In trigonometry, arctan refers to lớn 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.
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Arctan (tan-1x) is not the same as 1 / tung 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, và 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, & the integral as well as derivative value.
| 1. | What is Arctan? |
| 2. | Arctan Formula |
| 3. | Arctan Identities |
| 4. | Arctan Domain and Range |
| 5. | Properties of Arctan Function |
| 6. | Arctan Graph |
| 7. | Derivative of Arctan |
| 8. | Integral of Arctan |
| 9. | FAQs on Arctan |
What is Arctan?
Arctan is one of the important inverse trigonometry functions. In a right-angled triangle, the chảy 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 và the hypotenuse. Thus, arctan is the inverse of the tan function.
If the tangent of angle θ is equal to lớn x, that is, x = tung 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/2tan (π / 3) = √3 ⇒ arctan(√3) = π/3tan (0) = 0 ⇒ arctan(0) = 0Suppose we have a right-angled triangle. Let θ be the angle whose value needs to lớn be determined. We know that chảy θ will be equal to lớn the ratio of the perpendicular và the base. Hence, chảy θ = Perpendicular / Base. Lớn find θ we will use the arctan function as, θ = tan-1
Arctan Formula
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 lớn find the value of angle θ in terms of degrees or radians. We can also write this formula as θ = tan-1
Arctan Identities
There are several arctan formulas, arctan identities & 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 ∈ Rtan (arctan x) = x, for all real numbers xarctan (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 sin(arctan x) = x / √(1 + x2)cos(arctan x) = 1 / √(1 + x2)arctan(x) = 2arctan(left ( fracx1 + sqrt1 + x^^2 ight )).arctan(x) = (int_0^xfrac1z^2 + 1dz)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 và 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°.
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 tên miền of tan x must be restricted otherwise the inverse cannot exist. In other words, the trigonometric function must be restricted khổng lồ its principal branch as we desire only one value.
The domain of chảy x is restricted to lớn (-π/2, π/2). The values where cos(x) = 0 have been excluded. The range of chảy (x) is all real numbers. We know that the domain & range of a trigonometric function get converted lớn the range & domain of the inverse trigonometric function, respectively. Thus, we can say that the tên miền of tan-1x is all real numbers và the range is (-π/2, π/2). An interesting fact to lưu ý is that we can extend the arctan function to lớn complex numbers. In such a case, the domain name of arctan will be all complex numbers.
Arctan Table
Any angle that is expressed in degrees can also be converted into radians. To vì chưng so we multiply the degree value by a factor of π/180°. Furthermore, the arctan function takes a real number as an input và outputs the corresponding chất lượng 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.
| -∞ | -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 |
| 0 | 0° | 0 |
| 1/2 | 26.565° | 0.4636 |
| 1/√3 | 30° | π/6 |
| 1 | 45° | π/4 |
| √3 | 60° | π/3 |
| 2 | 63.435° | 1.1071 |
| 3 | 71.565° | 1.2490 |
| ∞ | 90° | π/2 |
Arctan x Properties
Given below are some useful arctan identities based on the properties of the arctan function. These formulas can be used lớn simplify complex trigonometric expressions thus, increasing the ease of attempting problems.
tan-1x + tan-1y = tan-1<(x + y)/(1 - xy)>, when xy tan-1x - tan-1y = tan-1<(x - y)/(1 + xy)>, when xy > -1We have 3 formulas for 2tan-1x2tan-1x = sin-1(2x / (1+x2)), when |x| ≤ 12tan-1x = cos-1((1-x2) / (1+x2)), when x ≥ 02tan-1x = tan-1(2x / (1-x2)), when -1 tan-1(-x) = -tan-1x, for all x ∈ Rtan-1(1/x) = cot-1x, when x > 0tan-1x + cot-1x = π/2, when x ∈ Rtan-1(tan x) = x, only when x ∈ R - x : x = (2n + 1) (π/2), where n ∈ Zi.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 name of arctan is R (all real numbers) và 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/2When x = √3, y = π/3When x = 0, y = 0When x = -√3, y = -π/3When x = -∞, y = -π/2Using these we can plot the graph of arctan.
Arctan Derivative
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 rã (arctan x) = x
tan y = x
Now on differentiating both sides và using the chain rule we get,
sec2y dy/dx = 1
⇒ dy/dx = 1 / sec2y
According khổng lồ the trigonometric identity we have sec2y = 1 + tan2y
dy/dx = 1 / (1 + tan2y)
On substitution,
Thus, d(arctan x) / dx = 1 / (1 + x2)
Integral of Arctan x
The integral of arctan is the antiderivative of the inverse tangent function. Integration by parts is used to lớn evaluate the integral of arctan.
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Here, f(x) = tan-1x, g(x) = 1
The formula is given as ∫f(x)g(x)dx = f(x) ∫g(x)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.
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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 / rã 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