Math CentralQuandaries & Queries


Question from Usama, a student:


There are many modern calculators and softwares that can plot the functions.
I have two very simple equations y=x-1 and y=sin(x), if i draw them manually on paper i can not get the accurate value. But i think a computer can easily draw perfect straight and curve lines so it should be easy to find the intersection point of two lines.
Is there any way or software that can tell the exact value of intersection point of two lines?


A drawing program is not what you need, you need a computer program that will algebraically or numerically solve sin(x) = x - 1. There is one such program that I know of online, it's called WolframAlpha, and uses the computer program Mathematica. If you enter

sin(x) = x - 1

the program returns a graph but also a numerical approximation to the solution to the equation. It is assumed here that the units of x are radians.

I hope this helps,



This equation, like many that mix polynomials & trig functions, does not have any "closed form" solution. A numerical approximation is the best that you will be able to obtain.

If you only have a drawing program but it is capable of zooming, you can get a high-accuracy numerical approximation by zooming in on the point of intersection.

With a very basic nongraphing calculator you can do the same by evaluating sin(x)-x+1 for values of x until you pass from positive to negative or vice versa. Then evaluate it halfway between those two points; in one of the two half-intervals the function changes sign. Keep doing this till the value is as accurate as you need.

(If you know calculus you can use "Newton's method" which is faster but more difficult.)

Good Hunting!

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