How do you calculate the exact value of sin(pi/5)?

Question

sin(pi) = 0
sin(pi/2) = 1
sin(pi/3) = sqrt(3)/2
sin(pi/4) = sqrt(2)/2
sin(pi/5) = ?
sin(pi/6) = 1/2
sin(pi/7) = ?
sin(pi/8) = [involves the half angle identity, (1 + cos(2[pi/8]))/2 ]

What identities are used to solve for an answer of sin(pi/5) expressed in radicals?

Answer 1

Try solving z^5 = 1 in the complex domain (by factoring the polynomial z^5 - 1). That will give you the 5 complex quintic roots of unity in the form x+iy (IOW, the Cartesian coords of a regular pentagon). On the other hand, these roots are exp(i2kPi/5), k=0,1,2,3,4. So the real and imaginary parts of the first complex root are cos and sin of 2pi/5. Write them out, and then you can get sin(pi/5) as well.

More detail: z^5 - 1 = (z-1)(z^4+z^3+z^2+z+1).

Divide the eqn z^4+z^3+z^2+z+1 = 0 by z^2, and substitute w = z + (1/z) to get the quadratic eqn w^2+w-1=0. Solve for w, then for z. Take the one complex root in the first quadrant; its real part is (1/4)sqrt(5) - (1/4). So that's cos(2pi/5). And so

sin(pi/5) = sqrt{[1-cos(2pi/5)]/2} = (1/4) sqrt(2) sqrt(5-sqrt(5)).

Answer 2

There is a formula for sin 5x:

sin5x = 16sin^5(x) − 20sin^3(x )+ 5sinx

So let x = pi/5 . Then 5x = pi and sin pi = 0

Factor out sinx from the above formula and solve the remaining 4th degree equation for sin^2 x, then do the square root

Answer 3

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Measure the circumference of a circle. Measure it's diameter. Divide the first by the second on a really big calculator.