How do you solve cosx - xsinx equals 0?

Answer 1

To solve this, you need to find values of x where cos(x)

=

xsin(x).

First of all, 0 is not a solution because cos(0) =

1, and sin(0) =

0. Since 0 is not a solution, divide both sides of the equation by sin(x)

to get cot(x)

=

x (remember that cos divided by sin is the same as cot). The new question to answer is, when is cot(x)

=

x? Using Wolfram Alpha, the results are

x ±9.52933440536196...
x ±6.43729817917195...
x ±3.42561845948173...
x ±0.860333589019380... there will be an infinite number of solutions.

If you'd like to do the calculation yourself (not asking WolframAlpha)

then there's a trick which almost always works, even for equations which cant be done analytically.

Starting with the basic equation, cos(x)

=

x*sin(x),

transpose it to a form starting with "x =".

In this case you could get: x =

1/tan(x), x =

cot(x)

or from tan(x)

=

1/x you get x =

Arctan(1/x).

Because I like to do my calcs

on an old calculator which only has Arctan

and not Arccotan

(Inverse cotangent(x))

I use the last above - x =

Arctan(1/x)

Starting with a value like 0.5, hit the 1/x key then shift tan keys. Just keep repeating those two operations and the display will converge on 0.860333. Too easy. This example of the method is not a good one as it takes about 25 iterations to converge to within 0.0000001 of the right answer. It is unusually slow.

And finally, this method has only 50% chance of working first try. We were lucky picking x =

Arctan(1/x). x =

1/tan(x) diverges ind the iterations do not converge on the answer.

So if you try this method on another problem and it diverges, just transpose the equation again and have another go.

Starting with x^2 + x - 3 =

0,

and iterating x =

3-x^2, you find it diverges, so

try x =

sqr(3-x) which (with care and about 25 iterations) converges on 1.302775638.