Arctan

Arctan is defined as the inverse of the tangent function. Arctan(x) is denoted as tan^-1(x). There are six trigonometric functions and the inverse of all six functions is repressed as, sin^-1x, cos^-1x, tan^-1x, cosec^-1x, sec^-1x, and cot^-1x.

Arctan (tan^-1x) is not similar to 1 / tan x. tan^-1 x is the inverse of tan x whereas 1/ tan x is the reciprocal of tan x. tan^-1 x is used to solve various trigonometric equations. In this article, we will study the arctan function formula, graph, properties, and others in detail.

What is Arctan?

Arcatan is the inverse of the trigonometric function tan x. The ratio of the perpendicular and the base in a right angle triangle is called the trigonometric function and taking its inverse gives the arctan function. This is explained as,

tan (π/4) = 1

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

If we have a right angle triangle with an angle θ then tan θ is perpendicular/base, then the arctan function is,

θ = tan^-1(perpendicular/base)

Learn More, Inverse Trigonometric Function

What is Arctan Formula?

Tangent is a trigonometric function and in a right-angled triangle, the tangent function equals the ratio of perpendicular and base (perpendicular/base).

Arctan is a reference to the inverse function of the tangent. Symbolically, arctan is represented by tan^-1x in trigonometric equations.

Arctan Formula Definition

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.

Suppose, the tangent of the angle θ equals x.

x = tan θ ⇒ θ = tan^-1x

Let us take a right-angled triangle ABC with angle BCA as θ. Side AB is perpendicular(p) and side BC is base(b). Now, as we studied that tangent equals perpendicular by the base.

i.e. tan θ = Perpendicular/Base = p/b

And, by using the above expression,

θ = tan^-1(p/b)

Arctan Identities

There are various Arctan identities that are used to solve various trigonometric equations. Some of the important arctan identities 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
• 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) = ∫_o^x 1/√(1+z²)dz

How To Apply Arctan Formula?

Arctan Formula is used in solving various trigonometric problems and the same is explained in the example added below.

Example: In the right-angled triangle PQR, if the height of the triangle is √3 units and the base of the triangle is 1 unit. Find the angle.

To find the angle (θ)

θ = arctan (perpendicular/height)

θ = arctan (√3/1)

θ = 60°

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.

• Domain of arctan x is Real Number
• Range of arctan (x) is (-π/2, π/2)

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 (x) Properties

Arctan x properties are used for solving various trigonometric equations. There are various trigonometric properties that need to be studied for studying trigonometry. Some important properties of the arctan function are given 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

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.

As we studied above that the value of arctan can be derived by degrees or radians. So, the below-given table illustrates 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°	0
1/√3	30°	π/6
1	45°	π/4
√3	60°	π/3
∞	90°	π/2

Arctan Graph

The graph of the Arctan function is the infinite graph. The domain of arctan is R (real numbers) and the range of the Arctan function is (-π/2, π/2). The graph of the Arctan function is discussed below in the image below:

The graph is made using the value 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

Arctan x Derivative

Derivative of arctan is very important for studying mathematics. The derivative of the arctan function is calculated 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

Integral of arctan is defined as the antiderivative of the inverse tangent function. Integration of Arctan x is derived using the concept given 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

Arctan 0

The arctan of 0 is 0. We can also say that, tan^-1(x) = 0. Thus, Arctan(0) = 0

Arctan 2

The arctan of 2 is 63.435. We can also say that, tan^-1(2) = 63.435. Thus, Arctan(2) = 63.435.

Arctan Infinity

The arctan infinity is given as lim_x→∞ tan^-1x = π/2.

Arctan Examples

Example 1: Evaluate tan^-1(1).

Solution:

tan^-1(1)

Value 1 can also be written as,

1 = tan(45°)

Now,

tan^-1(1) = tan^-1( tan 45°) = 45°

Example 2: Evaluate tan^-1(1.732).

Solution:

tan^-1(1.732)

Value of 1.732 can also be written as

1.732 = tan(60°)

Now, 

tan^-1(1.732) = tan^-1( tan 60°) = 60°

Example 3: Solve tan^-1x + tan^-11/x

Solution:

• We know that, tan^-1x + tan^-1y = tan^-1[(x + y)/(1 – xy)]

= tan^-1x + tan^-11/x

= tan^-1[(x + 1/x)/(1 – x × 1/x)]

= tan^-1[(x + 1/x)/(1 – x × 1/x)]

= tan^-1[(x + 1/x)/(1 – 1)]

= tan^-1[(x + 1/x)/(0)]

= tan^-1[∞]

= π/2

Example 4: Find the derivative of  tan^-1√x

Solution:

We know that, d/dx (tan^-1x) = 1 / (1 + x²)

⇒ d/dx (tan^-1√x)

Using Chain Rule

= 1 / (1 + [√x]²)

= 1 / (1+x) × d/dx(1/√x)

= 1/(1+x) × 1/2√x

= √x/{2x(x+1)}

Thus, derivative of d/dx (tan^-1√x) is √x/{2x(x+1)}

Arctan Practice Questions

Q1. Find the derivative of  tan^-1(2x² + 3)

Q2. Find the Integral of  tan^-1√x

Q3. Evaluate tan^-1(10)

Q4. Solve tan^-1(x) + tan^-1(x²)

Arctan-FAQs

1. What is the Arctan?

Inverse of the tangent function is called Arctan. It is denoted as arctan x or tan^-1x. The formula used to determine the value of arctan is θ = tan^-1(x)

2. Find the Derivative of Arctan.

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

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

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

4. Is Arctan similar to Cot?

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

5. What is Arctan of Infinity?

As, we already know that the value of tan (π/2) = ∞. Arctan is the inverse function of tan then, we can say that arctan(∞) = π/2.

6. Is Arctan and tan^-1 the same?

Yes, Arctan and tan^-1 is the same as, Arctan is other name of tan^-1(x)

7. Why is Arctan (1) pi over 4?

The value of sin^-1(π/4) is 1/√2 and the value of cos^-1(π/4) is 1/√2 and we know that, tan^-1(π/4) is sin-¹(π/4)/cos^-1(π/4) and the value of arcsin and arccos is equal then the value of arctan (1) is π/4.