Is it possible to find the exact value of sin 10?

Question

I'm not speaking of the approximate value

sin 10=0.173648177...

We know that sin 9=[(sqrt 2)(sqrt 5+1)]/8-[sqrt (5-sqrt 5)]/4

which is an exact value.

Similarly sin 12=[sqrt (10+2 sqrt 5)]/8-[(sqrt 3)(sqrt 5-1)]/8.

But what about sin 10?

All angles are measured in degrees.

Answer 1

Well, by doing something like:

sin 10 = cos 80 = 2cos^2(40) - 1

= 2(2cos^2(20) - 1)^2 - 1

= 2(2(1 - 2sin^2(10))^2 - 1)^2 - 1

We find that it's a solution to the algebraic equation

x = 2(2(1 - 2x^2)^2 - 1)^2 - 1

But this will be a degree 8 polynomial... you can work with it if you want, but I stop here.

Answer 2

There is a formula for sin(3x) in function of x :

sin(3x) = 3sinx - 4sin³x

So if y = sin10°, we have

0.5 = sin 30° = 3y - 4y³

So we have the cubic equation

4y³ - 3y + 0.5 = 0

If you can solve this cubic equation, you have

the value for y = sin10°

We have several methods for solving a cubic

equation.  One of them is the method of

Cardano.  This will give y in function of roots

and cube roots of rational numbers.

Too bad, we have the irreducible case with

the negative discriminant.  We need to take

the cube root of cos(120°) + i sin(120°), which

is amongs others cos(40°) + i sin(40°), so we

are running around in circles : we need cos and

sin of 40° to calculate the one of 10°.

Too bad !!!!