Tan Theta Formula

Dhristi JEE 2022-24

Tan Theta

The tangent is defined in right triangle trigonometry as the ratio of the opposite side to the adjacent side (It is applicable for acute angles only because it's only defined this way for right triangles).  To find values of the tangent function at different angles when evaluating the tangent function, we first define the reference angle created by the terminal side and the x-axis. Then we calculate the tangent of this reference angle and determine whether it is positive or negative based on which quadrant the terminal side is in. In the first and third quadrants, the tangent is positive. In the second and fourth quadrants, the tangent is negative. The slope of the terminal side is also equal to the tangent. 

Let us discuss an introduction to Trigonometry in detail before looking at the formula. Trigonometry is a branch of mathematics concerned with the application of specific functions of angles to calculations. In trigonometry, there are six functions of an angle that are widely used. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc) are their names and abbreviations. In relation to a right triangle, these six trigonometric functions. The sine of A, or sin A, is defined as the ratio of the side opposite to A and the side opposite to the right angle (the hypotenuse) in a triangle. The other trigonometric functions are defined similarly. These functions are properties of the angle that are independent of the triangle's size, and measured values for several angles were tabulated before computers made trigonometry tables outdated. In geometric figures, trigonometric functions are used to calculate unknown angles and distances from known or measured angles. Trigonometry has a wide range of applications, from specific fields such as oceanography, where it is used to measure the height of tides in oceans, to the backyard of our home, where it can be used to roof a building, make the roof inclined in the case of single independent bungalows, and calculate the height of the roof etc. Here, we will discuss the tan theta formula in detail. 

How to Find the Tangent?

You must first locate the hypotenuse to find the tangent. The hypotenuse is typically the right triangle's longest side. The next task is to decide the angle. There are only two angles to choose from. You cannot choose the right angle. After you've chosen an angle, you will mark the sides. The side opposite to this angle will be the opposite side and the side next to the angle is the adjacent side. After labeling the sides, you can take the required ratio. Let’s discuss ratios, what is tan theta and it’s practical applications?

What is Tan Theta? 

The length of the opposite side to the length of the adjacent side of a right-angled triangle is known as the tangent function or tangent ratio of the angle between the hypotenuse and the base.

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As discussed, the tangent function is one of the three most common trigonometric functions, along with sine and cosine. The tangent of an angle in a right triangle is equal to the length of the opposite side (O) divided by the length of the adjacent side (A). It is written simply as 'tan' in a formula.

⇒ tan x = O/A

tan(x) is the symbol for the tangent function which is also called the tan x formula. It is one of the six trigonometric functions that are commonly used. Sine and cosine are most often associated with the tangent. In trigonometry, the tangent function is a periodic function that is very useful.

The tan formula is as follows:

⇒ Tan = Opposite/Adjacent 

What is tan theta in terms of sine and cos?

⇒ tan x = sin x/cos x

or, tan theta = sin theta/cos theta   (here, theta is an angle)

The sine of an angle is equal to the length of the opposite side divided by the length of the hypotenuse side, while the cosine of an angle is the ratio of the adjacent side to the hypotenuse side.

Hence, sin x = Opposite Side/Hypotenuse Side

cos x = Adjacent Side/Hypotenuse Side

Therefore, (tan formula) tan x = Opposite Side/Adjacent Side

Finding the Tangent of the Triangle

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Angles A and B are the two angles we will deal with in this triangle. To find our tangent, we must first find out the hypotenuse. Our right angle is clear to me. Can you see it? That means our hypotenuse is directly across it, and the side that measures 5 is the hypotenuse. Can you see it? Okay, as we have our hypotenuse, let's choose an angle to work with. We'll choose angle B. As  B is our angle, our opposite side is the side that measures 3. Our adjacent side is the one that measures 4 because it is the only side next to angle B which is not the hypotenuse. 

This means that our tangent of angle B will be the ratio of the opposite side over the adjacent side or we can write it as 3/4 which will be equal to 0.75. Similarly, If we choose angle A, our sides will change and the tangent will be 4/3 which will be equal to 1.33. 

Tangent to Find the Missing Side

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In a few problems, we will have to find the missing side of the triangle. This particularly has real-life applications like when construction companies are building on hills. In these kinds of problems, an angle is given to us. To solve these problems, we first have to locate the side that is missing. In the triangle given above, the side missed is the adjacent side to the angle given. Right?  What we have to do now is just to write the equation from the definition of the tangent. Once we have the equation, we can move ahead, use algebra to solve for the variable which will be missing. 

By multiplying both sides by x and then dividing both sides by tan 66, we've identified our variable. And by using the calculator to measure tan 66, we get the answer as 2.22.

Arctan - Inverse Tangent Function

There is an inverse function for any trigonometry function, for example, tan has arctan, which works in a reverse manner. These inverse functions have the same name as the originals, but with the word 'arc' added to the start. So, arctan is the opposite of tan. If we know the tangent of an angle and want to know the actual angle, we use the inverse function.

Large & Negative Angles

The two variable angles in a right triangle are always less than 90 degrees. However, we can find the tangent of any angle, irrespective of its height, as well as the tangent of negative angles. We can also graph the tangent function.

Tangents will be Used to Calculate the Height of a Building or a Mountain:

You can easily find the height of a building if you know the distance from where you observe it and also the angle of elevation. Similarly, you can find another side of the triangle if you know the value of one side and the angle of depression from the top of the house. All you need to know is a side and an angle of the triangle.


In fields like astronomy, mapmaking, surveying, and artillery range finding, trigonometry emerged from the need to compute angles and distances. Plane trigonometry deals with issues involving angles and distances in a single plane. Spherical trigonometry considers applications of related problems in more than one plane of three-dimensional space. 

All You Need to know About Tan theta

Students in the regular classes learn about the Triangles and various geometric measurements and operations performed over them. Trigonometry is another branch of mathematics concerned with the measurements of angles and sides corresponding to it in any right-angle triangle. Most commonly heard of functions in introductory chapters of Trigonometry are Sine theta (sin), Cosine theta (cos), tangent theta (tan), cotangent theta (cot), secant theta (sec), and cosecant theta (codec). As we all know a right angle triangle has three sides namely base, height and hypotenuse. These 6 functions of Trigonometry are nothing more than the different combinations of the three sides of the triangle in pairs concerning their proportions with each other. The functions are independent of the size or length of sides but are determined by the angle produced by the two corresponding sides.

Tangent theta or Tan theta is the ratio of the height of a right-angled triangle over the length of the base. Locating the hypotenuse of any triangle other than the right triangles sometimes becomes the difficult part of this approach. Tan theta is also used for obtaining the length of the missing side after measuring the distance between the point of observation and the origin of that side or video versa. But in this case, the value of the angle formed by the base and the line projecting to the top of the other side is required to be known before calculation.

FAQs on Tan Theta Formula

1. What are the other functions of Trigonometry and their values?

Theta is the term used for the angle of the triangle. It can be anything between 0 to 360 degrees. Likewise, the Trigonometric function values correspond to each value of theta. For example, the value of Sin 30° is ½ while the value of Sin 45° is 1/√2. Similarly, another example is that the value of Tan 30° is  1/√3, Tan 60° is √3 and Tan 45° is 1. Students can get the values of all the functions from any standard Book or reference book with an explanation. Vedantu also provides study materials for the same which you can download and read by yourself.

2. When is the right to learn about the concepts of Trigonometry?

Students learn about various types of triangles such as Acute angled Triangle, Obtuse angled Triangle, Isosceles  Triangle, Equilateral Triangle or Right-angled Triangle very early in the classes for secondary education. But Trigonometry as a chapter is introduced from class 9 or 10 in the CBSE or ICSE pattern of the curriculum. Students can learn about triangles before arriving in the 9 class but do have to be conceptually clear about the different types of triangles and the knowledge about the concept and use of angles. 

3. Do I need to Practice questions on the Tan Theta chapter?

Trigonometry Is essentially a chapter of Mathematics. So mere reading of the subject matter will not be sufficient to get a good grasp of the concepts. Every student. Mathematics needs to work on the exercise questions provided after each topic. It not only clears the concept of the subject read but gives you an understanding of how to apply the concept to solve the problems in real life. You can also arrange for extra questions to practice by following any reference or complement book publications available in the market and online.

4. Do Vedantu Online courses make available Practice Questions for its Students?

Vedantu Online Course has been started with the motto of helping out the students of both CBSE and ICSE preparing for their term exams, board exams of any class or any other competitive exams after 12th board. As most of the students in India can't afford the expensive coaching classes or special books in the market. So Vedantu provides for them all types of courses to learn the subject in just one click. We also have compiled question sets other than those in the standard Book of every topic. Any student with a smartphone and internet connection can avail of the facilities.

5. Where Can I obtain all the study materials and practice questions set available on Vedantu? And what is the total cost?

The materials available on the Vedantu have been arranged according to the proper course structure and relevant web pages for the subject. The students are required to get enrolled to avail the study materials and other products available on this website. Registration is very easy and free with some basic data to be filled in. Then students can get access to all the pages they want. Students can also download the same materials available in PDF format by clicking on the link provided and without paying any cost to do so.

6. What is Tan Theta Equal to?

In the right triangle, the tangent is defined as the ratio of the opposite side to the adjacent side. In other words, the length of the opposite side to the length of the adjacent side will be known as the tangent function or tangent ratio of an angle.

7. Can Trigonometry Be Used in Everyday Life?

While trigonometry does not have clear applications in solving practical problems, it is used in a variety of things that we appreciate. For example, sound travels in waves in music, and this pattern, although not as normal as a sine or cosine function, is still useful in computer music creation. Since a machine can't listen to or comprehend music as we can, it's represented mathematically by its constituent sound waves. As a result, sound engineers must have a basic understanding of trigonometry. And, thanks to trigonometry, the good music that these sound engineers make is used to help us recover from our hectic, stressful lives.

8. What is the Use of the Tan Theta Formula?

The most efficient and progressive application of trigonometry is the analysis and simplification of equations using various trigonometric functions such as sine, cosine, and tangent, among others. In engineering fields such as mechanical engineering, electronics, and mechatronics, the analytical use of trigonometry is important. So, with the basics of the tan theta formula we can easily solve and simplify higher-order problems such as 1 tan theta, tan 3 theta, tan 3 theta, tan 3 theta 1 tan 2 theta, tan3x formula and so on.