Tangent Formula

Tangent Formula

Mathematics is divided into numerous categories based on the sorts of calculations performed and the disciplines covered. Arithmetic, percentages, exponentials, geometry, algebra, and other topics are covered in the branches. Furthermore, derived equations are frequently used in mathematics to assure the precision of computations and operations. The following article contains all of the fundamental equations found in the many areas or domains of mathematics. The principles of mathematics demonstrate how to apply specific equations, such as the equation of forces, accelerations, or work done, to solve mathematical issues. They are also used to provide mathematical solutions to problems that emerge in everyday life. Equations can take many different forms and are utilised in many different fields of mathematics. The methods employed to research them, however, differ depending on their kind. It might be as simple as utilising the basic addition formula or as sophisticated as integrating differentiation.

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Trigonometry is a field of mathematics that studies the connection between angles and side lengths of a right-angled triangle. Sine, cosine, tangent, cosecant, and secant are the six trigonometric ratios or functions, and a trigonometric ratio is a ratio between the sides of a right-angled triangle. The sine, cosine, and tangent functions are the reciprocal functions of the sine, cosine, and tangent functions, respectively.

sin θ = Opposite side/Hypotenuse

cos θ = Adjacent side/Hypotenuse

tan θ = Opposite side/Adjacent side

cosec θ = Hypotenuse/Opposite side

sec θ = Hypotenuse/Adjacent side 

cot θ = Adjacent side/Opposite side

The Tangent Formula is a trigonometric formula that deals with the tangent function. The tangent function (sometimes known as “tan”) is one of six trigonometric functions that is the ratio of the opposing side to the adjacent side. There are several tangent function formulae that may be obtained from various trigonometric identities and formulas.

What Are Tangent Formulas?

The Tangent formula discusses the tangent (tan) function. Consider a right-angled triangle with x as one of its sharp angles. The Tangent Formula is tan x = (opposite side) / (adjacent side), where “opposite side” refers to the side opposite the angle x and “adjacent side” refers to the side next to the angle x. Aside from this generic formula, there are other different formulae in trigonometry that will define a tangent function.

Tangent Formulas Using Reciprocal Identity

Students already know that the tangent function (tan) and the cotangent function (cot) are reciprocals. In other words, if tan x = a / b, then cot x = b / a. As a result, the Tangent Formula employing one of the reciprocal identities is, 

The Tangent Formula tan x = 1 / (cot x) (cot x)

Tangent Formula Using Sin and Cos

Students know that sin x = (opposite)(hypotenuse)  ,

 cos x = (adjacent) (hypotenuse)

tan x = (opposite)(hypotenuse) .

Now, students will divide sin x by cos x.

tan x= (sin x)(cos x) = [ (opposite)(hypotenuse) ] / [  (adjacent) (hypotenuse)] =  (opposite)(adjacent)  

Thus, The sine and cosine are the Tangent Formula terms 

the Tangent Formula tan x = (sin x)(cos x)

Tangent Formulas Using Pythagorean Identity

The link between sec and tan is discussed in one of the Pythagorean identities. It says, sec2x – tan2x = 1, for any x. Students can solve this for tan x. They see how.

sec2x – tan2x = 1

Subtracting sec2x from both sides,

-tan2x = 1 – sec2x

Multiplying both sides by -1,

tan2x = sec2x – 1

Taking square root on both sides,

The Tangent Formula tan x = ± √ (sec2x – 1)

Tangent Formula Using Cofunction Identities

The cofunction identities establish the relationship between the co functions sin and cos, sec and csc, and tan and cot. Taking use of one of the cofunction identities,

The Tangent Formula tan x = cot (90o – x) 


The Tangent Formula tan x = cot (/2 – x)

Tangent Formulas Using Sum/Difference Formulas

For each trigonometric function, students have sum/difference formulae that deal with the sum of angles (A + B) and the difference of angles (A – B). Tangent function sum/difference formulae are as follows:

tan (A + B) =(tan A + tan B)(1 – tan A tan B) 

tan (A – B) =(tan A – tan B)(1 + tan A tan B) 

Tangent Formula of Double Angle

In trigonometry, students have double angle formulae that deal with twice the angle. The tan double angle formula is

tan 2x = (2 tan x) / (1 – tan2x) 

Tangent Formula of Triple Angle

All trigonometric functions have triple-angle formulations. The triple angle formula of the tangent function is one of them.

 tan 3x =(3 tan x – tan3x) (1 – 3 tan2x)

Tangent Formula of Half Angle

In trigonometry, half-angle formulae are used to deal with half of the angles (x/2). Tangent function half-angle formulae are,

tan (x/2) =± √[(1 – cos x) / (1 + cos x)]

tan (x/2) = (1 – cos x) / (sin x)

Examples Using Tangent Formulas

The Extramarks come up with various types of example questions that help students to learn easily. The Tangent Formula assists students in improving problem-solving abilities and answering problems in competitive exams. The example questions on Tangent provide a variety of higher-level application-based problems such as MCQs, Short Answer Questions, and so on so that students may completely answer and comprehend the Tangent Formula. The example questions on the Tangent Formula are exclusively accessible to students. The Extramarks experts have curated study material to assist students in becoming acquainted with advanced-level ideas. 

Extramarks include several examples to help students fully comprehend the concepts. The Tangent Formula is a difficult concept to grasp. The Tangent examples may be quite useful for test preparation. Before trying to tackle Tangent problems, it is necessary to first properly comprehend the Tangent Formula.

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FAQs (Frequently Asked Questions)

1. What is the Tangent Function?

Tangent in a right-angled triangle is the ratio of the opposing side divided by the adjacent side. In trigonometry, there are six possible ratios. A ratio is a comparison of two numbers, such as the sides of a triangle. The Greek letter will be used to denote the reference angle in the right triangle. These six ratios can be used to compare two sides of a right triangle in various ways. Tan A formula is generally useful for calculating the angle of a right triangle. The tangent of an angle in a right triangle or right-angled triangle is a simple ratio between the lengths of the opposing side and the lengths of the adjacent side. Tangent is sometimes abbreviated as ‘tan,’ however, it is not. Tangent is commonly written as ‘tan,’ although it is pronounced as a tangent. This function is useful for determining the length of a triangle’s side. When someone understands at least one side of the triangle and one of the sharp angles, it is doable.