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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?


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Please don't use display style $$ ... $$ in titles. It messes up the question list. – Antonio Vargas Aug 29 '13 at 19:09
Since no one seemed to mention it directly: $\sin x = 2$ is not "undefined". It is an equation whose solution set is empty. Ie, there is no real number $x$ so that $\sin x = 2$. Or: $$\{x \in \mathbb{R} \mid \sin x = 2\} = \varnothing$$ – alecb Aug 30 '13 at 4:34
But there are complex numbers with $\sin{x} = 2$. – Dan Aug 30 '13 at 5:54
Taking the arctangent of both sides for this problem (which has simple, exact solutions) is a bad idea because you will miss some solutions. This usually happens with inverse trig functions. If you have a good textbook it may explain why. It is similar to solving $x^2 = 4$ by taking square roots and concluding that the only solution is $x=2$. – Stefan Smith Aug 31 '13 at 13:31
@StefanSmith how else can you solve the equation? – salman Aug 31 '13 at 13:51
up vote 7 down vote accepted

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 $\sin x=1$.

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@brian-m-scott this is exactly wanted to know. With respect to notation, if I ever run into an equation such as $\sin x = 2$ or $e^x = 0$ how would you mathematically show that the equation has no solution? And also why must you we use the absolute sine? – salman Aug 29 '13 at 19:23
@user90771 Those examples depend greatly on your definitions of $\sin$ and $e^x$. – Potato Aug 29 '13 at 19:31
@user90771: You’re welcome. Potato is right about your question: it really will depend on what information you already have available about the sine and exponential functions. If you’ve defined $\sin\theta$ as the $x$-coordinate of the point on the unit circle whose polar coordinates are $\langle 1,\theta\rangle$, for instance, then the fact that the unit circle is described by $x^2+y^2=1$ lets you infer easily that $|\sin x|\le 1$ for all real $x$. – Brian M. Scott Aug 29 '13 at 19:39

$\tan{x}$ 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?

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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.

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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 $\{x|x=\frac{\pi}{4}+k\pi ,\, k\in \mathbb{Z}\}$.

enter image description here

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