# Inverse Tangent Formula

## Inverse Tangent Formula

Every function in trigonometry, including sine, cosine, and tangent, has an inverse function. In a right-angled triangle, the angle’s tan value is calculated using the tangent formula. When students know the side opposite to that angle and the adjacent side, they can apply the Inverse Tangent Formula to get the angle. Arctan or tan-1 is used to denote the inverse of a tangent. The trigonometric function “tangent” has an inverse called “inverse tan.” The opposite side of the right triangle is divided by the adjacent side, and this relationship is used to compute the angle by using the tangent ratio of the angle. This function can be used to calculate the value of tan 1, arctan 1, tan 10, etc. All inverse trigonometric functions must be named with the prefix “arc,” so Inverse Tangent Formula is abbreviated as “arctan.”

## What is Inverse Tan?

The fundamental trigonometric functions of trigonometry, sine, cosine, tangent, cotangent, secant, and cosecant, are used to evaluate angles in trigonometry. Each of these trigonometric functions has a unique trigonometric ratio for use in trigonometric calculations. These functions also have inverses, which are referred to as arcsin, arccos, arctan, arccot, arcsec, and arccosec. Students in their higher secondary classes need to study the Inverse Tangent Formula or arctan. This concept is fully explained by the Extramarks portal. It also provides some practice questions, as well as the Inverse Tangent Formula for evaluating angles. The trigonometric function Inverse Tangent Formula is the inverse of the trigonometric function tangent. It is also known as the arctan since in trigonometry, the word “-arc” denotes the inverse. The symbol for the Inverse Tangent Formula is tan-1x. The value of the angle is calculated using the inverse tangent function and the (perpendicular/base) ratio. The trigonometric functions/ratios are Sine, Cosine, Tangent, Secant, Cosecant, and Cotangent. These trigonometric functions have the inverses as inverse cotangent, inverse cosecant, inverse cosine, inverse tangent, inverse cosine and inverse sine.

### Inverse Tan Examples

One of the inverse trigonometric functions, the Inverse Tangent Formula, is the inverse of the tan function. It is also referred to as the arctan function, pronounced “arc tan.” The mathematical notation for it is “a tan x” (sometimes written as “tan-1x” or “arctan x”). “Tan-1 x” is interpreted as “tan inverse x.” When two functions f and f-1 are inverses of one another, one gets x = f-1(y) whenever f(x) = y. Therefore, tan x = y implies that x = tan-1 (y). To put it another way, “tan” becomes tan-1 when it moves from one side of the equation to the other. To further understand how the inverse tan function functions, students can look at a few examples, such as

tan 0 = 0 ⇒ 0 = tan-1(0)

tan π/4 = 1 ⇒ π/4 = tan-1(1)

## Inverse Tan Formula

The Inverse Tangent Formula is an inverse trigonometric function of the tangent, just as the tangent is an equivalent trigonometric function. The Inverse Tangent Formula, which can be represented in degrees or radians, is used to get the values for these inverse functions. For understanding the Inverse Tangent Formula, one can suppose an angle where the tangent is equal to x. The tangent’s inverse function will then be presented.

Since x = tan,

=> θ = tan-1 x

In Mathematics, the ratio of the perpendicular to the base yields the Inverse Tangent Formula.

The ratio of an angle’s opposing side to its adjacent side in a right-angled triangle is known as the tangent of the angle. Specifically, tan θ = (opposite side) / (adjacent side). The Inverse Tangent Formula is thus θ = tan-1[(opposite side) / (adjacent side)], according to the definition of the Inverse Tangent Formula. As a result, when the neighbouring side and the opposite side are known, the right-angled triangle’s angle can be calculated using the Inverse Tangent Formula.

## Domain, Range, and Graph of Inverse Tan

Students should know how to determine the inverse tan function’s domain and range while studying the Inverse Tangent Formula. They should also learn to plot the graph. One can determine the inverse tangent’s domain and range. The inverse tangent’s domain is (-inf, inf). This implies that the x-value must be contained in the domain of a function of the kind y = tan-1(x), that is, (-inf, inf). Also, (-pi/2, pi/2) is the range of the Inverse Tangent Formula. This indicates that the y-value must be within the range of (-pi/2, pi/2) if one has a function of the form y = tan-1(x). One needs to remember that there are some parentheses present. Therefore, while one may have numbers that are close to -pi/2 or pi/2, but not actually include them. When thinking about the range of the Inverse Tangent Formula on a circle, one only selects numbers from the appropriate area.

## Domain and Range of Inverse Tan

Students should be aware that on the tan function, n varies from R – {x: x = (2n + 1) (π/2), where n ∈ Z} to R. This is due to the fact that odd multiples of π/2 are not defined for the tan function. The inverse of the tan function, however, does not exist in this domain because it is not one-one on this domain. The domain of the tan function can be constrained to one of the intervals (-3π/2, -π/2), (-π/2, π/2), (π/2, 3π/2), etc. for it to be one-one. One obtains a branch of the Inverse Tangent Formula with regard to each of these intervals. However, in order to make the tan function one-one, the domain is typically limited to (-π/2, π/2). The inverse tan function’s domain and range are the range and domain of the tan function, respectively. Therefore, tan inverse x has a domain of R, whereas (-π/2, π/2) is the range of the tan inverse x.

## Graph of Inverse Tan

The x-coordinates and y-coordinates of the function are switched around in the graph, which is a characteristic of an inverse function. Thus, the x values of the inverse tangent function correspond to the tangent function’s y values and the y values to the x values. All real numbers are included in the inverse tangent function’s domain. This implies that any integer can be used as an input or independent variable (usually x). As a result, the angle formed by the Inverse Tangent Formula must fall within its range.

### Steps to Find Tan Inverse x

To determine the tan inverse of x, students should follow these steps. The solution should be within the range of (-π/2, π/2), which is the range of the tan inverse x. Suppose y = tan-1x. In that case, the Inverse Tangent Formula is defined as tan y = x. The solution is the value of y in the interval (-π/2, π/2) that fulfils the formula tan y = x.

### Examples of Finding Tan Inverse of x

The impact of the original function is reversed by an inverse function. The tangent function’s effect is reversed by the inverse tangent function. Students have to be aware that if they know the adjacent side and the angle of a right triangle, they can use the tangent function to get the opposite side. If they know the measurements of the adjacent and opposite sides, then they can use the inverse tangent function to determine the angle’s measure. In this topic, students shall practice using the inverse tangent function to solve some examples.

### Properties of Inverse Tan

The inverse tan function (arctan) has some basic properties to understand that make it easy to understand it. It is written as y = arctan (x). It is defined as x = tan (y). The ratio’s range includes all real values. The range of the principal value is -π/2 < y < π/2. The range of the principal value is -90° < y < 90°.

### Derivative of Inverse Tan

The Inverse Tangent Formula for the derivative of tan inverse x is (tan-1x)’ = 1/(1 + x2). Finding the derivative of tan inverse x with respect to x is done by the process of differentiation of tan inverse x. The rate of change of tan inverse x can also be used to understand the derivative of tan inverse x. The first principle of differentiation, the concept of the derivative of arctan, its proof using implicit differentiation, and the derivative of tan inverse x with respect to cot inverse x are all important concepts to understand the differentiation of inverse tangent function. The ideas of derivatives and inverse trigonometric functions can be used to determine the derivative of tan inverse x. The first principle of derivatives and implicit differentiation are two techniques that can be used to find the derivative of tan inverse x. The fact that the derivative of tan inverse x is the negative of the derivative of cot inverse x makes it simple to remember. The derivative of cot inverse x is, in other words, the negative of the derivative of tan inverse x.

### Integral of Inverse Tan

Students shall use the method of integration by parts to determine tan-1x dx. Students should practice examples based on the integrals of tan inverse x or arctan x.

### Examples on Tan Inverse x

Angles of depression and elevation are two examples of inverse trigonometric functions, such as tan inverse x. These two are a good fit because they apply sine, cosine, or tangent to calculate the angle of a person’s view to the top of a building. Another example would be to determine the angle of depression a person would be gazing down at an object if they were standing on a building that was a particular height and looked down at it from a certain distance away. In that case, one would set up all the variables and solve for the angle, which employs more of the inverse aspect of trigonometric functions. Builders, like architects and construction workers, are examples of persons who use inverse trigonometric functions. The construction of a bike ramp is an example of its application. One can determine the length and height. In many different domains, including Physics, Astronomy, Navigation, and Construction, inverse functions are applied. They are often used in daily life. For instance, if someone is planning a hike, they can see on the map the route they should take. The physical state of this problem can be represented by a triangle that can be constructed. In this example, arctan is used to determine the angle.

### Practice Questions on Tan Inverse x

In order to comprehend the Inverse Tangent Formula clearly, it is necessary to practise answering a variety of questions. For practice, Extramarks offers a number of practice questions on the Inverse Tangent Formula.