Trigonometric Formulas : Understanding Trigonometric Formulas In Detail : Complete List of Trigonometric Formulas

Trigonometry, a word derived from Greek word ‘Trigon’ and ‘Metron’ refers to ‘measuring the sides of a triangle’. This branch of Mathematics studies the relation between angles and sides of triangles. Trigonometry and its formulae is used widely in various fields and has been in use since the 3rd Century BC. From Navigation to Celestial Mechanics to Engineering, Architecture and other various fields trigonometry plays a vital role.

Various geometrical problems are solved using the trigonometric ratios Sine (Sin), Cosine (Cos), Tangent (Tan), Cotangent (Cot), Secant (Sec) and Cosecant (Csc), product identities, Pythagorean Identities etc. There are a number of Trigonometric functions, formulas and ratios like the sign of ratios in different quadrants, sum identities, difference identities, cofunction identities, double angle identities, half-angle identities etc. All of the above formulas, ratios and functions are to be remembered by a student. These are introduced from Class 10th and the concept is further explained in classes 11th & 12th.

Memorizing these formulas help students to solve trigonometry problems easily, which further leads to a good score in Mathematics. As the concept covers quite a large part of the maths syllabus. For easy understanding and access we have provided the trigonometry table and inverse trigonometry formulas.

Trigonometry Formulas List

As trigonometry is the study of relationships between angles and sides of the triangles. The primary triangle studied is the right triangle. In a right angled triangle the three sides are named as Hypotenuse, Opposite and Adjacent. The longest side is called the Hypotenuse, opposite side to the angle is called the Perpendicular (Opposite) and the side where both hypotenuse and opposite sits is the adjacent side.

Here is the complete list of Trigonometry:

Basic Formulas
Reciprocal Identities
Trigonometry Table
Periodic Identities
Co-function Identities
Sum and Difference Identities
Double Angle Identities
Triple Angle Identities
Half Angle Identities
Product Identities
Sum to Product Identities
Inverse Trigonometry Formulas

Basic Formulae

There are six basic ratios which are used for finding the elements in trigonometry which are called trigonometric functions. These are Sine (Sin), Cosine (Cos), Tangent (Tan), Cotangent (Cot), Secant (Sec) and Cosecant (Csc).

By taking right triangle as a reference, these Trigonometric functions are derived as following:

  • sin θ = Opposite Side/Hypotenuse
  • cos θ = Adjacent Side/Hypotenuse
  • tan θ = Opposite Side/Adjacent Side
  • sec θ = Hypotenuse/Adjacent Side
  • cosec θ = Hypotenuse/Opposite Side
  • cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

Reciprocal identities are given as following:

  • cosec θ = 1/sin θ
  • sec θ = 1/cos θ
  • cot θ = 1/tan θ
  • sin θ = 1/cosec θ
  • cos θ = 1/sec θ
  • tan θ = 1/cot θ

These are taken from a right angled triangle. As the height and base of the right triangle are given, we can find out the Sine (sin), Cosine (cos), Tangent (tan), Cotangent (cot), Secant (sec) and Cosecant (csc) values using trigonometric formulas. These reciprocal trigonometric identities can also be derived by using the trigonometric functions.

Trigonometry Table

Below is the table for trigonometry formulas for angles that are commonly used for solving problems.

Angles (In Degrees) 30° 45° 60° 90° 180° 270° 360°
Angles (In Radians) π/6 π/4 π/3 π/2 π 3π/2
sin 0 1/2 1/√2 √3/2 1 0 -1 0
cos 1 √3/2 1/√2 1/2 0 -1 0 1
tan 0 1/√3 1 √3 0 0
cot √3 1 1/√3 0 0
csc 2 √2 2/√3 1 -1
sec 1 2/√3 √2 2 -1 1


Periodicity Identities in Radian

Periodicity Identities are used in shifting the trigonometric functions by one period to the left or right. These are called co-function identities.

  • sin (π/2 – A) = cos A & cos (π/2 – A) = sin A
  • sin (π/2 + A) = cos A & cos (π/2 + A) = – sin A
  • sin (3π/2 – A)  = – cos A & cos (3π/2 – A)  = – sin A
  • sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A
  • sin (π – A) = sin A &  cos (π – A) = – cos A
  • sin (π + A) = – sin A & cos (π + A) = – cos A
  • sin (2π – A) = – sin A & cos (2π – A) = cos A
  • sin (2π + A) = sin A & cos (2π + A) = cos A

TAll trigonometric functions are repetitive in nature, repeating themselves after this periodicity constant. This periodicity constant is different for different trigonometric identities. tan 45° = tan 225° but this is also true for cos 45° and cos 225°.

Co-function Identities (in Degrees

The co-function or periodic identities can also be represented in degrees as:

  • sin(90°−x) = cos x
  • cos(90°−x) = sin x
  • tan(90°−x) = cot x
  • cot(90°−x) = tan x
  • sec(90°−x) = csc x
  • csc(90°−x) = sec x

Sum & Difference Identities

  • sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
  • cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
  • tan(x+y) = (tan x + tan y)/ (1−tan x •tan y)
  • sin(x–y) = sin(x)cos(y)–cos(x)sin(y)
  • cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
  • tan(x−y) = (tan x–tan y)/ (1+tan x • tan y)

Double Angle Identities

  • sin(2x) = 2sin(x) • cos(x) = [2tan x/(1+tan2 x)]
  • cos(2x) = cos2(x)–sin2(x) = [(1-tan2 x)/(1+tan2 x)]
  • cos(2x) = 2cos2(x)−1 = 1–2sin2(x)
  • tan(2x) = [2tan(x)]/ [1−tan2(x)]
  • sec (2x) = secx/(2-sec2 x)
  • csc (2x) = (sec x. csc x)/2

Triple Angle Identities

  • Sin 3x = 3sin x – 4sin3x
  • Cos 3x = 4cos3x-3cos x
  • Tan 3x = [3tanx-tan3x]/[1-3tan2x]

Trigonometry Formulas Major systems

Trigonometric formulas are divided into two significant systems
Trigonometric Identities – These are the formulas which involve trigonometric functions. They are true for all values of the occurring variables. Trigonometric identities are useful for simplifying the expressions involving trigonometric functions.

Trigonometric Ratios – Trigonometric ratios are only considered for right angled triangles. There are six ratios of a right triangle Sine (Sin), Cosine (Cos), Tangent (Tan), Cotangent (Cot), Secant (Sec) and Cosecant (Csc). Trigonometric ratios are used to find the relationship between the measurement of the angles and lengths of the sides of a right triangle.


Frequently Asked Questions – FAQs

What are the basic trigonometric ratios?

Sine, Cosine, Tangent, Cotangent, Secant and Cosecant.

What are formulas for trigonometry ratios?

Sin A = Perpendicular/Hypotenuse
Cos A = Base/Hypotenuse
Tan A = Perpendicular/Base

What are the three main functions in trigonometry?

Sin, Cos and Tan are three main functions in trigonometry.

What are the fundamental trigonometry identities?

The three fundamental identities are:
sin^2 A + cos^2 A = 1
1+tan^2 A = sec^2 A
1+cot^2 A = csc^2 A

Trigonometry formulas are applicable to which triangle?

Right-angled triangle

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