Answer: To find the exact value of tan(π/16) without using a calculator, we can use the trigonometric identity for the tangent of half-angle:
tan(θ/2) = (1 - cos(θ)) / sin(θ)
In this case, θ = π/8, so we can find the value of tan(π/16) by substituting this into the formula:
tan(π/16) = tan(π/8 / 2) = (1 - cos(π/8)) / sin(π/8)
Now, let's focus on finding the values of cos(π/8) and sin(π/8) without using a calculator. We can use the half-angle trigonometric identities for these values:
cos(θ/2) = ±√((1 + cos(θ)) / 2)
sin(θ/2) = ±√((1 - cos(θ)) / 2)
In this case, θ = π/4, so we can find the values of cos(π/8) and sin(π/8):
cos(π/8) = ±√((1 + cos(π/4)) / 2) = ±√((1 + √2/2) / 2)
To determine the sign of cos(π/8), we need to consider that π/8 is in the first quadrant, where cosine is positive. So, we take the positive square root:
cos(π/8) = √((1 + √2/2) / 2)
sin(π/8) = ±√((1 - cos(π/4)) / 2) = ±√((1 - √2/2) / 2)
Again, since π/8 is in the first quadrant, sine is positive. So, we take the positive square root:
sin(π/8) = √((1 - √2/2) / 2)
Now that we have the values of cos(π/8) and sin(π/8), we can find tan(π/16):
tan(π/16) = (1 - cos(π/8)) / sin(π/8)
tan(π/16) = (1 - √((1 + √2/2) / 2)) / √((1 - √2/2) / 2)
This is the exact value of tan(π/16) without using a calculator.