Tan2x Formula

Trigonometry is a field of mathematics that determines the angles and unknown sides of a triangle using trigonometric ratios. It simplifies the estimation of unknown dimensions of a right-angled triangle by using equations and identities based on this relation. The words ‘Trigonon’ and ‘Metron,’ which represent a triangle and a measurement, respectively, make up the term trigonometry. This branch of mathematics is studied using trigonometric ratios such as sine, cosine, tangent, cotangent, secant, and cosecant. It’s the study of the relationship between the sides and angles of a right-angled triangle.

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