Tangent 3 Theta Formula, Definition, Solved Examples

Tangent 3 Theta Formula: Trigonometry explores the relationships between the angles and side ratios within a right-angled triangle. This field uses trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant to analyze these connections. The term "Trigonometry" originates from 'Trigonon' and 'Metron,' denoting a triangle and measurement, respectively. By investigating these connections, trigonometry enables the computation of unknown dimensions in right-angled triangles through equations and identities based on these interrelations.

The trigonometric ratio belongs to the relationship between sides within a right triangle. Specifically, the tangent ratio is defined as the division of the length of the side opposite an angle by the length of the adjacent side.

When θ represents the angle formed by the base and the hypotenuse of a right-angled triangle,

tan θ = Perpendicular/Base = sin θ/ cos θ

Tangent 3 Theta Formula

The Tan 3θ formula is a significant triple angle identity within trigonometry, used extensively to address diverse trigonometric and integration problems. This trigonometric function computes the tangent function value for a triple angle, expressible as tan3θ = sin 3θ/cos 3θ because the tangent function is derived from the ratio of the sine and cosine functions. The value of tan3θ cycles after every π/3 radian, represented as tan3θ = tan (3θ + π/3). Its graph appears narrower compared to the graph of tan θ.

The Tangent 3θ formula is obtained through the application of the sum angle formula for Tangent θ and Tangent 2θ ratios.

To establish Tangent 3 Theta Formula tan 3θ = (3 tan θ – tan³θ) / (1 – 3 tan²θ), the process involves expressing 3θ as the sum of (2θ + θ).

Starting with the left-hand side (L.H.S.) = tan 3θ:

= tan (2θ + θ)

L.H.S. = tan 3θ

= tan (2θ + θ)

Use the formula tan (x + y) = (tan x + tan y) / (1 – tan x tan y)

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

Use the formula tan 2x = (2 tan x) / (1 – tan ² x) for tan 2θ.

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

= (tan θ – tan ³ θ + 2 tan θ) / (1 – tan ² θ – 2 tan ² θ)

= (3 tan θ – tan ³ θ) / (1 – 3 tan ² θ)

= R.H.S.

This derives the formula for tangent 3 theta ratio.

W hich equals the right-hand side (R.H.S.), completing the derivation of the formula for the tangent 3θ ratio.

Tangent 3 Theta Formula Solved Examples

Example 1: Given tan tan θ= 5 / 12 ​ .

Using the Tangent 3 Theta Formula

tan 3θ= 3tan θ−tan ³ θ / 1−3tan ² θ

Substituting the given value:

tan 3θ= 3 ⋅ 5 / 12 ​ −( 5 / 12 ​ ) ³ ​ /  1−3 ⋅( 5 / 12 ​ ) ²

tan 3θ= ​ (12 /15 ​ −  125 /1728) ​ ​ / (1− 125 / 144)

tan 3θ= ​ (2160−125 ​/1728 ) ​ / (144−125 / 144)

tan 3θ= ​  (2035/1728) / (19/144)   ​ ​

tan 3θ= 2035×144 ​/ 1728×19

tan 3θ= 293040 ​ / 32832

tan 3θ=  9150 ​ / 1023

Example 2: Given tan θ= −  7/ 24 ​ .

Using the formula tan

tan 3θ= 3tan θ−tan ³ θ ​ /  1−3tan ² θ

Substituting the given value:

tan 3θ= (3 ⋅(− 7/24 ​ )−(− 7/24 ​ ) ³ )​/  1−3 ⋅(−  7/24 ​ ) ²

This calculation will result in the value of tan 3θ using the provided formula.

Example 3: Given3 tanθ= 2/3 ​ .

Using the Tangent 3 Theta Formula tan3θ= 3tanθ−tan ³ θ ​ / 1−3tan ² θ

Substituting the given value:

tan3θ= (3 ⋅ 2/3 ​ −( 2/3 ​ ) ³ )/  (1−3 ⋅( 2/3 ​ ) ² )  ​ =

(2 − 8 /27)  / (1 − 3 ⋅ 4 /9)

46 /27 / 1 − 4 /3

= 46 /27 / -1/3

− 138 / 27

= − 5

Therefore, with the given value of tan ⁡tanθ= 2/3 ​ , the value of tan3θ using the formula is -5.

The Tangent 3 Theta Formula is a fundamental trigonometric identity used to find the tangent of a triple angle (3θ) based on the known tangent value of a single angle (θ). This formula, tan3θ= 3tanθ−tan ³ θ /1−3tan ² θ ​ , plays a crucial role in trigonometry, enabling solutions for various trigonometric and integration problems involving triple angles.

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