Introduction
The tangent angle formula is one of the formulas that are used to calculate the angle of the right triangle. The tangent function along with the sine and cosine is one of the three most common trigonometric functions. In any triangle the tangent of a triangle can be provided as follows:
Tan θ = \[\frac {\text{Opposite side}} {\text{Adjacent side}} \]
Where, O is the opposite side and A the adjacent side.
In such a way students can learn about Tangent Formula - Tangent Function, Derivation, Trigonometry Ratio, and Questions Solved via Vedantu.
Learn Tangent Formula
Sine, Cosine, and Tangent are the three basic functions of trigonometry through which trigonometric identities, trigonometry functions, and formulas are formed. The tangent is defined as the ratio of the length of the opposite side or perpendicular of a right angle to the angle and the length of the adjacent side. The tangent function in trigonometry is used to calculate the slope of a line between the origin and a point defining the intersection between hypotenuse and altitude of a right-angle triangle. In this article, we will discuss the tan formula, the formula of a tangent.
What is Trigonometry?
It is the study of the relationships which involve angles, lengths, and heights of triangles given. It also relates to the different parts of circles as well as other geometrical figures. Trigonometry has many trigonometric ratios which are very fundamental in mathematics. It has many identities that are very useful for learning and deriving the many equations and formulas in science. There are many fields where these identities of trigonometry and formulas of trigonometry are useful.
What is the Tangent Function?
Tangent is the ratio of the opposite side divided by the adjacent side in a right-angled triangle. In trigonometry, there are six possible ratios. A ratio is a comparison of two numbers i.e. sides of a triangle. The Greek letter,θ, will be used to represent the reference angle in the right triangle. These six ratios are useful in different ways to compare two sides of a right triangle.
Tan Formula is normally useful to calculate the angle of the right triangle. In a right triangle or the right-angled triangle, the tangent of an angle is a simple ratio of the length of the opposite side and the length of the adjacent side. Tangent is usually denoted as ‘tan’, but it is pronounced as a tangent. This function is useful to find out the length of a side of a triangle. It is possible when someone knows at least one side of the triangle and one of the acute angles.
Derivation of the Tan Formula
As we know, Sine, Cosine, and Tangent are the three basic functions of trigonometry. Let us briefly all the three basic functions with the help of a right-angle triangle.
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What is the Sine Function?
The Sine Function states that for a given right angle triangle, the Sin of angle θ is defined as the ratio of the length of the opposite side of a triangle to its hypotenuse.
Sin θ = \[ \frac{\text{Opposite side}}{Hypotenuse} \]
What is the Cosine Function?
The Cosine Function states that for a given right angle triangle, the Cosine of angle θ is defined as the ratio of the length of the adjacent side of a triangle to its hypotenuse.
Cos θ = \[ \frac {\text{Adjacent side}} {Hypotenuse} \]
What is the Tangent Function?
The Tangent Function states that for a given right angle triangle, the Cosine of angle ϴ is defined as the ratio of the length of the opposite side of a triangle to the angle and the adjacent side.
Tan θ =\[ \frac{\text{Opposite side}} {Hypotenuse}\]
Trigonometry Equations on the Basis of Tangent Function (Tangent Formulas)
Various tangent formulas can be formulated through a tangent function in trigonometry. The basic formula of the tangent which is mostly used is to solve questions is,
Tan θ = \[ \frac{Perpendicular}{Base}\] or Tanθ = \[ \frac{Sinθ}{Cosθ}\] or Tanθ = \[ \frac{1}{Cotθ}\]
Other Tangent Formulas are
Tan (a+b) equals Tan (a) + Tan (b)/1- Tan (a) Tan (b)
Tan (90 + θ) = Cot θ
Tan (90 - θ) = - Cot θ
Tan (-ϴ) = Tanθ
Trigonometry Ratio Table of Different Angles
Questions to be Solved
Question 1
Calculate the tangent angle of a right triangle whose adjacent side and opposite sides are 8 cm and 6 cm respectively?
Solution
Given, Adjacent side i.e. base = 8 cm
Opposite side i.e. perpendicular = 6 cm
Also, the tangent formula is: Tan θ=perpendicular/base
Tan θ=6/8 = 0.75
Therefore, tan θ=0.75
Thus the tangent value will be 0.75.
Question 2
Calculate the tangent angle of a right triangle whose adjacent side and opposite sides are 10 cm and 4 cm respectively?
Solution
Given, Adjacent side i.e. base = 10 cm
Opposite side i.e. perpendicular = 4 cm
Also, the tangent formula is: Tan θ=perpendicular/base
Tan θ=10/4 = 2.5
Therefore, tan θ= 2.5
Thus the tangent value will be 2.5.
FAQs on Tangent Formula
1. What is Tan Equal to?
The tangent of x is defined to be its sine divided by its cosine: tan x = sin x/ cos x. The cotangent of x can be defined to be the cosine of x divided by the sine of x: cot x = cos x/ sin x.
2. What is the Symbol for a Tangent?
The ratio of the adjacent side of a right triangle to the hypotenuse is called the cosine and given the symbol cos. Now finally, the ratio of the opposite side to the adjacent side is known as the tangent and given the symbol tan.
3. Where is Tangent 1?
The exact value of arctan(1) is equal to \[ \frac {π}{4}\]. The tangent function is known to be positive in the first and third quadrants. Now to find the second solution, we need to add the reference angle from π to find the solution in the fourth quadrant.
4. What is the Symbol for Pi?
Pi is denoted by the symbol π and is defined as the ratio of the circumference of a circle to its diameter, pi, or in symbol form, π, which seems a simple enough concept. But it rather turns out to be an "irrational number," which means that its exact value is inherently unknowable.
5. What are the applications of tangent formulas?
The tangent formulae talk about the tangent function also called the tan function. In order to understand this formula, you can consider a right-angled triangle with one of its acute angles as x. The tangent formula will then be tan x = (opposite side) / (adjacent side), where the opposite side will be the side opposite to angle x and the adjacent side will be the side adjacent to angle x. This formula has various forms. These forms can be used to check proof of a trigonometric identity or to find out the value of the tangent function with the information that is available.
6. What is the origin of trigonometric ratios?
Greek words trigonal which means “triangle” and metro on which means “to measure” when mixed together produce the word Trigonometry. It was until the 16th-century that trigonometry was highly concerned with the computation of the numerical values of the missing parts of a triangle for which the values for the other parts present are provided. For example, if there are lengths of two sides that are provided by a triangle then the two remaining angles can be calculated. These calculations tend to differentiate trigonometry from geometry.
7. How many tangent formulae are present?
There are a total of 8 tangent formulae that are present and can be provided as follows:
1. Tangent formulas using reciprocal identity
2. Tangent formula using sin and cos
3. Tangent formulas using Pythagorean identity
4. Tangent formulas using cofunction identities
5. Tangent formulas using sum/difference formulas
6. Tangent formulas of double angle
7. The tangent formula of triple angle
8. The tangent formula of half-angle
All of these tangent formulae can be used for various determination of functions.
8. Why is it important to learn regarding the Tangent Formula?
Tangent Formula is one of the topics that are important in Maths and can be used for solving a variety of questions. These formulae also help people to determine the various functions that need to be solved. It is also an important part of the JEE Main exams and hence it is important to study all of these topics well. These topics are also a part of other such exams that have various levels of questions being asked. Students can access the important questions related to JEE Main via Vedantu JEE Main sample papers.
9. Why are the tangent functions important and how many tangent formulae are the important ones that need to be remembered?
These tangent functions and tangent formulas are important due to the following reasons:
They let us find out the angles when the sides of the triangle are known
They also let us find out the sides of a triangle when the angles of the triangles are known.
The tangent formulas are related to tangent functions and these tangent formulas can be provided as follows:
Tan x = (opposite side) / (adjacent side)
Tan x = 1 / (cot x)
Tan x = (sin x) / (cos x)
Tan x = ± √( sec2 x - 1)