Tan3x - Formula, Proof & Examples (Triple Angle)

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Tan3x - Formula, Proof & Examples (Triple Angle)

TL;DR

The tan3x formula is ( \tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x} ), the triple-angle identity for tangent. This article derives it from the angle-addition and double-angle formulas, gives its derivative (( 3\sec^2 3x )) and integration, works through examples, and separates tan3x (triple angle) from tan³x (tangent cubed).

What Is Tan3x?

Tan3x is the tangent of three times an angle ( x ), and the tan3x formula expresses it entirely in terms of ( \tan x ):

( \tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x} )

It is one of the triple-angle identities, the tangent counterpart to the double-angle tan2x formula. Equivalently, since tangent is the ratio of sine to cosine, ( \tan 3x = \frac{\sin 3x}{\cos 3x} ) — but the form above is the one used for simplification, because it needs only ( \tan x ).

A quick warning that the notation invites: ( \tan 3x ) is not ( \tan^3 x ). The first is the tangent of the angle 3x; the second is (( \tan x ))^3, tangent cubed. They appear together inside this very formula, so reading them apart is essential. The identity sits in the broader set of trigonometric identities.

How Do You Prove the Tan3x Formula?

The derivation splits 3x into 2x+x and applies two identities you already have. Each step sits on its own line so nothing is skipped.

Step 1 — Split the angle and apply the addition formula

Using ( \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A\tan B} ) with ( A=2x, B=x ):

( \tan 3x = \tan(2x+x) = \frac{\tan 2x + \tan x}{1 - \tan 2x\tan x} )

Step 2 — Substitute the double-angle formula

Replace ( \tan 2x = \frac{2\tan x}{1 - \tan^2 x} ):

( \tan 3x = \frac{\frac{2\tan x}{1 - \tan^2 x} + \tan x}{1 - \frac{2\tan x}{1 - \tan^2 x}\cdot\tan x} )

Step 3 — Clear the inner fractions

Multiply numerator and denominator by (1−tan²x):

Numerator: ( 2\tan x + \tan x(1 - \tan^2 x) = 3\tan x - \tan^3 x )

Denominator: ( (1 - \tan^2 x) - 2\tan^2 x = 1 - 3\tan^2 x )

Step 4 — Assemble

( \tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x} )

Letting ( t=\tan x ), the compact form is ( \tan 3x = \frac{3t - t^3}{1 - 3t^2} ).

Tan3x in Calculus

Two operations come up often enough to state directly.

Derivative. By the chain rule on the angle 3x:

( \frac{d}{dx}(\tan 3x) = 3\sec^2 3x )

Integration. Writing ( \tan 3x = \frac{\sin 3x}{\cos 3x} ) and substituting ( u=\cos 3x ):

( \int \tan 3x , dx = -\frac{1}{3}\ln|\cos 3x| + C = \frac{1}{3}\ln|\sec 3x| + C )

Examples of Tan3x

Example 1

Find ( \tan 3x ) when ( \tan x = \frac{1}{\sqrt{3}} ) (i.e. x=30°).

Let ( t=\frac{1}{\sqrt{3}} ), so ( t^2=\frac{1}{3} ) and ( t^3=\frac{1}{3\sqrt{3}} ).

( \tan 3x = \frac{3\cdot\frac{1}{\sqrt{3}} - \frac{1}{3\sqrt{3}}}{1 - 3\cdot\frac{1}{3}} = \text{undefined} )

Example 2

Evaluate ( \tan 3x ) at ( \tan x = 1 ) (i.e. x=45°). Wrong path first.

The tempting move is: ( \tan 3x = 3\tan x = 3 ). Check it: 3x=135°; ( \tan 135° = -1 ), not 3. So, rewriting:

( \tan 3x = \frac{3(1) - (1)^3}{1 - 3(1)^2} = -1 )

Example 3

Compute ( \tan 135° ) using the formula with x=45°.

This is the same setup as Example 2:

( \tan 135° = -1 )

Example 4

Prove ( \tan 180° = 0 ) using the formula with x=60°.

Letting ( t=\sqrt{3} ):

( \tan 180° = \frac{3\sqrt{3} - 3\sqrt{3}}{1 - 3(3)} = 0 )

Example 5

Show that ( \tan 3x, \tan 2x, \tan x = \tan 3x - \tan 2x - \tan x ).

Starting from ( \tan 3x = \tan(2x+x) = \frac{\tan 2x + \tan x}{1 - \tan 2x\tan x} ), rearranging gives the identity.

Example 6

Find the period of ( y=\tan 3x ).

The period of ( \tan x ) is ( \pi ); therefore, Period = ( \frac{\pi}{3}).

Key Takeaways

Common Mistakes With Tan3x

  1. Confusing ( \tan 3x ) with ( \tan^3 x ).
  2. Assuming ( \tan 3x = 3 \tan x ).
  3. Dropping a sign during the expansion.

Frequently Asked Questions

What is the tan3x formula? ( \tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x} )

Is tan3x the same as tan³x? No.

What is the derivative of tan3x? ( 3\sec^2 3x )

What is the period of tan3x? ( \frac{\pi}{3} )

How is the tan3x formula derived? By applying the angle addition formula and substituting the double-angle identity.