

Combine using the product rule for radicals.

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

Multiply $3$ by $2$ .

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

Multiply $2$ by $2$ .

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

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

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

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.25881904 \dots$

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