Tangent 3 Theta Formula: Proof & Derivation

Tangent 3 Theta formula is, Tan 3 theta = 3 tan theta – tan ³ theta / 1 – 3 tan² . Trigonometric ratios (sin, cos, tan, sec, cosec, and tan), Pythagorean identities, and trigonometric identities are among the most commonly used ways to solve the Tan 3 theta. 

• The period of Tan 3 Theta is Period 3π.
• The tangent 3 theta formula can be used at times when we simplify a triple angle tan function or expand them as per requirement. 
• Herein, Tan is the trigonometric function, while theta is the unknown angle. 
• Based on this formula, either the unknown angle is determined, the trigonometric equation is simplified or any other trigonometric relation is derived.

Read More: Inverse Trigonometric Functions

Key terms: Sin, Sin Squared X, Triangle, Pythagoras Theorem, Cosine, Secant, Cosecant, Tangent, Cotangent, Perpendicular, Hypotenuse, Integrals, Trigonometric Ratio, Trigonometry

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Tangent 3 Theta

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Tan 3 θ can be expressed as a triple angle identity in trigonometry. Tangent 3 Theta is known to be a vital trigonometric identity used to solve a variety of trigonometric and integration issues.

• For a triple angle, it is a trigonometric function which returns the tan function value.
• It can also be written as tan 3 θ = sin 3 θ/cos 3 θ because the tangent function is a ratio of the sine functions and cosine functions.
• The tan 3 θ value repeats once after every π/3 radians, tan3θ = tan (3θ + π/3). In fact, the graph it has is thinner than tan θ’s graph.

Trigonometry is a discipline of mathematics that deals with the use of certain angle functions in computations. In trigonometry, the abbreviations used are sine (sin), cosine (cos), cotangent (cot), tangent (tan), secant (sec), and cosecant (CSC).

Trigonometric Functions Detailed Video Explanation

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Tan 3 Theta Formula

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Tan 3 theta is also known as tan triple angle identity, and it is implemented as a formula in the following two circumstances.

• The quotient of subtracting Tan cubed of angle from three times Tan of Angle by subtracting three times Tan squared of angle from one is the Tan of Triple angle.
• Tan of Triple Angle is the quotient of subtracting three times Tan of an Angle from three times Tan of an Angle by subtracting three times Tan squared of an angle from one.

The relationship that can be established between the basic trigonometric functions carrying 3 times the angle in the trigonometric functions of those angles itself typically yields the trigonometric triple-angle identities.

By the triple angle identity, we can expand or simplify the triple angle tan functions like tan 3 A, tan 3 x, tan 3 alpha etc:

Tan 3 Theta Derivation

Tangent 3 theta formula can be derived by using the sum angle formula used for the Tangent theta and Tangent 2 theta ratios.

In order to show that tan 3θ = (3 tan θ – tan³θ) / (1 – 3 tan²θ), we can express it as: 3θ as (2θ + θ).

Thus,

L.H.S. = tan 3θ 

= tan (2θ + θ)

By applying the formula, tan (x + y) = (tan x + tan y) / (1 – tan x tan y)

= (tan 2θ + tan θ)/(1 – tan 2θ tan θ)

Thus, by using the formula, we get: tan 2x = (2 tan x) / (1 – tan²x) for tan 2θ

= [(2 tan θ / (1 – tan²θ)) + tan θ] / [1 – (2 tan θ / (1 – tan²θ)) tan θ]

= (tan θ – tan³θ + 2 tan θ) / (1 – tan²θ – 2 tan²θ)

= (3 tan θ – tan³θ) / (1 – 3 tan²θ) 

= R.H.S.

Hence, it derives the formula for the tangent 3 theta ratio.

Read More: Introduction to Trigonometry

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How to Prove Tan 3 Theta Formula?

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We have,

Tan 3 = (3 tan θ − tan 3 θ) / (1 − 3 tan 2 θ)

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

We must apply the tan(A + B) formula and the tan(A + B) formula to obtain the RHS equation (2A)

As a result, we have:

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

Taking A to be 2θ and B to be θ and use the tan(A + B) formula, we have:

tan( 2θ + θ) =(tan 2 θ + tan θ) / (1 − tan 2 θ × tan θ)

Apply the tan 2 A formula now:

→ \(tan (2 \theta +\theta) = \frac{\frac{2tan\theta}{1-tan \theta} + tan \theta}{1-\frac{2 tan \theta}{1-tan \theta} tan \theta}\)

= 2 tan θ + tan θ

Read More: Heights and Distances

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Things to Remember

• The sign of ratios in different quadrants, involving co-function identities, sum and difference identities, and so on, are some of the trigonometric formulas. 
• Trigonometry is a discipline of mathematics that deals with the use of certain angle functions in computations. 
• There are six functions of an angle that are commonly used in trigonometry, namely: sin (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).
• The Tangent 3 Theta formula can be expressed as, Tan 3 theta = 3 tan theta – tan ³ theta / 1 – 3 tan² . 
• The domain, range, and value of a compound trigonometric function are determined using these trigonometric formulas. 

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Previous Year Questions

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