Tangent Formula

Tangent Formula

Tangent formulas are used for computing the tangent functions in mathematics. The tangent function is also abbreviated as a tan. The tangent function occurs by dividing the perpendicular by the base. 

About the Topic 

In trigonometry, there are six types of ratios or functions. The tangent function is an example of those functions. Various trigonometry identities and formulas compute the tangent function. There are various tangent function formulas computed from these identities and formulas. 

Assume the right-angled triangle. A right-angled triangle contains three sides: base, perpendicular, and hypotenuse. The simple formula of tan function is, 

tan = Perpendicular/Base 

On the other hand, another formula for the tan function is computed by considering the acute angle in the right-angled triangle. 

tan = opposite side / adjacent side

In this formula, the opposite side is the side on the right-angled triangle, opposite to the angle x. The adjacent side is the side on the right-angled triangle, adjacent to angle x. There are many formulas for adjacent angles. All these formulas are described deeply in the next section.

Formula 

General Formula: 

(Based on the sides of trigonometry) 

tan = Perpendicular/Base 

(Respective to the angle x on the right-angled triangle) 

tan = opposite side / adjacent side

Sin and cos Tangent Formula:

tan x = (sin x) / (cos x)

Tangent Formulas Using Reciprocal Identity:

tan x = 1 / (cot x)

Tangent Formulas Using Pythagorean Identity:

tan x = ± √( sec²x – 1)

Cofunction Identity Tangent Formula:

tan x = cot (90° – x) (OR)

tan x = cot (π/2 – x)

Sum and difference Tangent Formula:

tan (A – B) = (tan A – tan B) / (1 + tan A tan B)

tan (A + B) = (tan A + tan B) / (1 – tan A tan B)

Double angle Tangent Formula:

tan 2x = (2 tan x) / (1 – tan²x)

Triple Angle Tangent Formula:

tan 3x = (3 tan x – tan³x) / (1 – 3tan²x)

Half Angle Tangent Formula:

tan (x/2) = (1 – cos x) / ( sin x)

tan (x/2) =± √[(1 – cos x) / (1 + cos x)]

Solved Examples

1. How is the cos sin tangent formula derived? 

According to the ratios in trigonometry:

sin x = (perpendicular) / (hypotenuse)

cos = (base) / (hypotenuse)

tan = Perpendicular/base

(sin x) / (cos x) = [ (perpendicular) / (hypotenuse) ] / [ (base) / (hypotenuse) ]

tan x = Perpendicular/Base 

2. How to derive a tangent formula using Pythagoras theorem? 

The tangent formula is derived from the Pythagoras theorem, by using the identity:

sec²x – tan²x = 1

On subtracting 1 from both sides in the given identity:

-tan²x = 1 – sec²x

On performing the product of the whole expression with – 1.

tan²x = sec²x – 1

Taking the square root on both sides:

tan x = ± √( sec²x – 1)