Tangent 3 Theta Formula - Explanation, Proof, Solved Example, and FAQs

Introduction to Tangent 3 Theta Formula

In Trigonometry, we can solve different types of problems by using trigonometry formulas. These problems mostly include trigonometric ratios (sin, cos, tan, sec, cosec, and tan), Pythagorean identities, trigonometric identities, etc. Some of the trigonometric formulas include the sign of ratios in different quadrants, involving co-function identities (shifting angles), sum and difference identities, double angle identities, half-angle identities, etc. Here, we will discuss the tan 3 theta formula in detail. We will learn how to prove the tan 3 theta formula along with the solved example.


Let us discuss an introduction to Trigonometry in detail before looking at the formula. Trigonometry is a branch of Mathematics mainly 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of the triangle's size, and measured values for several angles were tabulated before computers made trigonometry tables outdated.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.


These formulas are very helpful for the students while solving problems based on these formulas or any trigonometric application. Along with these, basic trigonometric identities help us to derive the trigonometric formulas in the examination. These trigonometric formulae are used to determine the domain, range, and value of a compound trigonometric function.


Formula of Tan 3 Theta

Tan 3 theta is also called tan triple angle identity and it is used in the following two cases as a formula.

  • Tan of triple angle is expanded as the quotient of subtraction of tan cubed of angle from three times tan of angle by subtraction of three times tan squared of angle from one.

  • The quotient of subtraction of tan cubed of angle from three times tan of an angle by subtraction of three times tan squared of angle from one is simplified as tan of triple angle.

Tan 3 theta formula is \[\frac{3tan\theta-tan^{3}\theta}{1-3tan^{2}\theta}\]


How to Prove Tan 3 Theta Formula?

Tan3= \[\frac{3tan\theta-tan^{3}\theta}{1-3tan^{2}\theta}\]

Proof: Tanking Left-hand side = tan(3θ) = tan(2θ + θ)

To get RHS equation we have to use tan(A + B) formula and tan(2A)

So tan(A + B) = (tanA + tan B)/(1 - tanA tanB) and tan 2A = (2tanA)/(1 - tan2A)

Consider A as 2θ and B as θ apply tan(A + B) formula

tan( 2+ ) =\[\frac{(tan2\theta+tan\theta)}{(1-tan2\theta\times tan\theta)}\]

Now apply the tan2A formula

tan(2θ+θ) = \[\frac{\frac{2tan\Theta }{1-tan^{2}\Theta } + tan\Theta}{1-\frac{2tan\Theta }{1-tan^{2}\Theta }tan\Theta}\]

= 2tanθ+tanθθ

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FAQs (Frequently Asked Questions)

1. Where can I find Tangent 3 Theta Formula - Explanation, Proof, Solved Example, and FAQs? 

The students can easily find the study material for their math preparation at Vedantu.com absolutely free of charge. These study materials are easy to study and are made for the absolute convenience of the student. Once you start using the free PDFs available at Vedantu, then there is no going back. For this, the students need to sign in to Vedantu.com using the website or the app. The aforementioned study material includes PDFs of various mock tests, sample tests, revision notes and so much more. Which proves to be beneficial for a student's preparation. 

2. What should I know about the Tangent 3 Theta Formula?

The students must be aware that the Tan 3 theta is also known as Tan triple angle identity. Also, the value of Tan 3 degrees is approximately 0.0524. It is used in varied cases such as when the Tan of triple angle is simplified as the quotient of subtraction of tan cubed of angle from three times tan of angle by subtraction of three times tan squared of angle from one. Or The quotient of subtraction of tan cubed of angle from three times tan of an angle by subtraction of three times tan squared of angle from one is written as tan of triple angle.

3. What can I do to understand the Tangent 3 theta formula better?

The students can easily solve most of the trigonometric questions which are associated with Tan or Tangent theta with no problem if they practice and learn the formula of Tangent 3 theta. For using this formula a student needs to attempt the questions which are based on the usage of this formula. This helps them to get a better grip on the topic of trigonometry and also helps them to understand how to implement the formula into questions as required.

4. How can I understand the various trigonometric functions?

The students can understand the various concepts of trigonometry once they are properly introduced to the basic concepts. For starters, they should be aware of sin, cos, tan and other functions. Then they must look into the formulas of each of these functions. For example, one of the important formulas you need to know is Tan 3 theta which is widely used in various questions based on trigonometry in the math section. The students can also use short forms of various trigonometric ratios for their studies to become easier and more fun.

5. What is the importance of knowing the Tangent 3 theta formula?

The tangent 3 theta formula is used whenever there is some need to simplify the given triple angle tan functions or to expand them as per the need of the question. In these questions, Tan is the trigonometric function and theta is the unknown angle. Many Questions may arise based on this formula where the students have to either find the unknown angle, simplify the given trigonometric equation or derive any other trigonometric relation required. Thus, it is necessary for the students to know the Tangent 3 theta formula.

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