I have the cubic, x^3 - 3/4 x -1/8 = 0, but need a solution not involving the cube roots of 1 ± i√3. I have an approximation using Newton's method, but want exact values. Thanks!
2007-06-04 02:02:01 · 0 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
It turns out that you cannot write the exact answer in terms of radicals involving only real numbers; you have to go to a cube root of a complex number. There just isn't any way around it. The proof of this impossibility is not easy and uses Galois theory.
Even worse, you can find the cube root of that complex number using DeMorgan's theorem. But if you do, you will get an answer for the root of your cubic of....sin(20 degrees)! This is an example of the causus irreducibilis for cubic equations.
Sorry.
2007-06-04 02:11:54 · answer #1 · answered by mathematician 7 · 1⤊ 0⤋
Sin 20 Degrees
2016-10-04 02:50:30 · answer #2 · answered by ? 4 · 0⤊ 0⤋
You can see an expression involving an infinite set of nested radicals here, for what it's worth:
http://mathworld.wolfram.com/TrigonometryAnglesPi18.html
Unfortunately I don't think there's any way to get a nice finite algebraic expression for sin(20). This is sin(pi/18) = sin( (1/2) * (pi/9) ), so even if you use a half-angle formula you're stuck with some trig value evaluated at pi/9. This page explains it a little more:
http://mathworld.wolfram.com/TrigonometryAnglesPi9.html
2007-06-04 02:27:43 · answer #3 · answered by Anonymous · 1⤊ 0⤋
Sine 20 Degrees
2016-12-16 10:31:55 · answer #4 · answered by seeger 4 · 0⤊ 0⤋
The *exact* value of sin(20°) is sin(π/9).
The term 'exact' usua;;y means to put the argument of the trig function in the notation I've given....radian measure.
Is this what you were looking for?
2007-06-04 02:07:47 · answer #5 · answered by Anonymous · 3⤊ 1⤋
Think about it. 20° is 1/18 of a circle or (1/18)*2π Rad which is π/9 Rad. No way will you get a rational (i.e. 'exact') value for sin(20°) from that. (Think about the Taylor series for sin ☺)
Doug
2007-06-04 02:10:59 · answer #6 · answered by doug_donaghue 7 · 2⤊ 1⤋
No exact value
2007-06-04 07:20:15 · answer #7 · answered by Anonymous · 0⤊ 0⤋
The scientific calculator which comes with Windows says:
sin 20° = 0,34202014332566873304409961468226 (approx.)
2007-06-04 02:08:59 · answer #8 · answered by jcastro 6 · 0⤊ 4⤋
sin 20 = 0.34202014332566873304409961468226
2007-06-04 02:07:58 · answer #9 · answered by JR 2 · 0⤊ 4⤋
No. Sin 20º is an irrational number. It goes on forever.
2007-06-04 02:28:16 · answer #10 · answered by davec996 4 · 1⤊ 3⤋