## The Formula for Tan3A

Trigonometry is the branch of mathematics concerned with the functions of angles with their applications in calculations. In trigonometry, we will find six functions of an angle, namely sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These functions are interdependent on the size of the triangle. These functions are used to get unknown angles and distances from the measured angles in the geometric figures.

Trigonometry is used to compute distances and angles in different fields such as surveying, artillery, range finding, map making, and astronomy. Problems that consist of distances and angles in a plane are covered in plane trigonometry. The problems involving distances and angles in more than one plane of three-dimensional space are covered in spherical trigonometry. In this topic, we will study the derivations and formulas of tan3A.

### What is the Formula of Tan3A?

As said earlier, trigonometry has 6 functions of an angle: sin, cos, tan, sec, cot, and csc, expressed in terms of sides of a right-angled triangle for a particular angle θ.

sinθ = \[\frac{opposite side}{hypotenuse}\]

cotθ = \[\frac{adjacent side}{hypotenuse}\]

tanθ = \[\frac{opposite side}{adjacent side}\]

cosec θ = \[\frac{Hypotenuse}{opposite side}\]

secθ = \[\frac{Hypotenuse}{Opposite side}\]

cotθ = \[\frac{adjacent side}{Opposite side}\]

### Reciprocal Identities

The Reciprocal Identities are given as:

cosec θ = \[\frac{1}{sin\Theta }\]

sec θ = \[\frac{1}{cos\Theta }\]

cot θ = \[\frac{1}{tan\Theta }\]

sin θ = \[\frac{1}{cosec\Theta }\]

cos θ = \[\frac{1}{sec\Theta }\]

tan θ = \[\frac{1}{cot\Theta }\]

Let us see how to express multiple angles of tan3A concerning tan A or A.

The trigonometric function of tan3A concerning tan A is called a double angle formula.

If A is an angle, then the formula of 3A = \[\frac{3tanA-tan^{3}A}{1-3tan^{2}A}\]

Now latest prove the above formula step by step:

Tan 3A = \[tan\left ( 2A+A \right )\]

= \[\frac{tan2A-A}{1-tan2A.tanA}\]

= \[\frac{\frac{2tanA}{1-tan^{2}A}+tanA}{1-\frac{2tanA}{1-tan^{2}A}.tanA}\]

= \[\frac{2tanA+tanA-tan^{3}A}{1-tan^{3}A-2tan^{2}A}\]

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

Therefore, tan3A = \[\frac{3tanA-tan^{3}A}{1-3tan^{2}A}\]

In the above formula, the angle on the right-hand side of the formula is about one-third of the left-hand side’s angle. So

tan30\[^{\circ}\] = \[\frac{3tan10^{\circ}-tan^{3}10^{\circ}}{1-3tan^{2}10^{\circ}}\]

### Conclusion

In this topic, we got to know the detailed derivation and what is the formula of tan3A. Understand the different terms of these formulas to determine how to use them for solving trigonometric problems.

**1. What is the Value of Tan 3A After Including A = B = C in the Formula?**

From the formula of tan3A,

the value of tan(A+B+C) = \[\frac{tanA+tanB+tanC-tanAtanBtanC}{1-tanAtanB-tanBtanC-tanCtanA}\]

**2. What is the formula of Tan3A, Cos3A, and Sin3A?**

The formula of sin 3A = 3sinA – 4sin³A

The formula of cos 3A = 4cos³A – 3cosA

The formula of tan 3A = 3tanA - \[\frac{tan^{3}A}{1-3tan^{2}A}\]

**3. If Tan A = 4, then Find the Value of Tan 3A?**

The formula of tanA = 3tanA - \[\frac{tan^{3}A}{1-3tan^{2}A}\]

= 3(4) - \[\frac{4^{3}}{1}\] - 3(4)²

= 12 - \[\frac{64}{1}\] - 48

= \[\frac{-52}{-47}\]

= \[\frac{52}{47}\] = 1.10