Arctan Formula

Trigonometry is the science of evaluating and demonstrating the sides and angles of a right-angled triangle. Trigonometric operations are performed using sides, angles, and trigonometric ratios. These trigonometric ratios are the values of trigonometric functions calculated from the ratios of the triangle’s sides and angles.

Trigonometry has fundamental trigonometric functions, each with its standard trigonometric ratio value under different angles. The fundamental functions are sine, cosine, tangent, cotangent, cosecant, and secant. The inverse of several trigonometric functions, such as arcsin, arccos, arctan, arccot, arcsec, and arccosec, is denoted by the prefix ‘arc-‘.

Arctan is not equivalent to 1/tan x. tan-1 x is the inverse of tan x, while 1/tan x is the reciprocal of tan x. tan-1 x is used to solve numerous types of trigonometric equations. This post will look at the arctan function formula,  its properties, and more in-depth.

What is Arctan?

Tangent is a trigonometric function that equals the perpendicular to base ratio in a right-angled triangle. Arctan refers to the inverse function of the tangent. Arctan is symbolically represented as tan-1x in trigonometric formulae.

The formula for arctan is arctan (Perpendicular/Base) = θ, where θ represents the angle between a right-angled triangle’s hypotenuse and base. Use the arctan formula to calculate angle θ in degrees or radians.

Assume the tangent of the angle θ equals x.

x = tan θ ⇒ θ = tan-1x

Arctan Formula

Arctan is defined as the inverse of the tangent function. Arctan(x) is written as tan-1 x. There are six trigonometric functions, and their inverses are expressed as:

sin-1x, cos-1x, tan-1x, cosec-1x, sec-1x, and cot-1x.

Every function in mathematics has an inverse. Similarly, the trigonometric function includes an inverse. In trigonometry, the Arctan Formula is the inverse of the tangent function and is used to determine the angle measure from a right triangle’s tangent ratio (tan = opposite/adjacent). The Arctan Formula can be expressed in terms of degrees and radians.

The inverse of the trigonometric function tanx is Arcatan. The trigonometric function is the ratio of a right angle triangle’s perpendicular and base, and its inverse is the arctan function. This is explained as follows:

tan (π/4) = 1

⇒ π/4 = tan-1(1)…(this is Arctan Function)

In a right triangle with angle θ, tan θ is perpendicular/base. The arctan function is:

θ = tan-1(perpendicular/base)

Arctan Identities

There are several Arctan identities for solving trigonometric equations. Some of the key arctan identities are listed here.

  • 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
  • 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 {x/(1 + √(1+x2))}
  • arctan(x) = ∫ox 1/√(1+z2)dz

How To Apply Arctan x Formula?

The Arctan Formula is used to solve numerous trigonometric problems which are illustrated in the example below.

Example: In the right-angled triangle ABC, if the perpendicular of the triangle is 1/√3 units and the base of the triangle is 1 unit. Calculate the angle.

To find the angle (θ)

θ = arctan (perpendicular/base)

θ = arctan (1/√3/1)

θ = arctan (1/√3)

θ = 30°

Arctan Properties

Arctan x properties are used to solve trigonometric equations. Several trigonometric properties mustic properties that must be mastered when learning trigonometry. The main properties of the arctan function are listed below in this article:

  • tan (tan-1x) = x
  • tan-1(-x) = -tan-1x
  • tan-1(1/x) = cot-1x, when x > 0
  • 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
  • tan-1x + cot-1x = π/2
  • tan-1(tan x) = x [when x ∈ R – {x : x = (2n + 1) (π/2), where n ∈ Z}]
  • tan-1(tan x) = x [when x is NOT an odd multiple of π/2. else, tan-1(tan x) is undefined.]
  • 2 tan-1x = sin-1(2x / (1+x2)), when |x| ≤ 1
  • 2 tan-1x = cos-1((1-x2) / (1+x2)), when x ≥ 0
  • 2 tan-1x = tan-1(2x / (1-x2)), when -1 < x < 1

As previously stated, the fundamental of the Arctan Formula is arctan (Perpendicular/Base) =θ, where θ is the angle between the hypotenuse and the base of a right-angled triangle. Researchers use the  Arctan Formula to get the value of an angle θ in degrees or radians. One can also write the Arctan Formula as θ = tan-1[Perpendicular / Base].

Arctan Table

Any angle represented in degrees may be translated to radians. To do so, multiply the degree value by π/180°. Furthermore, the arctan function accepts a real integer as input and returns the associated unique angle value. The table below shows the arctan angle values for several real integers. These may also be used to plot an arctan graph.

As we previously discussed, the value of arctan may be calculated using degrees or radians. So, the table below shows the estimated values of arctan.

x arctan(x) (in degree) Arctan(x) (in radian)
-∞ -90° -π/2
-√3 -60° -π/3
-1 -45° -π/4
-1/√3 -30° -π/6
0 0
1/√3 30° π/6
1 45° π/4
√3 60° π/3
90° π/2

Arctan Graph

The Arctan function graph is an infinite graph. The Arctan function has a domain of R (real numbers) and a range of -π/2 to π/2. The graph of the Arctan function is shown in the figure below:

The graph is generated using the values of the known points for the function y = tan-1(x).

  • x = ∞ ⇒ y = π/2
  • x = √3 ⇒ y = π/3
  • x = 1/√3 ⇒ y = π/6
  • x = 0 ⇒ y = 0
  • x = -1/√3 ⇒ y = -π/6
  • x = -√3 ⇒ y = -π/3
  • x = -∞ ⇒ y = -π/2

Also Read: Graphs Of Trigonometric Functions Formula

Arctan Derivative

The derivative of arctan is extremely essential in mathematics. The derivative of the arctan function is computed using the following concept:

y = arctan x (let)…(1)

Taking tan both sides

tan y = tan (arctan x)    [we know that tan (arctan x) = x]

tan y = x

Differentiating both sides (using chain rule)

sec2y × dy/dx = 1

dy/dx = 1 / sec2y

dy/dx = 1 / (1 + tan2y)  {using, sec2y = 1 + tan2y}

d / dx (arctan x) = 1 / (1 + x2)

Arctan Integral

The integral of arctan is defined as the antiderivative of the inverse tangent function. The integration of Arctan x is derived using the notion described below.

Lets take f(x) = tan-1x, and g(x) = 1

We know that, ∫f(x)g(x)dx = f(x) ∫g(x)dx – ∫[d(f(x))/dx × ∫g(x) dx] dx

putting the value of f(x) and g(x) in above equation we get,

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

where C is the constant of integration

Related Read:

Solved Examples on Arctan Formula

Example 1 : Evaluate tan-1(1.732)

Solution : The given value is, tan-1(1.732)

From this given quantity, 1.732 can be written as a function of tan.

So, 1.732 = tan 60°

Therefore, tan-1(1.732) = tan-1 (tan 60°) = 60°

Example 2 : Evaluate tan-1(0.577)

Solution : The given value is, tan-1(0.577)

From this given quantity, 0.577 can be written as a function of tan.

So, 0.577 = tan 30°

Therefore, tan-1(0.577) = tan-1 (tan 30°) = 30°

FAQs (Frequently Asked Questions)

1. Evaluate the Derivative of Arctan?

The derivative of arctan is, d/dx (arctan x) = 1 / (1 + x2)

2. Is Arctan function the Inverse of the Tan function?

Yes, the inverse of the tan function is the arctan function. If, tan x = y than x = tan-1y

3. Is Arctan same to Cot?

No, arctan is not same to the cot. The reciprocal of the tan function is cot.  i.e. tan x = 1/cot x, whereas Arctan is inverse of tan function arctan x = tan-1 x

4. Is Arctan and tan-1 the identical?

Yes, Arctan is identical to the tan-1 as, Arctan is another name of tan-1(x)

5. 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.

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

Arctan’s value approaches π/2 as x approaches infinity. We also know that tan(π/2) equals ∞. As x approaches infinity, the limit of arctan becomes π/2.