Answer:
To solve sin(pi/16), we can use the half-angle formula for sine, which states that:
sin(x/2) = ±√[(1 - cos(x))/2]
Let's use this formula with x = pi/8:
sin(pi/16) = sin(pi/8)/2 = ±√[(1 - cos(pi/8))/2]
To determine the sign, we need to know in which quadrant pi/16 lies. Since pi/2 < pi/16 < pi, pi/16 lies in the second quadrant where sine is positive. Hence,
sin(pi/16) = √[(1 - cos(pi/8))/2]
Now, we need to find cos(pi/8). We can use the half-angle formula for cosine, which states that:
cos(x/2) = ±√[(1 + cos(x))/2]
Again, let's use this formula with x = pi/4:
cos(pi/8) = cos(pi/4)/2 = ±√[(1 + cos(pi/4))/2]
To determine the sign, we need to know in which quadrant pi/8 lies. Since pi/2 > pi/8 > 0, pi/8 lies in the first quadrant where cosine is positive. Hence,
cos(pi/8) = √[(1 + cos(pi/4))/2] = √[(1 + √2/2)/2]
Finally, we can substitute this expression for cos(pi/8) into the expression we found for sin(pi/16):
sin(pi/16) = √[(1 - √[(1 + √2/2)/2])/2] ≈ 0.1951
Therefore, sin(pi/16) is approximately equal to 0.1951.