What is arctan

What is an inverse function

So that you can understand what the arctan is, you should familiarize yourself with the inverse functions in general.

• A function is a relationship between a dependent and an independent variable. The function equation is usually represented as f (x) = term, whereby the dependent variable y can also be written instead of f (x); y = term.
• Uniqueness is important for a function. For each variable x, the term always results in exactly one variable y. Example f (x) = y = 2x + 3 or f (x) = y = 2 x² or f (x) = y = tan x.
• If you substitute any number for x, you will get exactly one result for y. However, it is entirely possible that you get the same function value y for two different x values. Example: For the function f (x) = 2 x
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  ² we have f (1) = 2 1² = 2 and f (-1) = 2 (-1)² = 2.
• Now it is conceivable that you have a value for the dependent variable y and want to know what value the independent variable x must have for y to have this value. If you set up a function equation that tells you which x-values ​​led to which y-values, then you need the inverse function. In principle you swap x and y and solve for y. For the function f (x) = 2x + 3 this means: x = 2 y + 3 => x - 3 = 2 y => y = 1/2 x - 3/2. f^-1(x) = 1/2 x -3.


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• For the function f (x) = 2 x² you run into two problems. There are the same y-values ​​for different x-values. To create an inverse function, you must divide the function into intervals in which there are no duplicate y-values. In the interval] -infinity, 0 [and in the interval [0, + infinite [there are for f (x) = 2 x² no double function values. So you can reverse the function in each of the two intervals, but not in the whole. The other problem is that you need a new arithmetic instruction if you want to reverse the function. For example, take the interval [0, + infinity [and the inverse x = 2 y², By dividing by 2, you get 1/2 x = y². Now you need a new arithmetic instruction, the root sign. The root indicates which number multiplied by itself results in the argument under the root. Example: Root 4 = 2 or Root 4 = -2. In that case you come to f^-1(x) = + root (1/2 x).

Arctane as the inverse of the tangent function

• The function f (x) = tan x repeats itself periodically. In the interval] - pi / 2, pi / 2 [there are no repetitions of the function value. Likewise in the interval] pi / 2,3 / 2 pi [etc., if you calculate in radians as usual. If you are calculating in degrees, the interval would be] -90 °, 90 ° [.
• Within the interval] - pi / 2, pi / 2 [you can swap the variables and resolve them again for y. You get x = tan y. Now you have a problem similar to that with the quadratic function equation. You need a new calculation instruction. This is called arctan; arctan indicates to which angle heard a specific numerical value. Example: tan x = 5 => arctan 5 = 0.43 pi. So if the angle is 0.43 pi then the tan of that is 5.

Clarification via the unit circle

• Imagine the angle alpha in such a way that it is the angle that the pointer z covers in a counterclockwise direction. The tan alpha is opposite side through adjacent side. The adjacent is - as you can see - 1. So the tan alpha corresponds to the length of the opposite side. As soon as the pointer turns past pi / 2, the opposite cathetus becomes shorter again and consequently takes on values ​​again that it had already assumed in the range between 0 and pi / 2. For this reason, you must no longer use the range after pi / 2 for the formation of the inverse function. If the pointer rotates clockwise, you will come to the angle -pi / 2 as the limit.
• The arctan means that you know the length of the opposite cathetus (blue sketch) and you have to find the corresponding angle. Connect the end of the opposite cathetus to the center point of the circle. Now you can see which angle alpha belongs to the given opposite side.