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 or arctan(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.

Arctan (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.

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 and the base, that is, "Perpendicular / Base". In contrast, the arctan of the ratio "Perpendicular / Base" gives us the value of the corresponding angle between the base and the hypotenuse. Thus, arctan is the inverse of the tan function.

If the tangent of angle θ is equal to x, that is, x = tan θ, then we have θ = arctan(x). Given below are some examples that can help us understand how the arctan function works:

• tan(π / 2) = ∞ ⇒ arctan(∞) = π/2
• tan (π / 3) = √3 ⇒ arctan(√3) = π/3
• tan (0) = 0 ⇒ arctan(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].

As discussed above, the basic formula for the arctan is given by, arctan (Perpendicular/Base) = θ, where θ is the angle between the hypotenuse and the base of a right-angled triangle. We use this formula for arctan to find the value of angle θ in terms of degrees or radians. We can also write this formula as θ = tan^-1[Perpendicular / Base].

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

• arctan(-x) = -arctan(x), for all x ∈ R
• tan (arctan x) = x, for all real numbers x
• arctan (tan x) = x, for x ∈ (-π/2, π/2)
• 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)
• arctan(x) = 2arctan\(\left ( \frac{x}{1 + \sqrt{1 + x^{^{2}}}} \right )\).
• arctan(x) = \(\int_{0}^{x}\frac{1}{z^{2} + 1}dz\)

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 x 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 = 56.3^°

Answer: The angle is 56.3^°.

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, respectively. 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 arctan graph.

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.

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.

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,

sec²y dy/dx = 1

⇒ dy/dx = 1 / sec²y

According to the trigonometric identity we have sec²y = 1 + tan²y

dy/dx = 1 / (1 + tan²y)

On substitution,

Thus, d(arctan x) / dx = 1 / (1 + x²)

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.

Related Articles:

• Arctan Calculator
• Trigonometric Functions
• Trigonometric Chart
• Sin cos tan

Important Notes on Arctan

• Arctan can also be written as arctan x or 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

What is the Arctan Function in Trigonometry?

Arctan function is the inverse of the tangent function. It is usually denoted as arctan x or tan^-1x. The basic formula to determine the value of arctan is θ = tan^-1(Perpendicular / Base).

Is Arctan the Inverse of Tan?

Yes, arctan is the inverse of tan. It can determine the value of an angle in a right triangle using the tangent function. Tan^-1x will only exist if we restrict the domain of the tangent function.

Are Arctan and Cot the Same?

Arctan and cot are not the same. The inverse of the tangent function is arctan given by tan^-1x. However, cotangent is the reciprocal of the tangent function. That is (tan x)^-1 = 1 / cot x

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 substitution and chain rule concepts. Thus, d(arctan x) / dx = 1 / (1 + x²), x ≠ i, -i.

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.

What is the Limit of Arctan x as x Approaches Infinity?

The value of arctan approaches π/2 as x approaches infinity. Also, we know that tan (π/2) = ∞. So, the limit of arctan is equal to π/2 as x tends to infinity.