Tan3x Formula

Trigonometry is a branch of mathematics that shows the relation between triangle side lengths and angles. There are six trigonometric functions which are sine function, cosine function, tangent function, cotangent function, secant function and cosecant function. Trigonometry has a lot of different identities these trigonometric functions are used to form various trigonometric identities. These trigonometric identities are frequently used to rewrite trigonometrical formulas in order to simplify them, find a more useful form of them, or solve an equation. 

Here tan 3x formula is one of the trigonometric identity.

The formula of tan3x is given as follows:

\[ tan 3x = \frac{3tan x - tan^{3} x}{1- 3tan^{2} x} \]

Let us discuss the step by step derivation of the tan3x formula.

Derivation of Tan 3x Formula

Step 1: Rewriting tan 3x as tan (2x+x)

tan 3x = tan (2x+x) …………..(1)

Step 2: Using the trigonometry identity tan (A+B)

The trigonometric identity tan (A+B) is given as follows:

\[ tan (A+B) = \frac{tan A + tan B}{1- tan A tan B} \]

Now expanding  tan (2x+x) using the trigonometry identity tan (A+B) the equation (1) simplifies as follows:

\[ tan 3x = tan (2x + x ) = \frac{tan 2x + tan x}{1- tan2x tan x} \]…………….(2) 

Step 3: Now use the trigonometric identity tan 2x formula to simplify the equation (2)

We know that \[ tan 2x = \frac{2tan x}{1-tan^{2} x} \]

Now substituting this value in equation (2) we get

\[ tan 3x = tan(2x + x) = \frac{\frac{2tanx}{1-tan^{2}x} + tanx } {1- \frac{2tanx}{1-tan^{2} x} tanx} \]

On solving we get

\[ \text{tan 3x} = tan (2x +x ) = \frac{2tan x + tan x (1-tan^{2} x )}{1- tan^{2} x - 2tan ^{2} x} \]

\[ \text{tan 3x} = \frac{3tan x - tan^{3} x}{1- 3tan^{2} x} \]

This formula is known as the tan3x formula. 

Further simplifying the formula of tan3x we will get tan cube x formula as follows:

\[ tan^{3} x = 3tanx - tan3x(1- 3tan^{2}x ) \]

Problems on Tan3x Formula

1. If tan 600= \[\sqrt{3}\]. Find the value of tan 1800 using tan 3x formula.

Ans: Here the value of x = 600

The tan 3x formula is given as follows:

\[ \text{tan 3x} = \frac{3tanx - tan^{3}x}{1- 3tan^{2} x} \]

Now substituting the value of x = 600 we get

\[ \text {tan180} = \frac{3tan60 - tan^{3} 60}{1- 3tan^{2} 60} \]

The value of tan 600= \[\sqrt{3}\].  So substituting this value we get

\[ \text{tan180} = \frac{3\times \sqrt{3} - (\sqrt{3})^{3}}{1-3(\sqrt{2})^{2}} \]

\[ \text{tan 180} = \frac{0}{1- 3(\sqrt{3})^{2}} \]

tan 1800 = 0

Therefore the value of tan 1800 is 0.

Conclusion

Trigonometric identities are always true trigonometry equations that are commonly utilised to solve trigonometry and geometry problems and comprehend numerous mathematical aspects. Knowing important trigonometric identities aids us in remembering and comprehending important mathematical principles as well as solving a variety of math problems.