I don't remember where I saw this equation but it was difficult to solve. The equation was:

$$\frac{\tan(x)}{1+\sin(x)} = q$$

where $q$ is a constant. When I started to solve this equation it made me baffled, so I am asking this question to check whether there are any alternative methods, rather than doing simple calculations.


Note that $\tan(x) = \sin(x)/\cos(x)$, and $\cos^2(x) + \sin^2(x) = 1$.

Henceforth, I just use $t, s, c$ as shorthand.

Your question is to solve for $q$ given $\frac{t}{1 + s} = q$. Observe:

$$\frac{t}{1+s} = \frac{s}{(1+s)c} = \frac{s}{(1+s)\sqrt{1 - s^2}}$$

At this point, you have an expression in one variable. And so setting it equal to a given number $q$ yields an equation in one variable, which you can then solve.

  • $\begingroup$ But how to express x in terms of q. That's the question?? $\endgroup$ – Creepy Creature Mar 12 '17 at 19:01
  • $\begingroup$ @CreepyCreature For a given $q$ you are solving for $s := \sin(x)$; so, take the inverse sine to find $x$. $\endgroup$ – Benjamin Dickman Mar 12 '17 at 19:02
  • $\begingroup$ Please explain this in your answer so I should understand! $\endgroup$ – Creepy Creature Mar 12 '17 at 19:04
  • $\begingroup$ And I need a general solution in terms of q. That's the problem!!! $\endgroup$ – Creepy Creature Mar 12 '17 at 19:06
  • $\begingroup$ @BenjaminDickman This yields to a nasty quartic equation ... $\endgroup$ – Michael Hoppe Mar 12 '17 at 19:32

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