# Exploring the Equivalence of r=arctan(tan(x)) and r=x

• sara_87
In summary, the conversation discusses whether a polar equation r=arctan(tan(x)) is the same as r=x and the use of graphing software to plot the graph of arctan(tan(x)). It is concluded that the two equations are not the same and the graph of arctan(tan(x)) should be plotted using a specific software due to the function's periodicity. It is also noted that the function has a period of pi and is not defined at certain points.
sara_87

## Homework Statement

if i have a polar equation:
r=arctan(tan(x))
is that the same as:
r=x ??

## The Attempt at a Solution

no!
-graph the function
-and try plugging in different numbers ... what if x = pi/2 ..
-find domain of x
-look at tan(x) function ... see the periodicity.

i am trying to draw the graphs but can't becuase some values for x give a math error.
so how would i draw the graph for r=arctan(tan(x))?

sara_87 said:
i am trying to draw the graphs but can't becuase some values for x give a math error.
so how would i draw the graph for r=arctan(tan(x))?

draw tan x and atan x separately, and you would find why it's giving you math error ... it should not give error but I am not sure what you used.
hint: when x = pi/2 tan(x) is inf when approached from left ...
so what's atan (x) when x is inf or when x is -inf?

you can draw atan(tan x) graph using this software:
nice, simple, and fast.
this helps me a lot

yeah but i have to lot the points on a table and i don't know what to put for r when x=pi/2 and 3pi/2.

http://img405.imageshack.us/img405/764/atantanxeo0.jpg

were you talking about the graph software?

No, you don't need to input numbers...see the image.
when x = pi/2-small number, tan(x) is inf, and atan(x) is pi/2 ..

so its pi/2

Last edited by a moderator:
The Tangent function fails the horizontal line test, ie There are many values for the independent variable that give the same Dependant variable. This means that The Inverse of the Tangent function is a relation but not a function. We can not find an inverse function for the usual tangent function, but we can find an inverse relation when we only take a little section of the function that does pass the horizontal line test. You will see that the function
$$f(x) = \tan x, |x|> \frac{\pi}{2}$$ does pass the test. It is a one-to one, monotonically increasing function. Let us also note that the range of this function is the Reals. Now let us define the inverse of this, and observe that it will also be a one-to-one monotonically increasing function, its domain will be the Reals and its range will be all values of y such that $|y| < \frac{\pi}{2}$. We commonly call this function the arctangent.

Now, by the definition of what it is to be an inverse function, $\arctan ( \tan x) = x$, but only for $|x| < \frac{\pi}{2}$. For values $$\frac{\pi}{2} \pm k\pi, k=1,2,3...$$, tan x is not defined, or, they are not in the domain of the tangent function, so arctan ( tan x) will not be defined either.

For all other real values, we use the fact that the tangent function has a period of pi, ie $$f(x) = f(x\pm k\pi), k=1,2,3...$$, for all values of x in its domain. Using these facts, we can conclude that $$\arctan (\tan x) = x, |x| < \frac{\pi}{2}$$ and that the function also has a period of pi, since the argument of arctan has a period of pi.

This function is somewhat inaccruately plotted on the Cartestian plain in that image rootX posted up (no offence intended). When drawing the graph, one should have drawn hollow white circles at the points $( \frac{\pi}{2}, \frac{\pi}{2}), (\frac{\pi}{2}, \frac{-\pi}{2})$ etc etc, to indicate that the graph does not include that point. Remember the function is not defined at those values.

In GibZ's post, the word "inaccurately" is spelled somewhat inaccruately.

Ahh I never pick up those spelling errors :( Ooh the Irony, it burns!

## 1. What is the mathematical concept behind the equation r=arctan(tan(x)) and r=x?

The equation r=arctan(tan(x)) and r=x explores the concept of equivalence in mathematics. It states that the polar coordinate r, which represents the distance from the origin, is equal to the tangent of the angle x when arctangent is applied, and also equal to the angle x itself.

## 2. How does the equation r=arctan(tan(x)) and r=x relate to trigonometric functions?

This equation relates to the trigonometric functions of tangent and arctangent, which are inverse functions of each other. The arctangent function "undoes" the tangent function, resulting in the original angle x. Therefore, when r=arctan(tan(x)), the r value is equal to the angle x in radians.

## 3. Can you provide an example of how r=arctan(tan(x)) and r=x can be used in real-world applications?

One example is in navigation, where polar coordinates are used to determine the position of an object or person. The equation r=arctan(tan(x)) and r=x can be used to convert between Cartesian coordinates (x,y) and polar coordinates (r,θ). This can be useful in situations such as locating a missing person in a search and rescue mission.

## 4. Is the equation r=arctan(tan(x)) and r=x always true?

The equation is true when x is within the range of values for which arctangent is defined, which is -π/2 to π/2 in radians. However, if x is outside of this range, the equation may not hold. For example, when x=π/2, r=π/2, but when x=3π/2, r=-π/2, which is not equal to x. Therefore, it is important to consider the range of x values when using this equation.

## 5. How can the equivalence of r=arctan(tan(x)) and r=x be proven?

The equivalence can be proven using the properties of inverse trigonometric functions and the definition of polar coordinates. By applying the inverse tangent function (arctan) to both sides of the equation, we can show that arctan(tan(x))=x. Then, by using the definition of tangent (opposite/adjacent), we can see that r=arctan(tan(x)) is equal to the angle x in radians. Therefore, r=x, proving the equivalence.

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