AlephOmega said:
I have learned a lot of formulas for converting trigonometric values, but when I looked up sin(pi/10) I got the exact answer (1/4)(√5 +1). I tried to arive at this using formulas, but I couldn't. How is this found? What other angles can be found exactly.
Ps. I already know about how taylor series and how they can be used.
this is actually an interesting question, but space prevents me from giving a complete answer here. the answer has to do with "constructible numbers", which is kind of an advanced algebraic topic, and one that relates to which polynomials can be solved "by taking roots".
perhaps you might wonder how this particular value was found. i can shed some light on this.
we start with a (not often used) trig identity:
\sin5x = 16\sin^5x - 20\sin^3x+5\sin x
now if x = \frac{\pi}{10} the left-hand side is 1. therefore, \sin(\frac{\pi}{10}) satisfies the 5-th degree polynomial:
16x^5 - 20x^3 + 5x - 1 = 0
the rational root test shows that x = 1 is a root, so we can factor out x - 1 to obtain:
(x - 1)(16x^4 + 16x^3 - 4x^2 - 4x + 1) = 0
we're not interested in the root 1 (since we know that's not what the sine is), so we are now just interested in a 4-th degree polynomial (well, it's an improvement) :S.
but luck is with us! the quartic polynomial is, in fact, a perfect square:
16x^4 + 16x^3 - 4x^2 - 4x + 1 = (4x^2 + 2x - 1)^2
we know the sine is positive (it's in the first quadrant), so we're only interested in the positive square root, so we have:
4x^2 + 2x - 1 = 0
from here, the quadratic formula gives us the possible solutions:
x = \frac{-1 \pm \sqrt{5}}{4}
since only one of these is positive, that leaves us with:
\sin(\frac{\pi}{10}) = \frac{\sqrt{5} - 1}{4}
