Find All Solutions of Tan 3x 1

Tan 3x

Tan 3x is one of the triple angle identities in trigonometry. It is an important trigonometric identity that is used to solve various trigonometric and integration problems. Tan 3x formula is given by tan 3x = (3 tan x - tan³x) / (1 - 3 tan²x) and it can be derived using angle sum formula of tan function. Tan 3x can also be expressed in terms of sin and cos as tan 3x = sin 3x/cos 3x.

In this article, we will explore the concept of the tan 3x formula, its application, and proof. We will also solve some examples based on tan 3x for a better understanding of the tan 3x identity.

1.	What is Tan 3x?
2.	Tan 3x Formula
3.	Proof of Tan 3x Formula
4.	Tan 3x Differentiation and Integration
5.	FAQs on Tan 3x

What is Tan 3x?

Tan 3x is a trigonometric function that gives the value of the tan function for a triple angle. The graph of tan 3x is narrower than 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 tan 3x is π/3 radians. So, the value of tan 3x repeats after every π/3 radians, that is, tan 3x = tan (3x + π/3). The formula for tan 3x can be derived using the tan (a + b) formula.

Tan 3x Formula

Tan 3x formula is an important trigonometric formula given as tan 3x = (3 tan x - tan³x) / (1 - 3 tan²x) that is used to solve various mathematical problems and complex integrations. The formula for tan 3x can also be written as tan 3x = sin 3x/cos 3x as tangent function is a ratio of the sine function and cosine function.

tan 3x formula

Proof of Tan 3x Formula

As we have studied, we know that the formula for tan 3x is (3 tan x - tan³x) / (1 - 3 tan²x). 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 to prove that tan 3x = (3 tan x - tan³x) / (1 - 3 tan²x):

tan (A + B) = (tan A + tan B) / (1 - tan A tan B)
tan 2x = (2 tan x) / (1 - tan²x)

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 - tan²x)}] / [1 - {(2 tan x) / (1 - tan²x)} tan x] {Substituing the value tan 2x = (2 tan x) / (1 - tan²x)}

On solving, we have

tan(2x + x) = (tan x - tan³x + 2 tan x) / (1 - tan²x - 2 tan²x)

= (3 tan x - tan³x) / (1- 3 tan²x)

Thus, tan 3x = (3 tan x - tan³x) / (1- 3 tan²x)

Hence we have derived the formula for tan 3x.

Tan 3x Differentiation and Integration

Next, we will determine the derivative and integral of tan 3x. First, let us differentiate tan 3x with respect to x using the chain rule method. We know that the derivative of tan x is sec²x and the derivative of ax is a, where a is a constant. Using this, we have

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

= sec²(3x) × 3

= 3 sec²(3x)

Therefore, the derivative of tan 3x is 3 sec²(3x).

Next, for tan 3x 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

∫tan 3x 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 Tan 3x

The formula for tan 3x is tan 3x = (3 tan x - tan³x) / (1- 3 tan²x).
The derivative of tan 3x is 3 sec²(3x).
The integral of tan 3x is (-1/3) ln |cos 3x| + C or (1/3) ln |sec 3x| + C.

Related Topics on Tan 3x

Trigonometric Formulas
Sin 3x
Cos 3x

Examples Using Tan 3x

Example 1: Calculate the value of tan 135° using tan 3x formula.

Solution: Assume 3x = 135° ⇒ x = 45°. We know that tan 45° = 1. Using tan 3x formula, we have

tan 135° = (3 tan 45° - tan³45°) / (1- 3 tan²45°)

= (3 × 1 - 1³)/(1 - 3 × 1²)

= (3 - 1)/(1 - 3)

= 2/(-2)

= -1

Answer: tan 135° = -1
Example 2: Prove that tan 180° is equal to 0 using tan 3x formula.

Solution: Assume 3x = 180° ⇒ x = 60°. We know that tan 60° = √3. Using tan 3x formula, we have

tan 180° = (3 tan 60° - tan³60°) / (1- 3 tan²60°)

= (3 × √3 - √3³)/(1 - 3 × √3²)

= (3√3 - 3√3)/(1 - 9)

= 0/(-8)

= 0

Answer: Hence tan 180° is equal to 0

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Practices Questions on Tan 3x

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FAQs on Tan 3x

What is Tan 3x Formula in Trigonometry?

The formula for tan 3x is given by tan 3x = (3 tan x - tan³x) / (1- 3 tan²x). It can also be written as tan 3x = sin 3x/cos 3x.

How to Find Tan 3x Formula?

Tan 3x 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 - tan²x) and substituting 3x = 2x + x.

What is Domain and Range of Tan 3x?

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

How Do You Find the Derivative of Tan 3x?

The derivative of tan 3x is 3 sec²(3x). It can be calculated using the chain rule method. Tan 3x differentiation can also be done using the quotient rule on tan 3x = sin 3x/cos 3x.

What is Tan 3x Integration?

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

What is the Period of Tan 3x?

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

Find All Solutions of Tan 3x 1

Source: https://www.cuemath.com/trigonometry/tan3x/

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