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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