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How to solve Sin(pi/16)

User Bshirley
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6 votes

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.

User David DV
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