I can solve for $x$ by taking the arctangent of both sides but I'm not understanding what the equation means. Does the equation represent the interesection between $y = \tan x$ and $y =1$? Likewise is $\sin x = 2$ said to be undefined as the two curves do not intersect for any real $x$ (they don't cut each other?). I feel that I am missing a key concept in trigonometric equations.
Also to find other solutions of $\tan x = 1$ do I just subtract or add $\pi$ to the principle value?
Thanks
$\endgroup$ 64 Answers
$\begingroup$Solving the equation $\tan x=1$ means finding every angle $x$ whose tangent is $1$. You can start with $\frac{\pi}4$, the principal arctangent of $1$; the tangent has period $\pi$, so (as you said) you need to add integer multiples of $\pi$ to that to get the entire set of solutions:
$$\left\{\frac{\pi}4+n\pi:n\in\Bbb Z\right\}\;.$$
If you’re asked to solve $\sin x=2$, you can safely say that the set of solutions is empty ($\varnothing$), since $|\sin x|\le 1$ for all $x\in\Bbb R$.
In neither case is it actually necessary to think in terms of graphs, though it’s certainly possible to do so: the set of solutions to $\tan x=1$ is indeed the set of $x\in\Bbb R$ where the graphs of $y=\tan x$ and $y=1$ intersect, and similarly for $\sin x=1$.
$\endgroup$ 3 $\begingroup$$\tan{x}$ is the ratio of $\sin{x}$ over $\cos{x}$. Having $\tan{x}=1$ means that $\sin{x}=\cos{x}$.
Now, what angles produce this result?
$\endgroup$ $\begingroup$What is your definition of the tangent function? It's the sine divided by the cosine, which is exactly the same thing as the famous slope formula "rise over run" So if x is the angle a line through the Origin makes with the positive horizontal axis, you are essentially asking for what angle the line has slope 1. Now when you solve the equation, you get 45° and so the line makes 45° with the positive axis.
$\endgroup$ $\begingroup$The solutions (also known as roots), if any, of an equation $f(x)=0$ can always be found graphically by plotting $y=f(x)$ and locating the $x$ intercepts.
Your equation $\tan x=1$ can be rewritten as $\tan x - 1 = 0$. It implies that $f(x)=\tan x -1$. The plot is as follows and the solutions ($x$ intercepts) are $\{x|x=\frac{\pi}{4}+k\pi ,\, k\in \mathbb{Z}\}$.
