# What are “tan” and “atan”?

As the title says, I'm confused on what tan and atan are. I'm writing a program in Java and I came across these two mathematical functions. I know tan stands for tangent but if possible could someone please explain this to me. I have not taken triginomotry yet (I've taken up to Algebra 1) so I don't really need a very in depth explanation since i wouldnt understand but just a simple one so i could move on with my program would be great! Thanks in advanced. Also if possible could someone possibly give me a link to an image/example of a tangent and atan.

• Why not simply youtube "tan"? – goblin May 4 '15 at 16:17
• did that, went to the website khanaccademy.com which usually has some very helpful videos but I needed further information still so I asked here. – Bob Mc Muffins. May 4 '15 at 16:50
• I will preserve this piece of information from my deleted answer; when programming you should be careful to check the precise conventions. Especially as regards the range of atan and the usage of radians vs. degrees. – quid May 4 '15 at 16:51
• see also atan2 w3schools.com/jsref/jsref_atan2.asp – John Joy May 5 '15 at 0:22
• @quid, I don't get why your answer was downvoted. – goblin May 5 '15 at 6:39

A quick google search of "java atan" would tell you that it stands for "arctangent", which is the inverse of tangent. Tangent is first understood as a ratio of non-hypotenuse sides of a right triangle. Given a non-right angle $x$ of a right triangle, $\tan(x)$ is the ratio $\frac{o}{a}$ where $o$ is the length of the leg of the triangle opposite $x$ and $a$ is the length of the leg adjacent to $x$. Arctangent takes the ratio as an input and returns the angle.

I don't think any single answer to your post will give you a complete understanding of tangent and arctangent. To get that, I recommend spending some time with sine, cosine, tangent and the unit circle. Here is a link to an image http://www.emanueleferonato.com/images/trigo.png that uses an angle $A$ and triangle sides $a,o,h$. In the context of that photo, $\arctan\left(\frac{\text{opposite}}{\text{adjacent}}\right) = A$

• Okay thank you, and thanks to everyone else who answered,this is basically what i needed to see, I will do further research at this point! – Bob Mc Muffins. May 4 '15 at 16:42
• @AshwinGupta good luck to you! I am happy to answer any other questions should you have any. A solid understanding of trigonometry will take you far in mathematics. – graydad May 4 '15 at 16:44

I think the following picture is useful [from Wikipedia]:

The tangent of a number $0<\theta<\pi/2$ is constructed by

• taking a circle of radius 1, centered at the origin
• drawing the tangent at the point on positive $x$-axis
• drawing the line (ray) at an angle of $\theta$ from the origin
• constructing the intersection of these two lines
• measuring the distance between the intersection and the positive $x$-axis.

The final distance is the tangent of $\theta$, denoted $\tan\theta$.

As others have mentioned, the arctangent of a number $0<d<\infty$ is the number $\theta$ for which $\tan\theta=d$. In other words, it is the angle for which, after performing the described procedure above, you get a distance of $d$. It is denoted in many ways, including atan $\theta$, arctan $\theta$, and $\tan^{-1}\theta$.

Note that I've only defined $\tan\theta$ for a limited collection of $\theta$: to get other values, you can use the identities $\tan(-\theta)=-\tan(\theta)$ and $\tan(\pi+\theta)=\tan\theta$. So for instance, if you knew that $\tan(\pi/4)=1$, then you would also know that

$$\tan\left(-\frac{7\pi}{4}\right) =-\tan\left(\frac{7\pi}{4}\right) =-\tan\left(\frac{3\pi}{4}\right) =-\tan\left(-\frac{\pi}{4}\right) =\tan\left(\frac{\pi}{4}\right) =1$$

From these facts you can also derive the rest of the arctangent values: it turns out that atan $-d$ = $-$atan $d$.

(This is not entirely true: the arctangent is technically a multivalued function, so you have to make some arbitrary choice of how to extend it to negative numbers. But the choice above is the most common one.)

• thanks for such a through answer! Im going to do some research on sin, cosine, ect like the answer above told me to than come back to this one. – Bob Mc Muffins. May 4 '15 at 16:46
• Also nice picture this helped – Bob Mc Muffins. May 6 '15 at 13:50

atan stands for "arc tangent." It is the inverse function of tangent - this means it undoes the tangent function. So atan(tan(30)) = 30.

• But the OP doesn't know what tangent is. – Eff May 4 '15 at 16:05
• @Eff yes thanks. Although, with the knowledge from the other answers, this does have some use. (It should just be a comment though..) – Bob Mc Muffins. Nov 23 '15 at 22:50