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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](/content/math/trigonometry/trigonometric-identities/index.html).

## 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

- The **tan3x formula** is  \( \tan 3x =  \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x} \).
- It is derived from \( \tan(2x + x) \) using addition and double-angle identities.
- Its derivative is \( 3\sec^2 3x \); its integral is \( \frac{1}{3}\ln|\sec 3x| + C \).
- \( \tan 3x \) (triple angle) is not \( \tan^3 x \) (tangent cubed).
- The period of \( \tan 3x \) is \( \frac{\pi}{3} \).

## 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.