sin x = 1/3
What is the exact value of x in radians and how do you figure this?
Off the top of my head I’m not absolutely positive, but I have a pretty good feeling that it’s impossible for that particular value. There’s no systematic way for finding the exact value of an inverse trig function; it’s not that there’s no known way, it’s just that it’s impossible.
There are various ways of approximating the value as accurately as you wish, such as Newton’s method, but you’re not going to get an exact value.
Well, I’d put 1/3 in my calculator and hit the arcsin button, after putting the calculator in radians mode.
Actually, looking at it again, I’m not clear on the question. Are you asking for the general algorithm for finding arcsins, a clever trick for finding THIS arcsin, or an explanation of radians?
Sounds like homework to me.
it is easy to write but not easy to type so i will leave it up to you. Bit i will give you a hint. Definations of the trig functions
I stuck it in my nifty Windows calculator and got
0.3398369094541 radians
Of course, you can figure this out using a power series for sin (x). (Or is it a Taylor series? I can never remember…)
The answer is an irrational number, roughly equal to 0.34 radians. I used a calculator, though back in the dark ages I used trig tables, or a slide rule. Nowadays, if I didn’t have a calculator, I’d probably write a little C program to calculate it using a run-time library function call.
Kevin B.
Well I got totally bitched at for not knowing that… I asked the math teacher what the answer was, he didn’t answer…
So, other than
sin x
where x = (integer)pi/12
Are there rational answers?
There are plenty of rational answers, but few that can be gotten analytically from a relatively simple number. For instance, sin(arcsin®) is just R, but good luck trying to express arcsin® in any other form.
If y=sin(x), x and y can’t both be rational unless they are zero.
You can use the infinite series for arcsin(x) with x=1/3:
arcsin(x) = x + (1)/(2)/3x[sup]3[/sup] + (13)/(24)/5x[sup]5[/sup] + (135)/(246)/7*x[sup]7[/sup] + …
If you want exact expressions for sine of rational multiples of pi other than Npi/12, you can use the half angle formula for values of the form Npi/24, and again for Npi/48, … A case unrelated to Npi/12 is
sin(pi/5) = sqrt( (5-sqrt(5))/8 )
and there are others like for arcsin(pi/17).
There are also identities like:
pi/4 = arctan(1/2) + arctan(1/3) = arctan(2/3) + arctan(1/5)
Uhh… so your answer is no?
jbird3000, if you mean the mathematical definition when you say “rational”, then manlob is right when he says the only case is x and y both zero.
If all you mean is that there is a simple expression for both x and y, then there are other cases. Some of these are:
x = pi / 2, y = 1
x = pi / 4, y = sqrt(2)
x = pi / 3, y = sqrt(3) / 2
x = pi / 6, y = 1/2
sqrt(2), sqrt(3), and pi are all irrational, so they wouldn’t count as rational solutions.
Basically you should not have been bitched at, because for sinx= 1/3 x is not only irrational but also not a product of any rational number and “pi”.
That’s what I meant. Rational number and pi.
As in arcsin 1/2= pi/6.
Let me rephrase the question…
Can “arcsin x” always be simplified in a way keeping the exact number and not an approximation, with or without the use of pi?
[QUOTE]
*Originally posted by jbird3000 *
**
Yes, for example that infinite series given above is exact. No, if you require a finite number of steps.
Values of sin(Npi/M) which can be expressed with a finite number of terms involving +,-,/, and sqrt() are closely related to constructing regular polygons with a straightedge and compass. This can be done for polygons with 3, 5, 17, 257, or 65537 sides. So sin(2*pi/17) can be simplified to a finite exact expression. Above I gave the value for sin(pi/5).
From those expressions you can create more by using half angle formulas, for example, sin(pi/10)= sqrt( (1-cos(pi/5))/2). You can also use the identity for sin(A+B), for example, sin(8*pi/15)= sin(pi/3+pi/5).