What is the Formula of tan 3x

What is the formula of tan 3x? 

The formula of tan 3x = (3 tan x – tan³x)/ (1- 3 tan²x).

Steps for calculating:

1. Start by calculating the value of tan x.

2. Then calculate the numerator by multiplying 3 with tan x and subtracting the whole square of tan x from it.

3. For calculating the denominator, multiply 3 with the whole square of tan x and subtract the answer from 1.

4. Divide the result obtained in the 2nd and 3rd steps, and you will get the value of tan 3x.

Features of tan 3x:

• tan 3x is important because it is used in different integrals and trigonometric problems.

• The graph of tan 3x is also smaller than the tan x graph because the period of tan x and tan 3x is different.

• tan 3x can also be written as Sin 3x/ Cos 3x.

• The derivative of tan 3x comes to be 3Sec² 3x.

How to derive the formula of tan 3x:

We can use the angle sum formula of trigonometric equations to derive the formula.

1. Write tan 3x as tan (x+2x).

2. Now we’ll put the formula for tan (A+B), which is (tan A + tan B)/ 1- tan A tan B.

3. Using the formula given in point no. 2, tan (2x +x) will become (tan x + tan 2x)/ 1- tan x tan 2x.

4. Again tan 2x can be broken as tan (x+x). So the equation looks like:

In the numerator: tan x + (2 tan x/ 1 – tan²x).

In the denominator: 1-(2tan²x / 1 – tan²x)

Note: You can also directly put the formula for tan 2x if you remember.

5. Solving the numerator, we get (tan x – tan³x + 2 tan x)/ (1- tan²x) and in the denominator, the answer comes out to be (1- tan²x – 2tan²x)/ (1- tan²x).

6. Dividing the numerator and denominator, we get (3 tan x- tan³x)/ 1-3 tan²x.