# Tan2x Formula

5/5 - (1 bình chọn)

Tan2x is an important trigonometric function. Tan2x formula is one of the very commonly used double angle trigonometric formulas and can be expressed in terms of different trigonometric functions such as tan x, cos x, and sin x. As we know that tan x is the ratio of sine and cosine function, therefore the tan2x identity can also be expressed as the ratio of sin 2x and cos 2x.

In this article, we will learn the tan2x and tan^2x formula, its proof, and express it in terms of different trigonometric functions. We will also explore the graph of tan2x and its period along with the concept of tan square x and solve examples for a better understanding.

Sin 2x, Cos 2x, Tan 2x are the trigonometric formulas known as double angle formulas because, they have double angles in their trigonometric functions. Let’s understand it by practising it through the solved examples.

Mục Lục

## What is Tan2x in Trigonometry?

Tan2x is a trigonometric function and has a formula that is used to solve various problems in trigonometry. Tan2x is an important double angle formula, that is, a trigonometry formula where the angle is doubled. It can be expressed in terms of tan x and also as a ratio of sin2x and cos2x. Since the reciprocal of tan x is cot x, therefore we can write tan2x as the reciprocal of cot 2x, that is, tan2x = 1/cot2x. Let us see the tan2x formula:

## Tan2x Formula

We can express the tan2x formula in two different forms. It can be expressed in terms of tangent function only and as a combination of the sine function and cosine function. The formula for tan2x identity is given as:

• tan2x = 2tan x / (1−tan2x)
• tan2x = sin 2x/cos 2x

## Tan2x Formula Proof

Tan2x formula can be derived using two different methods. First, we will use the angle addition formula for the tangent function to derive the tan2x identity. Note that we can write the double angle 2x as 2x = x + x. We will use the following trigonometric formula to prove the formula for tan2x:

• tan (a + b) = (tan a + tan b)/(1 – tan a tan b)

We have

tan2x = tan (x + x)

= (tan x + tan x)/(1 – tan x tan x)

= 2 tan x/(1 – tan2x)

Hence, we have derived the tan2x formula using the angle sum formula of the tangent function.

### Tan2x Identity Proof Using Sin and Cos

Now, we will derive the tan2x formula by expressing tan as a ratio of sin and cos. We will use the following trigonometric formulas:

• tan x = sin x/ cos x
• sin 2x = 2 sin x cos x
• cos 2x = cos2x – sin2x

Using the above formulas, we have

tan2x = sin 2x/cos 2x

= 2 sin x cos x/(cos2x – sin2x)

Divide the numerator and denominator of 2 sin x cos x/(1 – 2 sin2x) by cos2x

tan2x = [2 sin x cos x/cos2x]/[(cos2x – sin2x)/cos2x]

= (2 sin x/cos x)/(1 – sin2x/cos2x)

= 2 tan x/(1 – tan2x)

Hence we have derived the tan2x formula by expressing it as a ratio of sin 2x and cos 2x.

## Tan2x Graph

The graph of tan2x looks similar to the graph of tan x. We know that the period of tan x is π. Since the period of tan bx is given by π/|b|, the period of tan2x is π/2. Given below is the graph of tan2x and as we can observe from the graph, the value of tan2x repeats after every π/2 radians. Also, the value of tanx is equal to zero whenever x is an integral multiple of π, therefore tan2x is equal to zero whenever 2x = nπ, where n is an integer which implies the below graph has x-intercepts at x = nπ/2.

## Tan^2x (Tan Square x)

Tan^2x is the square of the trigonometric function tanx. We can derive the tan square x formulas using the trigonometric identities and formulas which consist of tan^2x. As we know that tan x can be expressed as the ratio of sinx and cosx, therefore we can express tan^2x can as the ratio of sin square x and cos square x. We use the tan^2x formula to solve complex integration and differentiation problems and simplify trigonometric expressions. In the next section, we will derive and discuss the formula of tan square x.

## Tan^2x Formula

Now, we have a trigonometric identity 1 + tan^2x = sec^2x which implies tan^2x = sec^2x – 1. Since tan x can be expressed as the ratio of sine function and cosine function, therefore we can write tans square x as the ratio of sin square x and cos square x, therefore we have tan^2x = sin^2x / cos^2x. Also, we know that tan x is the reciprocal of cot x, therefore we can write tan^2x = 1/cot^2x. Hence, the list of tan^2x formula is:

• tan^2x = sec^2x – 1 ⇒ tan2x = sec2x – 1
• tan^2x = sin^2x / cos^2x ⇒ tan2x = sin2x/cos2x
• tan^2x = 1/cot^2x ⇒ tan2x = 1/cot2x

## Tan2x in Terms of Cos

We can derive the tan2x formula in terms of cos. We will use the following trigonometric formulas to express tan2x in terms of cos x.

• tan x = sin x/ cos x
• sin 2x = 2 sin x cos x
• cos 2x = 2 cos2x – 1
• sin x = √(1 – cos2x)

Using the above formulas, we have

tan2x = sin 2x/ cos 2x

= 2 sin x cos x/(2 cos2x – 1)

= [2 √(1 – cos2x) cos x/(2 cos2x – 1)]

Similarly, we can write tan2x in terms of sin using the trigonometric identities.

tan2x = [2 sin x/(1 – 2 sin2x)]√(1 – sin2x)

Important Notes on Tan 2x Formula

• tan2x = 2tan x / (1 − tan2x)
• tan2x = sin 2x/cos 2x
• The derivative of tan2x is 2 sec2(2x)
• The integral of tan2x is (-1/2) ln |cos 2x| + C or (1/2) ln |sec 2x| + C.

## Tan2x Examples

Example 2: Find the value of tan2x if sin x = 12/13 and cos x = 5/13

Solution: We know that tan2x = sin 2x/ cos 2x

= 2 sin x cos x /(cos2x – sin2x)

= [2 × (12/13) × (5/13)] / [(25/169) – (144/169)]

= (120/169) / (-119/169)

= -120/119

Answer: tan2x = -120/119

Example 3: Evaluate the derivative and integral of tan2x.

Solution: We know that the derivative of trigonometric function tan x is given by sec2x. The derivative of tan2x can be calculated using different methods such as the chain rule and quotient rule. Let us determine the derivative of tan2x using the chain rule.

d(tan2x)/dx = d(tan 2x)/d(2x) × d(2x)/dx

= sec22x × 2

= 2 sec2(2x)

Now, we will determine the integral of tan2x. We know that the integral of tan x is -ln |cos x| + C or ln |sec x| + C. Using the formulas of integration, the integral of tan2x is given by,

∫ tan2x = (-1/2) ln |cos 2x| + C

= (1/2) ln |sec 2x| + C

Answer: Hence the derivative of tan2x is 2 sec2(2x) and the integral of tan2x is given by (-1/2) ln |cos 2x| + C or (1/2) ln |sec 2x| + C.

## FAQs on Tan2x Formula

### What is Tan2x in Trigonometry?

Tan2x formula is one of the very commonly used double angle trigonometric formulas and can be expressed in terms of different trigonometric functions such as tan x, cos x, and sin x. The commonly used formula for tan2x is tan2x = 2tan x / (1 − tan2x).

### What is Tan2x Formula?

Tan2x formula can be expressed in different ways such as:

• tan2x = 2tan x / (1 − tan2x)
• tan2x = sin 2x/cos 2x
• tan2x = [2 cos x/(2 cos2x – 1)]√(1 – cos2x)
• tan2x = [2 sin x/(1 – 2 sin2x)]√(1 – sin2x)

### What is the Domain and Range of Tan2x?

The domain of tan2x consists of all real numbers except those where tan2x is not defined. So, the domain of tan 2x is R – {(2n + 1)π/4, n ∈ Z}. The range of tan2x is all real numbers, that is, R.

### How to Find the Derivative of Tan2x?

The derivative of tan2x can be calculated using different methods such as the chain rule and quotient rule. The derivative of tan2x is 2 sec2(2x).

### What is the Integral of Tan2x?

The integral of tan2x is given by (-1/2) ln |cos 2x| + C or (1/2) ln |sec 2x| + C.

### What is the Formula of Tan2x in Terms of Cos x and Sin x?

Tan2x formula in terms of sin x and cos x is given by tan2x = 2 sin x cos x /(cos2x – sin2x)

### What is Tan Square x?

Tan square x is the square of the trigonometric function tan x and is mathematically written as tan^2x. We use the tan^2x formula to solve complex integration and differentiation problems and simplify trigonometric expressions.

### What is Tan^2x Formula?

The formula for tan^2x are:

• tan^2x = sec^2x – 1 ⇒ tan2x = sec2x – 1
• tan^2x = sin^2x / cos^2x ⇒ tan2x = sin2x/cos2x
• tan^2x = 1/cot^2x ⇒ tan2x = 1/cot2x

### What is the Difference Between tan2x and tan^2x?

Tan2x is a double angle trigonometric formula which gives the value of the tangent function for the compound angle 2x. On the other hand, tan^2x is the whole square of the trigonometric function tanx. The value of tan2x can be both positive and negative whereas the value of tan^2x is always non-negative as the square of a number is never negative.

### Tangent Trigonometric Ratio

A trigonometric ratio is defined as the ratio of the lengths of any two sides of a right triangle. These ratios relate the ratio of sides of a right triangle to the angles in trigonometry. The tangent ratio is calculated by computing the ratio of the length of the opposite side of an angle divided by the length of the adjacent side. It is denoted by the abbreviation tan.

If θ is the angle that lies between the base and hypotenuse of a right-angled triangle then,

tan θ = Perpendicular/Base = sin θ/ cos θ

Here, perpendicular is the side opposite to the angle and base is the side adjacent to it.

### Tan2x Formula

In trigonometry, Tan2x is a double angle identity. Because the tangent function is a ratio of the sine and cosine functions, it may also be represented as tan2x = sin 2x/cos 2x. It’s an important trigonometric identity that’s utilized to solve a wide range of trigonometric and integration problems. After every 2π radians, the value of tan2x repeats, tan2x = tan (2x + 2π). Its graph is noticeably thinner than tan x’s. It’s a trigonometric function that returns a double angle’s tan function value.

tan 2x = 2 tan x/(1 – tan2 x)

The formula can also be expressed in the terms of sine and cosine functions.

We know,

tan 2x = sin 2x / cos 2x

tan 2x = 2 sin x cos x/(cos2 x – sin2 x)

Derivation

The formula for tan 2x can be derived by using the double angle formulas for sine and cosine functions.

We already know, tan x = sin x/cos x

Substituting x with 2x in the equation, we get

tan 2x = sin 2x/cos 2x ⇢ (1)

Put sin 2x = 2 sin x cos x and cos 2x = cos2 x – sin2 x in the equation (1).

tan 2x = 2 sin x cos x/(cos2 x – sin2 x)

Dividing numerator and denominator on R.H.S. by cos2 x, we get

tan 2x = [(2 sin x cos x)/cos2 x]/[(cos2 x – sin2 x)/(cos2 x)]

tan 2x = [(2 sin x)/cos x]/(1 – sin2 x/cos2 x)

tan 2x = 2 tan x/(1 – tan2 x)

Thus the formula for tan 2x ratio is derived.

### Sample Problems

Problem 1: If tan x = 3/4, find the value of tan 2x using the formula.

Solution:

We have, tan x = 3/4.

Using the formula we get,

tan 2x = 2 tan x/(1 – tan2 x)

= (2 (3/4))/(1 – (3/4)2)

= (6/4)/(1 – 9/16)

= (6/4)/(7/16)

= 24/7

Problem 2: If tan x = 12/5, find the value of tan 2x using the formula.

Solution:

We have, tan x = 12/5.

Using the formula we get,

tan 2x = 2 tan x/(1 – tan2 x)

= (2 (12/5))/(1 – (12/5)2)

= (24/5)/(1 – 144/25)

= (24/5)/(-119/25)

= -120/119

Problem 3: If sin x = 4/5, find the value of tan 2x using the formula.

Solution:

We have, sin x = 4/5.

Clearly cos x = 3/5. Hence we have, tan x = 4/3.

Using the formula we get,

tan 2x = 2 tan x/(1 – tan2 x)

= (2 (4/3))/(1 – (4/3)2)

= (8/3)/(1 – 16/9)

= (8/3)/(-7/9)

= -24/7

Problem 4: If cos x = 12/13, find the value of tan 2x using the formula.

Solution:

We have, cos x = 12/13.

Clearly sin x = 5/13.

Hence we have, tan x = 5/12.

Using the formula we get,

tan 2x = 2 tan x/(1 – tan2 x)

= (2 (5/12))/(1 – (5/12)2)

= (5/6)/(1 – 25/144)

= (5/6)/(119/144)

= 120/119

Problem 5: If sec x = 17/8, find the value of tan 2x using the formula.

Solution:

We have, sec x = 17/8.

Find the value of tan x using the formula sec2 x = 1 + tan2 x.

tan x = √((289/64) – 1)

= √(225/64)

= 15/8

Using the formula we get,

tan 2x = 2 tan x/(1 – tan2 x)

= (2 (15/8))/(1 – (15/8)2)

= (15/4)/(1 – 225/64)

= (15/4)/(-161/64)

= -240/161

## Introduction

Tan 2x is a double-angle trigonometric function that can be represented as tan x, sin x, and cos x. The tan angle formula is used to calculate the angle of a right triangle. It has been found that the tangent to the angle of a right triangle is equal to the length of the opposite side divided by the length of the adjacent side. It can be expressed as tan 2x or as a ratio of sin 2x to cos 2x. Since the reciprocal of tan x is cot x, we can write tan 2x as the reciprocal of cot 2x, that is, tan 2x = 1/cot 2x.

### What is Tan Function?

Tangent Angle Formula is normally used to calculate the angle of the right-angle triangle. In any given right triangle, the tangent of an angle is the length of the opposite side divided by the length of the adjacent side.

The tangent function, along with sine and cosine, is known to be one of the three most common trigonometric functions. In any given right angle, the tangent of an angle is the length of the opposite side (O) divided by the length of the adjacent side (A).

In a formula, we can simply write it as ‘tan’.

Now tan θ = O/ A

Where,

O = Opposite side

A = Adjacent side

### Tan 2x Formula

Trigonometric Formulas like Sin 2x, Cos 2x, and Tan 2x are known as double angle formulas because these formulas have double angles in their trigonometric functions.

The tan 2x expression can be expressed in two different formats. This can only be expressed as a tangent function and as a combination of sine and cosine functions. The formula for the tan 2x identity is:

Let’s know how to derive the double angle tan 2x formula.

### Tan 2x Formula Derivation

Tan 2x expressions can be derived by using two different methods. First, we derive tan 2x identity using the angle addition formula of the tangent function. Note that you can write a double angle 2x as 2x = x + x. Prove the expression for tan 2x using the following trigonometric expression.

Therefore, we have derived the tan 2x formula using the angle sum formula of the tangent function.

### Values of Tan in Different Values

• For angles with their terminal arm in Quadrant II, since the sine function is positive and the cosine is negative, the tangent is negative.

• For angles with their terminal arm in Quadrant III, since the sine function is negative and the cosine is negative, the tangent is positive.

• For angles with their terminal arm in Quadrant IV, since sine function is negative and cosine is positive, the tangent is negative.

### Tan 2x Identity Proof in Form of sinx and cosx

Derive the tan 2x formula by expressing tan as a ratio of sin and cos by using the following trigonometric formulas:

tan x = sin x/cos x

sin 2x = 2 sin x . cos x

cos 2x = cos 2x – sin2x

Using the formulas given above, we get

tan 2x = sin 2x/cos 2x

= (2 sin x.cos x)/(cos2x–sin2x)

Dividing both the numerator and denominator of the equation 2 sin x . cos x/(1 –2 sin2x) by cos2x

tan 2x = [2 sin x. cos x/cos2x]/[(cos2x–sin2x)/cos2x]

= (2 sin x/cos x)/[1– (sin2x/cos2x)]

= 2 tan x/(1 –tan2x)

Hence, the tan 2x formula is derived and expressed as a ratio of sin 2x and cos 2x.

### List of all Tan 2x formulas:

1. tan(2x)= 2tanx/1–tanx.tanx

2. tan 2x = sin 2x/cos 2x

3. Derivative of tan 2x is 2 sec2(2x)

4. Integral of tan2x is (–1/2) ln |cos 2x| + C or (1/2) ln |sec 2x| + C.

### Solved Examples

Question 1) Calculate the tangent angle of a right triangle whose adjacent side and opposite sides are 8 cm and 6 cm, respectively?

Solution Given,

Adjacent side (A)= 8 cm

Opposite side (O)= 6 cm

Using the formula of a tangent:

Tan θ = O/A

Tan θ = 6/8

Tan θ = 0.75

Question 2) Calculate the tangent angle of a right triangle whose adjacent sides and opposite sides are 10 cm and 6 cm, respectively?

Solution  Given,

Adjacent side (A) = 10 cm

Opposite side (O) = 6 cm

Using the formula of a tangent:

Tan θ = O/A

Tan θ = 6/10

Tan θ = 0.6

Question 3) If tan x = 3/4, find the value of tan 2x.

Solution  I know the formula tan 2x = 2.tan x / (1–tan2x).

Therefore, if tan x = ¾, then  tan2x = 9/16.

Tan 2x = 2 (3/4) / 1(9/16)

= (3/2) / (7/16)

= 24/7

Question 4) If sin x = 12/13 and cos x = 5/13, determine the value of tan 2x.

Solution  tan 2x = 2×(12/13)×(5/13)

/ (25/169)(144/169)

= (120/169) /–(119/169)

= –120/119

Answer: tan 2x = – 120/119

## FAQs on Tan 2x Formula

1. If tan x =2/3 , then find the value of tan 2x.

The formula we know is tan 2x = 2.tan x / (1–tan 2x).

Therefore, if tan x = 2/3,

then  tan2x = 4/9.

Substituting in the formula tan 2x = 2.tan x / (1 − tan2x)

= 2 (2/3) / (1–4/9)

= (4/3) / (5/9)

= 36/15

1. How would you define tan 2x?

Tan 2x is a double-angle trigonometric function that  can be represented as tan x, sin x, cos x. The tan angle formula is used to calculate the angle of a right triangle. It has been found that the tangent to the angle of a right  triangle is equal to the length of the opposite side divided by the length of the adjacent side. It can be expressed as tan x or as a ratio of sin 2x to cos 2x. Since the reciprocal of tan x is cot x, we can write tan 2x as the reciprocal of cot 2x, that is tan 2x = 1 / cot 2x.

1. What is the difference between Tan 2x and Tan2x?

Tan 2x is a double-angle formula for trigonometric functions that specifies the value of the tangent function with a composite angle of 2x. On the other hand, tan2x is the entire square of the trigonometric function tan x. The value of tan 2x can be both positive and negative, but the value of tan2x is always positive because the square of the number can never be negative.

1. What is the formula for integral of Tan 2x ?

The formula for the integral of tan 2x is given by,

(–1/2) ln |cos 2x| + C or (1/2) ln |sec 2x| + C.

1. List out all the Tan 2x formulas.
• Tan(2x)= 2tanx/(1–tanx.tanx)
• Tan 2x = sin 2x/cos 2x
• Derivative of tan 2x is 2 sec2(2x)
• Integral of tan 2x is (–1/2) ln |cos 2x| + C or (1/2) ln |sec 2x| + C.

## Tan 2x Formula Proof

Now, let’s look at the tan 2x trigonometric formulae derivation in detail.

With the help of sine and  cosine function, tan 2x formula can be derived as follows:

We know that tan x = sin x/cos x

Hence, tan 2x = sin 2x / cos 2x …(1)

Using sine and cosine double angle formula,

Sin 2x = 2 sin x cos x

Cos 2x = cos2 x – sin2x

Now, substitute the above two formulas in (1), we get

Tan 2x = (2 sin x cos x) /(cos2 x – sin2 x)

Now divide both numerator and denominator of  (2 sin x cos x) /(cos2 x – sin2 x) by cos2 x, we get

Tan 2x = [(2 sin x cos x)/cos2x ]/[(cos2 x – sin2 x)/cos2x]

Tan 2x = [2 sin x / cos x] /[1-(sin2x/cos2x)]

Since (sin x/ cos x) = tan x and  (sin2x / cos2x) = tan2x, the above equation can be written as:

Tan 2x = 2 tan x / (1-tan2x)

Hence, the tan 2x formula can be derived with the help of sine and cosine functions.

### Examples on Tan 2x Formula

Example 1:

Find the value of tan 2x, if tan x = 5.

Solution:

Given: Tan x = 5

To find: Tan 2x

We know that the formula for tan 2x is:

Tan 2x = 2 tan x / (1-tan2x)

Now, substitute the known values in the formula, we get

Tan 2x = 2(5) / (1-(5)2)

Tan 2x = 10/ (1-25)

Tan 2x = 10/-24

Tan 2x = -5/12.

Hence, the value fo tan 2x is -5/12, if tan x = 5.

Example 2:

Find the value of tan 2x, if sin x = 5/12 and cos x = 3/12.

Solution:

Given: sin x = 5/12 and cos x = 3/12.

So, we have sin2x = 25/144, and cos 2x = 9/144

The formula for tan 2x, in terms of sine and cosine functions is:

Tan 2x = Sin 2x / Cos 2x

Tan 2x = (2 sin x cos x) /(cos2 x – sin2 x)

Now, substitute the known values in the formula, we get

Tan 2x = [2(5/12)(3/12)] / [(9/144) – (25/144)]

Tan 2x = (30/144)/(-16/144)

Tan 2x = -30/16, which on simplification, we get

Tan 2x = -15/8.

Hence, the value of tan 2x is -15/8.

## Tan2x Formula in terms of Tanx, Sinx, Cosx [with Proof]

Tan2x formula in terms of tanx is as follows:

formula in terms of sinx and cosx along with some examples. These formulas are very useful to solve trigonometric equations and simplify trigonometric expressions.

## Tan2x in Terms of Tanx

To derive the tan 2x formula, we will use the trigonometric formula below:

At first, we will write 2x as x+x, and then we will apply the above formula. By doing so, we get that

## Tan2x Formula Proof

Now, we will prove the tan2x formula using sin2x and cos2x formulas which are given below.

## Tan2x in Terms of Sinx

Now we will derive the tan2x formula, in terms of sinx. In order to do so, we will go through the following way:

## Tan2x in Terms of Cosx

Now we will derive the tan2x formula, in terms of cosx. In order to do so, we will go through the following way:

### The Importance of Mathematics Formulas for Students

Some of the most brilliant minds create mathematical equations for a reason. They help students respond to questions quickly and properly. It also makes it much easier to discover a solution to a sum than beginning from scratch. Mathematics formulae have the following advantages:

It is students’ responsibility to follow the school’s time-sensitive curriculum. Students’ knowledge is assessed regularly through various assessments such as unit, half-yearly, and final exams. Mathematics equations are essential to guarantee that students prepare the subject matter on time and with a buffer for revision.

While reviewing, a student is unlikely to solve a large number of problems with a pen and paper. As a result, to get a quick overview of sums and how to solve them, students must be familiar with formulas, which are the keys to finding the correct solutions.

Students do not have the luxury of deriving a complete formula to solve a question during examinations, implying that they cannot begin at step 1. To finish their question paper in the allocated time, they must memorise and remember the formula, which helps students with time management and scheduling.

##### FAQs (Frequently Asked Questions)
1. What is the Tan2x Formula in trigonometry?

The Tan2x Formula is a fairly popular double-angle trigonometric formula that can be stated in terms of several trigonometric functions such as tan x, cos x, and sin x. Tan2x is generally defined as

The Tan2x Formula  = 2tan x / (1 tan2x).

2. What Is the Distinction Between Tan^^2x and Tan2x?

The Tan2x Formula is a trigonometric double-angle formula that calculates the tangent function for the compound angle 2x. Tan^2x, on the other hand, is the whole square of the trigonometric function tanx. The Tan2x Formula can have both positive and negative values, whereas tan2^x is always non-negative, since the square of an integer is never negative.

3. What is a Tan Function?

Normally, the Tangent Angle Formula is used to compute the angle of a right-angle triangle. The tangent of an angle in any given right triangle is the length of the opposing side divided by the length of the neighbouring side. The tangent function is one of the three most popular trigonometric functions, along with sine and cosine. The tangent of an angle in any given right angle is the length of the opposing side divided by the length of the neighbouring side.

Students can just write it as ‘tan’ in a formula.

The tan formula is calculated by dividing opposed sides by adjacent sides.

Trigonometric formulae such as Sin 2x, Cos 2x, and Tan 2x are known as double-angle formulas because their trigonometric functions include double angles.

There are two ways to represent the tan 2x expression. This can only be written as a tangent function and as a sine and cosine function combination. The formula for the tan 2x identity is as follows:

Tan 2x equals sin 2x/cos 2x

GIA SƯ TOÁN BẰNG TIẾNG ANH

GIA SƯ DẠY SAT

Math Formulas

Mọi chi tiết liên hệ với chúng tôi :
TRUNG TÂM GIA SƯ TÂM TÀI ĐỨC
Các số điện thoại tư vấn cho Phụ Huynh :
Điện Thoại : 091 62 65 673 hoặc 01634 136 810
Các số điện thoại tư vấn cho Gia sư :
Điện thoại : 0902 968 024 hoặc 0908 290 601