**Tan3x** is one of the triple angle identities in trigonometry. It is an important trigonometric identity that is used lớn solve various trigonometric and integration problems. Tan3x formula is given by tan3x = (3 tan x - tan^{3}x) / (1 - 3 tan^{2}x) and it can be derived using angle sum formula of tan function. Tan3x can also be expressed in terms of sin and cos as tan3x = sin 3x/cos 3x.

In this article, we will explore the concept of the tan3x formula, its application, and proof. We will also solve some examples based on tan3x for a better understanding of the tan3x identity.

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1. | What is Tan3x? |

2. | Tan3x Formula |

3. | Proof of Tan3x Formula |

4. | Tan3x Differentiation and Integration |

5. | FAQs on Tan3x |

## What is Tan3x?

Tan3x is a trigonometric function that gives the value of the tan function for a triple angle. The graph of tan3x is narrower than vãn the graph of tan x. We know that the period of tan x is π radians and the period of tan bx is given by π/|b|. Hence the period of tan3x is π/3 radians. So, the value of tan3x repeats after every π/3 radians, that is, tan3x = tan (3x + π/3). The formula for tan3x can be derived using the tan (a + b) formula.

## Tan3x Formula

Tan3x formula is an important trigonometric formula given as tan3x = (3 tan x - tan^{3}x) / (1 - 3 tan^{2}x) that is used lớn solve various mathematical problems and complex integrations. The formula for tan3x can also be written as tan3x = sin 3x/cos 3x as tangent function is a ratio of the sine function and cosine function.

## Proof of Tan3x Formula

As we have studied, we know that the formula for tan3x is (3 tan x - tan^{3}x) / (1 - 3 tan^{2}x). Now, we will prove this formula using the angle sum formula for tan and tan 2x formula. Note that we can write 3x as 3x = 3x + x. Also, we will use the following formulas lớn prove that tan3x = (3 tan x - tan^{3}x) / (1 - 3 tan^{2}x):

- tan (A + B) = (tan A + tan B) / (1 - tan A tan B)
- tan 2x = (2 tan x) / (1 - tan
^{2}x)

Let us represent tan(3x) as follows,

⇒ tan(3x) = tan (2x + x)

Using the above trigonometric formula for tan (A + B), we have

tan (2x + x) = [tan(2x) + tan(x)] / [1 - tan(2x) tan(x)]

tan(2x + x) = [tan x + {(2 tan x) / (1 - tan^{2}x)}] / [1 - {(2 tan x) / (1 - tan^{2}x)} tan x] {Substituing the value tan 2x = (2 tan x) / (1 - tan^{2}x)}

On solving, we have

tan(2x + x) = (tan x - tan^{3}x + 2 tan x) / (1 - tan^{2}x - 2 tan^{2}x)

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

Thus, tan3x = (3 tan x - tan^{3}x) / (1- 3 tan^{2}x)

Hence we have derived the formula for tan3x.

## Tan3x Differentiation and Integration

Next, we will determine the derivative and integral of tan3x. First, let us differentiate tan3x with respect lớn x using the chain rule method. We know that the derivative of tan x is sec^{2}x and the derivative of ax is a, where a is a constant. Using this, we have

d(tan3x)/dx = d(tan3x)/d(3x) × d(3x)/dx

= sec^{2}(3x) × 3

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= 3 sec^{2}(3x)

Therefore, the derivative of tan 3x is 3 sec^{2}(3x).

Next, for tan3x integration, we will express tan 3x as a ratio of sin 3x and cos 3x, that is, tan 3x = sin 3x/cos 3x. Also, we will use the fact that the derivative of cos 3x is -3 sin 3x. Using these facts and formulas, we have

∫tan3x dx = ∫(sin 3x / cos 3x) dx

= ∫(3 sin 3x / 3 cos 3x) dx [Multiplying the numerator and denominator by 3]

Assume cos 3x = u ⇒ -3 sin 3x = du ⇒ 3 sin 3x dx = -du. We have

∫tan 3x dx = ∫(3 sin 3x / 3 cos 3x) dx

= (-1/3) ∫ (1/u) du

= (-1/3) ln |u| du

= (-1/3) ln |cos 3x| + C

= (1/3) ln |sec 3x| + C

Thus, the tan 3x integration is (-1/3) ln |cos 3x| + C or (1/3) ln |sec 3x| + C.

**Important Notes on Tan3x**

- The formula for tan3x is tan 3x = (3 tan x - tan
^{3}x) / (1- 3 tan^{2}x). - The derivative of tan3x is 3 sec
^{2}(3x). - The integral of tan3x is (-1/3) ln |cos 3x| + C or (1/3) ln |sec 3x| + C.

**☛ Related Topics:**

- Trigonometric Formulas
- Sin 3x
- Cos 3x

## FAQs on Tan3x

### What is Tan3x Formula in Trigonometry?

The formula for **tan3x** is given by tan3x = (3 tan x - tan^{3}x) / (1- 3 tan^{2}x). It can also be written as tan 3x = sin 3x/cos 3x.

### How lớn Find Tan3x Formula?

Tan3x formula can be derived using the formulas tan (A + B) = (tan A + tan B) / (1 - tan A tan B) and tan 2x = (2 tan x) / (1 - tan^{2}x) and substituting 3x = 2x + x.

### What is the Domain and Range of Tan3x?

We know that the domain name of tan x is all real numbers except nπ + π/2 and the range of tan x is all real numbers. So, the domain name of tan3x is all real numbers except (1/3)(nπ + π/2) = π/6 + nπ/3, where n is an integer and the range of tan3x is all real numbers.

### How Do You Find the Derivative of Tan3x?

The derivative of tan3x is 3 sec^{2}(3x). It can be calculated using the chain rule method. Tan3x differentiation can also be done using the quotient rule on tan 3x = sin 3x/cos 3x.

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### What is Tan3x Integration?

The integral of tan3x is (-1/3) ln |cos 3x| + C. It is also equal lớn (1/3) ln |sec 3x| + C using the properties of the logarithmic function.

### What is the Period of Tan3x?

The period of tan bx is given by π/|b|. Therefore, the period of tan3x function is π/3 and the period of tan x is π.

### What is Tan^3x?

Tan^3x is a trigonometric function that gives the value of the whole cube of the tangent function. It is called tan cube x and its formula can be determined using the tan3x formula. On simplifying the tan3x formula, the formula for tan^3x is 3tanx - tan3x (1 - 3tan^2x).

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