ArcTan Formula: Derivation, Domain, Range & Properties

Arctan is considered as the inverse tangent function in trigonometry. Inverse trigonometric functions usually occur by the prefix-arc. In mathematics, we represent arctan or the inverse tangent function as tan^-1x. There are a total of 6 trigonometric functions, so there will also be six inverse trigonometric functions such as sin^-1x, cos^-1x, tan^-1x, cosec^-1x, sec^-1x, cot^-1x. Tan^-1x is not the same as 1/tanx. This means that an inverse trigonometric function will not be reciprocal of the respective trigonometric function. The purpose of arctan is to find the value of an unknown angle by using the value of tangent trigonometric ratio.

Key Takeaways: Trigonometry, Ratio, Angles, Function, Identities, Inverse functions, Trigonometric functions, 

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Arctan Formulas

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In a right-angle triangle, the “tan” of an angle describes the ratio of the “perpendicular/base”. The arctan of the ratio of “perpendicular/base” gives the value of an angle. Therefore, arctan is the inverse of “tan” function. If the tangent of θ is equal to x:

x = tan θ

θ = tan^-1x

Here are some arctan formulas, identities that are used to solve simple as well as complicated sums on inverse trigonometry. 

• θ = arctan(perpendicular/base)
• arctan(-x) = -arctan(x), for all x ∈ R
• tan (arctan x) = x, for all real numbers x
• arctan(x) = π/2 - arctan(x) = arccot(x), if x>0 or,
• arctan(x) = -π/2 - arctan(x) = arccot(x)-π, if x<0
• sin (arctan x) = x/√(1+x)²
• cos (arctan x) = 1/√(1+x)²
• arctan(x) = 2arctan(x/1+√1+x²)
• arctan(x) = ∫^x₀ 1/z²+1 dz

We do have some formulas of arctan for π. Here are given below:

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

Arctan Formula Graph

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Derivation of Arctan Formula

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The derivation of arctan formula is as follows:

Let y = arctan x

Taking tan on both sides, 

tany = tan (arctan x)

We already know that, tan (arctan x) = x,

Tan y = x

Now we will differentiate both sides we get, 

dy/dx = 1/sec²y

According to the trigonometric identity sec²y = 1 + tan²y

dy/dx = 1/ (1 + tan²y)

On substitution, we get

d(arctan(x))/dx = 1/(1+x²)

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Arctan Domain and Range

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All of the trigonometric functions have a many-to-one relation. Hence, the inverse of function can only exist if it will be having a one-to-one and onto relation. In simple words, we can say that the trigonometric function must be restricted to its principal branch as we need only one value.

Domain and Range of Arctan

The domain of tanx is restricted to (-π/2, π/2). The range of tanx is always real numbers. The domain and range of trigonometric function converted into range and domain of inverse trigonometric function. Therefore, we can say that the domain of tan^-1x are all real numbers and range is (-π/2, π/2). 

Also Read: Difference between Variance and Standard Deviation

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Integral of Arctan

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

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

The formula will be 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 expression we will get the integral of arctan as, 

∫tan^-1x dx = xtan^-1x - ½ ln|1+x²| + c 

where, c is the constant of integration. 

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Properties of Arctan

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Here are some properties of Arctan:

• tan(tan^-1x) = x, for all real numbers x
• tan^-1x + tan^-1y = tan^-1[(x+y)/(1 – xy)], if xy < 1
• tan^-1x - tan^-1y = tan^-1[(x-y)/(1 + xy)], if xy> – 1

There are 3 properties 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(tanx) = x, only when x∈R - {x:x = (2n+1) (π/2), where n∈Z} i.e. tan^-1(tanx) = x only when x is not an odd multiple of π/2. Hence, tan^-1(tanx) will be undefined. 

Also Read: Decimal Expansion of Rational numbers

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ArcTan Table

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Any angle which is expressed in degrees can also be converted into radians. We should multiply the degree value by a factor of π/180. The arctan function takes a real number as an input and outputs the corresponding unique angle value. The table gives details of the arctan angle values for some real numbers. 

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Things to Remember

• The integral of arctan is the antiderivative of the inverse tangent function. Integration by parts is used to solve the integral of arctan.
• Arctan is the reverse of tan. It can be used to calculate the value of an angle in a right-angled triangle using the tangent function. 
• Tan^-1x will exist if and only if we restrict the domain of tangent function. 
• The basic formula for calculating arctan is given by θ = tan-1(perpendicular/base). Here, θ is the angle between the hypotenuse and base of the right-angled triangle.
• Arctan of infinity is the value of tan(π/2) = sin(π/2)/cos(π/2) = 1/0 = ∞. So, we can say arctan (∞) = π/2. 

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