Arctan Formula

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Arctan

In trigonometry, arctan refers to the inverse tangent function. Inverse trigonometric functions are usually accompanied by the prefix – arc. Mathematically, we represent arctan or the inverse tangent function as tan-1 x. As there are a total of six trigonometric functions, similarly, there are 6 inverse trigonometric functions, namely, sin-1x, cos-1x, tan-1x, cosec-1x, sec-1x, and cot-1x.

Tan-1x is not the same as 1 / tan x. That means an inverse trigonometric function is not the reciprocal of the respective trigonometric function. The purpose of arctan is to find the value of an unknown angle by using the value of the tangent trigonometric ratio. Navigation, physics, and engineering make widespread use of the arctan function. In this article, we will learn about several aspects of tan-1x including its domain, range, graph, and the integral as well as derivative value.

What is Arctan?

Arctan is one of the important inverse trigonometry functions. In a right-angled triangle, the tan of an angle determines the ratio of the “Perpendicular / Base”. In contrast, the arctan of the ratio of the “Perpendicular / Base” gives us the value of an angle. Thus, arctan is the inverse of the tan function.

If the tangent of θ is equal to x:

⇒ x = tan θ
⇒ θ = tan-1x

Given below are some examples that can help us understand how the arctan function works:

  • tan(π / 2) = ∞ ⇒ tan-1(∞) = π/2
  • tan (π / 3) = √3 ⇒ tan-1(√3) = π/3
  • tan (0) = 0 ⇒ tan-1(0) = 0

Suppose we have a right-angled triangle. Let θ be the angle whose value needs to be determined. We know that tan θ will be equal to the ratio of the perpendicular and the base. Hence, tan θ = Perpendicular / Base. To find θ we will use the arctan function as, θ = tan-1[Perpendicular / Base].

Arctan Formula

There are several arctan formulas, identities and properties that are helpful in solving simple as well as complicated sums on inverse trigonometry. A few of them are given below:

  • θ = arctan(Perpendicular / Base)
  • arctan(-x) = -arctan(x), for all x ∈ R
  • tan (arctan x) = x, for all real numbers x
  • arctan(1/x) = π/2 – arctan(x) = arccot(x), if x > 0 or,
    arctan(1/x) = – π/2 – arctan(x) = arccot(x) – π, if x < 0
  • sin(arctan x) = x / √(1 + x2)
  • cos(arctan x) = 1 / √(1 + x2)

We also have certain arctan formulas for π. These are given below.

  • π/4 = 4 arctan(1/5) – arctan(1/239)
  • π/4 = arctan(1/2) + arctan(1/3)
  • π/4 = 2 arctan(1/2) – arctan(1/7)
  • π/4 = 2 arctan(1/3) + arctan(1/7)
  • π/4 = 8 arctan(1/10) – 4 arctan(1/515) – arctan(1/239)
  • π/4 = 3 arctan(1/4) + arctan(1/20) + arctan(1/1985)

How To Apply Arctan Formula?

We can get an in-depth understanding of the application of the arctan formula with the help of the following examples:

Example: In the right-angled triangle ABC, if the base of the triangle is 2 units and the height of the triangle is 3 units. Find the base angle.

Solution:

To find: base angle

Using arctan formula, we know,
⇒ θ = arctan(3 ÷ 2) = arctan(1.5)
⇒ θ = 56o

Answer: The angle is 56o.

Arctan Domain and Range

All trigonometric functions including tan (x) have a many-to-one relation. However, the inverse of a function can only exist if it has a one-to-one and onto relation. For this reason, the domain of tan x must be restricted otherwise the inverse cannot exist. In other words, the trigonometric function must be restricted to its principal branch as we desire only one value.

The domain of tan x is restricted to (-π/2, π/2). The values where cos(x) = 0 have been excluded. The range of tan (x) is all real numbers. We know that the domain and range of a trigonometric function get converted to the range and domain of the inverse trigonometric function. Thus, we can say that the domain of tan-1x is all real numbers and the range is (-π/2, π/2). An interesting fact to note is that we can extend the arctan function to complex numbers. In such a case, the domain of arctan will be all complex numbers.

Arctan Table

Any angle that is expressed in degrees can also be converted into radians. To do so we multiply the degree value by a factor of π/180. Furthermore, the arctan function takes a real number as an input and outputs the corresponding unique angle value. The table given below details the arctan angle values for some real numbers. These can also be used while plotting the tan-1x graph.

xarctan(x)(°)arctan(x)(rad)
-∞-90°-π/2
-3-71.565°-1.2490
-2-63.435°-1.1071
-√3-60°-π/3
-1-45°-π/4
-1/√3-30°-π/6
-1/2-26.565°-0.4636
00
1/226.565°0.4636
1/√330°π/6
145°π/4
√360°π/3
263.435°1.1071
371.565°1.2490
90°π/2

Arctan Properties

Given below are some useful arctan identities based on the properties of the arctan function. These formulas can be used to simplify complex trigonometric expressions thus, increasing the ease of attempting problems.

  • tan (tan-1x) = x, for all real numbers x
  • tan-1x + tan-1y = tan-1[(x + y)/(1 – xy)], when xy < 1
    tan-1x – tan-1y = tan-1[(x – y)/(1 + xy)], when xy > -1
  • We have 3 formulas for 2tan-1x
    2tan-1x = sin-1(2x / (1+x2)), when |x| ≤ 1
    2tan-1x = cos-1((1-x2) / (1+x2)), when x ≥ 0
    2tan-1x = tan-1(2x / (1-x2)), when -1 < x < 1
  • tan-1(-x) = -tan-1x, for all x ∈ R
  • tan-1(1/x) = cot-1x, when x > 0
  • tan-1x + cot-1x = π/2, when x ∈ R
  • tan-1(tan x) = x, only when x ∈ R – {x : x = (2n + 1) (π/2), where n ∈ Z}
    i.e., tan-1(tan x) = x only when x is NOT an odd multiple of π/2. Otherwise, tan-1(tan x) is undefined.

Arctan Graph

We know that the domain of arctan is R (all real numbers) and the range is (-π/2, π/2). To plot the arctan graph we will first determine a few values of y = arctan(x). Using the values of the special angles that are already known we get the following points on the graph:

  • When x = ∞, y = π/2
  • When x = √3, y = π/3
  • When x = 0, y = 0
  • When x = -√3, y = -π/3
  • When x = -∞, y = -π/2

Using these we can plot the graph of arctan.


Derivative of Arctan

To find the derivative of arctan we can use the following algorithm.

Let y = arctan x

Taking tan on both the sides we get,

tan y = tan(arctan x)

From the formula, we already know that tan (arctan x) = x

tan y = x

Now on differentiating both sides and using the chain rule we get,

Integral of Arctan

The integral of arctan is the antiderivative of the inverse tangent function. Integration by parts is used to evaluate the integral of arctan.

Here, f(x) = tan-1x, g(x) = 1

The formula is given as ∫f(x)g(x)dx = f(x) ∫g(x)dx – ∫[d(f(x))/dx × ∫g(x) dx] dx

On substituting the values and solving the expression we get the integral of arctan as,

∫tan-1x dx = x tan-1x – ½ ln |1+x2| + C

where, C is the constant of integration

Important Notes on Arctan

  • Arctan can also be written as tan-1x. However, tan-1x is not equal to (tan x)-1 = 1 / tan x = cot x.
  • The basic formula for arctan is given as θ = arctan(Perpendicular / Base).
  • The derivative of arctan is d/dx(tan-1x) = 1/(1+x2).
  • The integral of arctan is ∫tan-1x dx = x tan-1x – ½ ln |1+x2| + C

Arctan definition

The arctangent of x is defined as the inverse tangent function of x when x is real (x∈ℝ).

When the tangent of y is equal to x:

tan y = x

Then the arctangent of x is equal to the inverse tangent function of x, which is equal to y:

arctan x= tan-1 x = y

Example

arctan 1 = tan-1 1 = π/4 rad = 45°

Graph of arctan

Arctan rules

Rule nameRule
Tangent of arctangenttan( arctan x ) = x
Arctan of negative argumentarctan(-x) = – arctan x
Arctan sumarctan α + arctan β = arctan [(α+β) / (1-αβ)]
Arctan differencearctan α – arctan β = arctan [(αβ) / (1+αβ)]

Arctan table

xarctan(x)(rad)arctan(x)(°)
-∞-π/2-90°
-3-1.2490-71.565°
-2-1.1071-63.435°
-√3-π/3-60°
-1-π/4-45°
-1/√3-π/6-30°
-0.5-0.4636-26.565°
00
0.50.463626.565°
1/√3π/630°
1π/445°
√3π/360°
21.107163.435°
31.249071.565°
π/290°

Large and negative angles

Recall that we can apply trig functions to any angle, including large and negative angles. But when we consider the inverse function we run into a problem, because there are an infinite number of angles that have the same tangent. For example 45° and 360+45° would have the same tangent. For more on this see Inverse trigonometric functions.

To solve this problem, the range of inverse trig functions are limited in such a way that the inverse functions are one-to-one, that is, there is only one result for each input value.

Range and domain of arctan

Recall that the domain of a function is the set of allowable inputs to it. The range is the set of possible outputs.

For y = arctan x :

By convention, the range of arctan is limited to -90° to +90° *.

So if you use a calculator to solve say arctan 0.55, out of the infinite number of possibilities it would return 28.81°, the one in the range of the function.

* Actually, -90° and +90° are themselves not in the range. This is because the tan function has a value of infinity at those values. But the values just below them are in the range, for example +89.9999999. But for simplicity of explanation, we say the range is ±90° .

Use of Arctan

The arctan is trigonometric functions and these functions are used to define values related to right triangles. In practical we use these functions to resolve the distance or height of the object which is difficult to measure.

Also, this is resolute using the measure of one angle (not right angle) and the ratio of two sides of the triangle. Besides, we determine trigonometric functions by the sides of the triangle that we use in the ratio of these formulas:

Sine = opposite ÷ hypotenuse

Cosine = adjacent ÷ hypotenuse

Tangent = opposite ÷ adjacent

Most noteworthy, we can use the inverse of these functions to resolute the angle measures when we knew the sides of the triangle. Also, you can use it to decide the measure of an angle when we knew the side opposite and the side adjacent to the angle.

Besides, it has practical application in many fields like landscaping, building, architecture, physics, engineering, and amidst other mathematical and scientific areas.

Most noteworthy, the best method to use the arctan is a scientific calculator. Furthermore, we can locate the arctan button just above the tangent on the calculator. Also, we can use a data table to resolve arctan; but this can be a tiresome and bulky method, however, it is effective if you don’t have a scientific calculator.

Now let’s look at some of the examples of it to find the angle measure.

Examples on Arctan

What is the Formula for Arctan?

The basic arctan formula can be given by θ = tan-1(Perpendicular / Base). Here, θ is the angle between the hypotenuse and the base of a right-angled triangle.

What is the Derivative of Arctan?

The derivative of arctan can be calculated by applying the

How to Calculate the Integral of Arctan?

We will have to use integration by parts to find the value of the integral of arctan. This value is given as ∫tan-1x dx = x tan-1x – ½ ln |1+x2| + C.

What is the Arctan of Infinity?

We know that the value of tan (π/2) = sin(π/2) / cos (π/2) = 1 / 0 = ∞. Thus, we can say that arctan(∞) = π/2.

Inverse Tangent

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