# Is there a solution to this transcendental equation ?

1. Sep 29, 2014

### Monsterboy

This is not a homework problem, this equation simply occurred to my mind and my math teacher said such an equation either can't exist or he doesn't know the answer.
sinX ➕ cosX = lnX

I don't know how to start....

2. Sep 29, 2014

### economicsnerd

This looks ugly. To simplify it a bit, notice that the LHS is equal to $\frac12 \sin (2x)$. In particular, this lies in $\left[-\frac12,\frac12\right]$, so that any $x$ solving your equation would have to belong to $\left[\frac1{\sqrt{e}},\sqrt{e}\right]$.

Using some knowlege of the functions $\sin$ and $\ln$ (i.e. knowing where each of them is positive/negative and where each of them is positively/negatively sloped, and using that $2\in (0,\pi)$), you can use intermediate value theorem to show that the equation has a unique solution $x^*>0$, and that it satifies $x^*\in (1, \sqrt{e})$.

As for an explicit solution, good luck.

3. Sep 29, 2014

### PeroK

You could have started by drawing a graph of each function. Note that:

$sin(x) + cos(x) = \sqrt{2}sin(x + \frac{\pi}{4})$

Which simplifies things.

4. Sep 29, 2014

### PeroK

$sin(x) + cos(x) \ne \frac12 \sin (2x)$

$sin(x)cos(x) = \frac12 \sin (2x)$

5. Sep 29, 2014

### economicsnerd

Was it "+" rather than "*" ?

On my computer, it's showing up as "sin(x) [black square with a white X in it] cos(x)". If it was addition, my bad. Sorry for adding confusion.

6. Sep 29, 2014

### Staff: Mentor

I think it is some special character, something like a large "+" sign. So it is clearly an addition.
This does not change the result - there is a unique solution, but probably no closed form for it.

7. Sep 30, 2014

### Monsterboy

with intermediate value method X is something like 1.8893....

8. Sep 30, 2014

### Char. Limit

Indeed. Which is the sole numerical solution to sin(x)+cos(x) = ln(x).

It has no closed form, though. Numerical solution is the best you're going to get.