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

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

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.

The exact value of is .

The exact value of is .

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.

The exact value of is .

The exact value of is .

Multiply .

Multiply and .

Multiply by .

Multiply .

Multiply and .

Combine using the product rule for radicals.

Multiply by .

Multiply by .

Combine the numerators over the common denominator.

Factor out of .

Factor out of .

Factor out of .

Simplify the expression.

Rewrite as .

Move the negative in front of the fraction.

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

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