Simplify the expression. sin ((5pi)/(8))=

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Express in terms of known values: Express sin((5pi)/8) in terms of known sine values. Since (5pi)/8 is not a standard angle for which we know the sine value directly, we can express it in terms of angles for which we do know the sine values. We can write (5pi)/8 as (4pi)/8 + (pi)/8, which simplifies to (pi)/2 + (pi)/8.Use sine addition formula: Use the sine addition formula. The sine addition formula is sin(A + B) = sin(A)cos(B) + cos(A)sin(B). We will use this formula with A = (pi)/2 and B = (pi)/8.Calculate sine and cosine values: Calculate the sine and cosine values for A and B. We know that sin((pi)/2) = 1 and cos((pi)/2) = 0. We also need to find the values for sin((pi)/8) and cos((pi)/8). Since (pi)/8 is not a standard angle, we can use the half-angle formula to find its sine and cosine values. The half-angle formulas are sin(x/2) = sqrt((1 - cos(x))/2) and cos(x/2) = sqrt((1 + cos(x))/2). We can use these with x = (pi)/4, for which we know sin((pi)/4) = sqrt(2)/2 and cos((pi)/4) = sqrt(2)/2.Apply half-angle formulas: Apply the half-angle formulas. Using the half-angle formulas, we get sin((pi)/8) = sqrt((1 - cos((pi)/4))/2) = sqrt((1 - sqrt(2)/2)/2) and cos((pi)/8) = sqrt((1 + cos((pi)/4))/2) = sqrt((1 + sqrt(2)/2)/2).Substitute into sine addition formula: Substitute the values into the sine addition formula. Substituting the values into the sine addition formula, we get sin((5pi)/8) = sin((pi)/2)cos((pi)/8) + cos((pi)/2)sin((pi)/8) = 1 * sqrt((1 + sqrt(2)/2)/2) + 0 * sqrt((1 - sqrt(2)/2)/2) = sqrt((1 + sqrt(2)/2)/2).Simplify expression: Simplify the expression. Simplifying the expression, we get sin((5pi)/8) = sqrt((2 + sqrt(2))/4) = sqrt(2 + sqrt(2))/2.

Simplify the expression. $\sin \frac{5 \pi}{8}$ =

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Simplify the expression. $\sin \frac{5 \pi}{8}$ =