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Tangent 3 Theta Formula

Tangent 3 Theta or tan 3 theta formula is tan 3θ = (3tanθ – tan3θ)/ (1 – 3tan2θ). It is an important trigonometric formula, that is used to solve various trigonometric problems. In this article we have covered, the Tangent 3 Theta (Tan 3θ) Formula, its derivation and others in detail.

Before, starting with Tangent 3 Theta Formula, let’s first learn in brief about what is a trigonometric ratio.

Trigonometric Ratios Definition

Trigonometric ratios are ratios of sides in a triangle and there are six trigonometric ratios. In a right-angle triangle, the six trigonometric ratios are defined as:

Right-angle-Triangle(1)

Trigonometric Ratios

  1. sin θ = (Opposite Side/Hypotenuse = AB/AC
  2. cos θ = Adjacent Side/Hypotenuse = BC/AC
  3. tan θ = Opposite side/adjacent side = AB/BC
  4. cosec θ = 1/sin θ = Hypotenuse/Opposite Side = AC/AB
  5. sec θ = 1/cos θ = Hypotenuse/Adjacent Side = AC/BC
  6. cot θ = 1/tan θ = Adjacent Side/Opposite Side = BC/AB

Tangent 3 Theta (Tan 3θ) Formula

Tan3θ is a triple angle identity in trigonometry. It is a crucial trigonometric identity that is used to solve a variety of trigonometric and integration issues. It is a trigonometric function that returns the tan function value for a triple angle. It may alternatively be written as tan3θ = sin 3θ/cos 3θ since the tangent function is a ratio of the sine and cosine functions.

Tangent-3-Theta-Formula

Tangent 3 Theta Formula

The value of tan 3θ repeats after every π/3 radians, tan 3θ = tan (3θ + π/3). Its graph is thinner than graph of tan θ and the graph of tan x and tan 3x is added below:

Tan 3θ Formula

Tan 3θ Graphs

[Tex]\tan3\theta=\frac{3\tan\theta-\tan^3\theta}{1-3\tan^2\theta}[/Tex]

Derivation on Tangent 3 Theta Formula

Formula for Tangent 3 theta is derived by using the sum angle formula for Tangent theta and Tangent 2 theta ratios.

To demonstrate that tan 3θ = (3 tan θ – tan3θ) / (1 – 3 tan2θ), we write 3θ as (2θ + θ).

L.H.S

= tan 3θ = tan (2θ + θ)

Use the formula tan (x + y) = (tan x + tan y) / (1 – tan x tan y)

= (tan 2θ + tan θ)/ (1 – tan 2θ tan θ)

Use the formula tan 2x = (2 tan x) / (1 – tan2x) for tan 2θ.

= [(2 tan θ / (1 – tan2θ)) + tan θ] / [1 – (2 tan θ / (1 – tan2θ)) tan θ]

= (tan θ – tan3θ + 2 tan θ) / (1 – tan2θ – 2 tan2θ)

= (3 tan θ – tan3θ) / (1 – 3 tan2θ) 

= R.H.S.

This derives the formula for tangent 3 theta ratio.

Important Notes on Tan 3x Formula:

tan3x

(3 tan θ – tan3θ) / (1 – 3 tan2θ) 

d/dx (tan3x)

3 sec2(3x)

∫tan3x dx

(1/3) ln |sec 3x| + C

Article Related to Tan3x Formula:

Problems on Tan 3x Formula

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

Solution:

We have, tan θ = 3/4.

Using the formula we get,

tan 3θ = (3 tan θ – tan3θ) / (1 – 3 tan2θ)

= (3 (3/4) – (3/4)3) / (1 – 3 (3/4)2)

= (9/4 – 27/64) / (1 – 3 (9/16))

= (117/64) / (-11/16)

= -117/44

Problem 2. If tan θ = 12/5, find the value of tan 3θ using the formula.

Solution:

We have, tan θ = 12/5.

Using the formula we get,

tan 3θ = (3 tan θ – tan3θ) / (1 – 3 tan2θ)

= (3 (12/5) – (12/5)3) / (1 – 3 (12/5)2)

= (36/5 – 1728/125) / (1 – 3 (144/25))

= (-828/125) / (-407/25)

= 828/2035

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

Solution:

We have, sin θ = 4/5.

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

Using the formula we get,

tan 3θ = (3 tan θ – tan3θ) / (1 – 3 tan2θ)

= (3 (4/3) – (4/3)3) / (1 – 3 (4/3)2)

= (4 – 64/27) / (1 – 3 (16/9))

= (44/27) / (-13/3)

= -44/117

Problem 4. If cos θ = 12/13, find the value of tan 3θ using the formula.

Solution:

We have, cos θ = 12/13.

Clearly sin θ = 5/13. Hence we have, tan θ = 5/12.

Using the formula we get,

tan 3θ = (3 tan θ – tan3θ) / (1 – 3 tan2θ)

= (3 (5/12) – (5/12)3) / (1 – 3 (5/12)2)

= (5/4 – 125/1728) / (1 – 3 (25/144))

= (2035/1728) / (19/144)

= 2035/228

Problem 5. If sec θ = 17/8, find the value of tan 3θ using the formula.

Solution:

We have, sec θ = 17/8.

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

tan θ = √((289/64) – 1)

= √(225/64)

= 15/8

Using the formula we get,

tan 3θ = (3 tan θ – tan3θ) / (1 – 3 tan2θ)

= (3 (15/8) – (15/8)3) / (1 – 3 (15/8)2)

= (45/8 – 3375/1728) / (1 – 3 (225/64))

= (72/25) / (64/675)

= 243/8

Problem 6. Find the value of tan 135° using the tan 3x formula.

Solution:

We have to find the value of tan 135°.

Let us take 3x = 135

=> x = 135/3

=> x = 45°

We know, tan 45° = 1.

Using the tan 3x formula, we get

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

= (3(1) – 13) / (1 – 3 (12))

= (3 – 1) / (1 – 3)

= 2 / (-2)

= -1

Problem 7. Find the value of tan 75° using the tan 3x formula.

Solution:

We have to find the value of tan 75°.

Let us take 3x = 75

=> x = 75/3

=> x = 25°

We know, tan 25° = 0.47.

Using the tan 3x formula, we get

tan 75° = (3 tan 25° – tan325°) / (1- 3 tan225°)

= (3(0.47) – (0.47)3) / (1 – 3 (0.47)2)

= (1.41 – 0.10) / (1 – 3 (0.22))

= (1.31) / (0.34)

= 3.85

FAQs on Tangent 3 Theta Formula

How to prove tan 3θ?

Tan 3θ can be proved using tan (A + B) identity.

What is the formula of tan 3x?

tan 3x formula is: tan3x = (3 tan x – tan3x) / (1 – 3 tan2x)

What is Tanx formula?

The formula for tanx is: tan x = sin x/ cos x.


 


Reffered: https://www.geeksforgeeks.org

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