Find the Exact Value cos((7pi)/12)cos((5pi)/12)+sin((7pi)/12)sin((5pi)/12)

$\cos (\frac{7 π}{12}) \cos (\frac{5 π}{12}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Step 1

The exact value of $\cos (\frac{7 π}{12})$ is $- \frac{\sqrt{2 - \sqrt{3}}}{2}$ .

Rewrite $\frac{7 π}{12}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cos (\frac{\frac{7 π}{6}}{2}) \cos (\frac{5 π}{12}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$(\pm \sqrt{\frac{1 + \cos (\frac{7 π}{6})}{2}}) \cos (\frac{5 π}{12}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Change the $\pm$ to $-$ because cosine is negative in the second quadrant.

$- \sqrt{\frac{1 + \cos (\frac{7 π}{6})}{2}} \cos (\frac{5 π}{12}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Simplify $- \sqrt{\frac{1 + \cos (\frac{7 π}{6})}{2}}$ .

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 third quadrant.

$- \sqrt{\frac{1 - \cos (\frac{π}{6})}{2}} \cos (\frac{5 π}{12}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$- \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \cos (\frac{5 π}{12}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Write $1$ as a fraction with a common denominator.

$- \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{3}}{2}}{2}} \cos (\frac{5 π}{12}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Combine the numerators over the common denominator.

$- \sqrt{\frac{\frac{2 - \sqrt{3}}{2}}{2}} \cos (\frac{5 π}{12}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Multiply the numerator by the reciprocal of the denominator.

$- \sqrt{\frac{2 - \sqrt{3}}{2} \cdot \frac{1}{2}} \cos (\frac{5 π}{12}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Multiply $\frac{2 - \sqrt{3}}{2} \cdot \frac{1}{2}$ .

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

$- \sqrt{\frac{2 - \sqrt{3}}{2 \cdot 2}} \cos (\frac{5 π}{12}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Multiply $2$ by $2$ .

$- \sqrt{\frac{2 - \sqrt{3}}{4}} \cos (\frac{5 π}{12}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Rewrite $\sqrt{\frac{2 - \sqrt{3}}{4}}$ as $\frac{\sqrt{2 - \sqrt{3}}}{\sqrt{4}}$ .

$- \frac{\sqrt{2 - \sqrt{3}}}{\sqrt{4}} \cos (\frac{5 π}{12}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$- \frac{\sqrt{2 - \sqrt{3}}}{\sqrt{2^{2}}} \cos (\frac{5 π}{12}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{\sqrt{2 - \sqrt{3}}}{2} \cos (\frac{5 π}{12}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Step 2

The exact value of $\cos (\frac{5 π}{12})$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

Split $\frac{5 π}{12}$ into two angles where the values of the six trigonometric functions are known.

$- \frac{\sqrt{2 - \sqrt{3}}}{2} \cos (\frac{π}{6} + \frac{π}{4}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

$- \frac{\sqrt{2 - \sqrt{3}}}{2} (\cos (\frac{π}{6}) \cos (\frac{π}{4}) - \sin (\frac{π}{6}) \sin (\frac{π}{4})) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$- \frac{\sqrt{2 - \sqrt{3}}}{2} (\frac{\sqrt{3}}{2} \cos (\frac{π}{4}) - \sin (\frac{π}{6}) \sin (\frac{π}{4})) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

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

$- \frac{\sqrt{2 - \sqrt{3}}}{2} (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (\frac{π}{6}) \sin (\frac{π}{4})) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$- \frac{\sqrt{2 - \sqrt{3}}}{2} (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (\frac{π}{4})) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

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

$- \frac{\sqrt{2 - \sqrt{3}}}{2} (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Simplify $\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ .

Simplify each term.

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

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

$- \frac{\sqrt{2 - \sqrt{3}}}{2} (\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Combine using the product rule for radicals.

$- \frac{\sqrt{2 - \sqrt{3}}}{2} (\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Multiply $3$ by $2$ .

$- \frac{\sqrt{2 - \sqrt{3}}}{2} (\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Multiply $2$ by $2$ .

$- \frac{\sqrt{2 - \sqrt{3}}}{2} (\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Multiply $- \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ .

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

$- \frac{\sqrt{2 - \sqrt{3}}}{2} (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Multiply $2$ by $2$ .

$- \frac{\sqrt{2 - \sqrt{3}}}{2} (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Combine the numerators over the common denominator.

$- \frac{\sqrt{2 - \sqrt{3}}}{2} \cdot \frac{\sqrt{6} - \sqrt{2}}{4} + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Step 3

Multiply $- \frac{\sqrt{2 - \sqrt{3}}}{2} \cdot \frac{\sqrt{6} - \sqrt{2}}{4}$ .

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4}$ by $\frac{\sqrt{2 - \sqrt{3}}}{2}$ .

$- \frac{(\sqrt{6} - \sqrt{2}) \sqrt{2 - \sqrt{3}}}{4 \cdot 2} + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Multiply $4$ by $2$ .

$- \frac{(\sqrt{6} - \sqrt{2}) \sqrt{2 - \sqrt{3}}}{8} + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Step 4

Apply the distributive property.

$- \frac{\sqrt{6} \sqrt{2 - \sqrt{3}} - \sqrt{2} \sqrt{2 - \sqrt{3}}}{8} + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Step 5

Combine using the product rule for radicals.

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{2} \sqrt{2 - \sqrt{3}}}{8} + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Step 6

Combine using the product rule for radicals.

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \sin (\frac{7 π}{12}) \sin (\frac{5 π}{12})$

Step 7

The exact value of $\sin (\frac{7 π}{12})$ is $\frac{\sqrt{2 + \sqrt{3}}}{2}$ .

Rewrite $\frac{7 π}{12}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \sin (\frac{\frac{7 π}{6}}{2}) \sin (\frac{5 π}{12})$

Apply the sine half-angle identity.

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + (\pm \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{2}}) \sin (\frac{5 π}{12})$

Change the $\pm$ to $+$ because sine is positive in the second quadrant.

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{2}} \sin (\frac{5 π}{12})$

Simplify $\sqrt{\frac{1 - \cos (\frac{7 π}{6})}{2}}$ .

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 third quadrant.

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \sqrt{\frac{1 - - \cos (\frac{π}{6})}{2}} \sin (\frac{5 π}{12})$

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \sqrt{\frac{1 - - \frac{\sqrt{3}}{2}}{2}} \sin (\frac{5 π}{12})$

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

Multiply $- 1$ by $- 1$ .

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \sqrt{\frac{1 + 1 \frac{\sqrt{3}}{2}}{2}} \sin (\frac{5 π}{12})$

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

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} \sin (\frac{5 π}{12})$

Write $1$ as a fraction with a common denominator.

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{3}}{2}}{2}} \sin (\frac{5 π}{12})$

Combine the numerators over the common denominator.

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{2}} \sin (\frac{5 π}{12})$

Multiply the numerator by the reciprocal of the denominator.

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \sqrt{\frac{2 + \sqrt{3}}{2} \cdot \frac{1}{2}} \sin (\frac{5 π}{12})$

Multiply $\frac{2 + \sqrt{3}}{2} \cdot \frac{1}{2}$ .

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

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \sqrt{\frac{2 + \sqrt{3}}{2 \cdot 2}} \sin (\frac{5 π}{12})$

Multiply $2$ by $2$ .

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \sqrt{\frac{2 + \sqrt{3}}{4}} \sin (\frac{5 π}{12})$

Rewrite $\sqrt{\frac{2 + \sqrt{3}}{4}}$ as $\frac{\sqrt{2 + \sqrt{3}}}{\sqrt{4}}$ .

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}}}{\sqrt{4}} \sin (\frac{5 π}{12})$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}}}{\sqrt{2^{2}}} \sin (\frac{5 π}{12})$

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}}}{2} \sin (\frac{5 π}{12})$

Step 8

The exact value of $\sin (\frac{5 π}{12})$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

Split $\frac{5 π}{12}$ into two angles where the values of the six trigonometric functions are known.

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}}}{2} \sin (\frac{π}{6} + \frac{π}{4})$

Apply the sum of angles identity.

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}}}{2} (\sin (\frac{π}{6}) \cos (\frac{π}{4}) + \cos (\frac{π}{6}) \sin (\frac{π}{4}))$

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}}}{2} (\frac{1}{2} \cos (\frac{π}{4}) + \cos (\frac{π}{6}) \sin (\frac{π}{4}))$

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

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}}}{2} (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (\frac{π}{6}) \sin (\frac{π}{4}))$

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}}}{2} (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (\frac{π}{4}))$

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

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}}}{2} (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2})$

Simplify $\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$ .

Simplify each term.

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

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

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}}}{2} (\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2})$

Multiply $2$ by $2$ .

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}}}{2} (\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2})$

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

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

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}}}{2} (\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2})$

Combine using the product rule for radicals.

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}}}{2} (\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2})$

Multiply $3$ by $2$ .

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}}}{2} (\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2})$

Multiply $2$ by $2$ .

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}}}{2} (\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4})$

Combine the numerators over the common denominator.

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}}}{2} \cdot \frac{\sqrt{2} + \sqrt{6}}{4}$

Step 9

Multiply $\frac{\sqrt{2 + \sqrt{3}}}{2} \cdot \frac{\sqrt{2} + \sqrt{6}}{4}$ .

Multiply $\frac{\sqrt{2 + \sqrt{3}}}{2}$ by $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}} (\sqrt{2} + \sqrt{6})}{2 \cdot 4}$

Multiply $2$ by $4$ .

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}} (\sqrt{2} + \sqrt{6})}{8}$

Step 10

Apply the distributive property.

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{2 + \sqrt{3}} \sqrt{2} + \sqrt{2 + \sqrt{3}} \sqrt{6}}{8}$

Step 11

Combine using the product rule for radicals.

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{(2 + \sqrt{3}) \cdot 2} + \sqrt{2 + \sqrt{3}} \sqrt{6}}{8}$

Step 12

Combine using the product rule for radicals.

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{(2 + \sqrt{3}) \cdot 2} + \sqrt{(2 + \sqrt{3}) \cdot 6}}{8}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{6 (2 - \sqrt{3})} - \sqrt{(2 - \sqrt{3}) \cdot 2}}{8} + \frac{\sqrt{(2 + \sqrt{3}) \cdot 2} + \sqrt{(2 + \sqrt{3}) \cdot 6}}{8}$

Decimal Form:

$0.86602540 \dots$

Find the Exact Value cos((7pi)/12)cos((5pi)/12)+sin((7pi)/12)sin((5pi)/12)

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