Question

How to find x in the equation: x*cos(x)=0.3 ?

Answer 1

Please see below.

Explanation:

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I can think of two ways to solve this. One is algebraically and the second graphically.

To solve it algebraically, we know that `cosx` can only have values between `1` and `-1`.

`xcosx=0.3`

`cosx=0.3/x`

We can try values between `1` and `-1` for `cosx` and solve for `x`. There will be infinite number of answers. Alternatively, you can rewrite the equation as:

and perform complex number algebra to solve for the roots some of which will be:

Or you can solve it graphically by graphing `cosx` as a function and `0.3/x` as a separate function and examine the points of intersection. You can use trial and error method of plugging in values to the right and left of each intersection to arrive at the actual value of `x` that best satisfies the equation.

If you graph the function as one function, i.e. `y=xcosx-0.3` you can see the roots where the graph crosses the `x`-axis:

If you graph it as two separate functions, i.e. `cosx` and `0.3/x`, you will see the roots where the graphs intersect:

The figure below shows the intersections of `cosx`: and `a/x` for `a=1, 5, 10, and 20`:

The figure below shows the intersections of `cosx`: and `a/x` for `a=0.01, 0.07, 0.1, and 0.2`:

As you can see, there is a different set of solutions for each group of values for `a`.