Derivation on Tangent 3 Theta Formula
Formula for Tangent 3 theta is derived by using the sum angle formula for Tangent theta and Tangent 2 theta ratios.
To demonstrate that tan 3θ = (3 tan θ – tan3θ) / (1 – 3 tan2θ), we write 3θ as (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 – 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.
This derives the formula for tangent 3 theta ratio.
Important Notes on Tan 3x Formula:
tan3x | (3 tan θ – tan3θ) / (1 – 3 tan2θ) |
|---|---|
d/dx (tan3x) | 3 sec2(3x) |
∫tan3x dx | (1/3) ln |sec 3x| + C |
Article Related to Tan3x Formula:
Tangent 3 Theta Formula
Tangent 3 Theta or tan 3 theta formula is tan 3θ = (3tanθ – tan3θ)/ (1 – 3tan2θ). It is an important trigonometric formula, that is used to solve various trigonometric problems. In this article we have covered, the Tangent 3 Theta (Tan 3θ) Formula, its derivation and others in detail.
Before, starting with Tangent 3 Theta Formula, let’s first learn in brief about what is a trigonometric ratio.
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