त्रिकोणमिति तालिका और सूत्र

त्रिकोणमिति तालिका और सूत्र



Trigonometry Ratios Table
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
cosec 2 √2 2/√3 1 -1
sec 1 2/√3 √2 2 -1 1



Trigonometry Formulas Involving Periodic Identities(in Radians)

Trigonometry formulas involving periodic identities are used to shift the angles by π/2, π, 2π, etc. All trigonometric identities are cyclic in nature which means that they repeat themselves after a period. This period differs for different trigonometry formulas on periodic identities. For example, tan 30° = tan 210° but the same is not true for cos 30° and cos 210°. You can refer to the trigonometry formulas given below to verify the periodicity of sine and cosine functions.


First Quadrant:


  • sin (π/2 – θ) = cos θ
  • cos (π/2 – θ) = sin θ
  • sin (π/2 + θ) = cos θ
  • cos (π/2 + θ) = – sin θ

Second Quadrant:


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

Third Quadrant:


  • sin (π – θ) = sin θ
  • cos (π – θ) = – cos θ
  • sin (π + θ) = – sin θ
  • cos (π + θ) = – cos θ


Fourth Quadrant:

  • sin (2π – θ) = – sin θ
  • cos (2π – θ) = cos θ
  • sin (2π + θ) = sin θ
  • cos (2π + θ) = cos θ


Trigonometry Formulas Involving Co-function Identities(in Degrees)

The trigonometry formulas on cofunction identities provide the interrelationship between the different trigonometry functions. The co-function trigonometry formulas are represented in degrees below:


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


Trigonometry Formulas Involving Sum and Difference Identities

The sum and difference identities include the trigonometry formulas of sin(x + y), cos(x - y), cot(x + y), etc.


  • 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)


Trigonometry Formulas For Multiple and Sub-Multiple Angles

Trigonometry formulas for multiple and sub-multiple angles can be used to calculate the value of trigonometric functions for half angle, double angle, triple angle, etc.


Trigonometry Formulas Involving Half-Angle Identities

The half of the angle x is presented through the below few trigonometry formulas.


  • sin (x/2) = ±√[(1 - cos x)/2]
  • cos (x/2) = ± √[(1 + cos x)/2]
  • tan (x/2) = ±√[(1 - cos x)/(1 + cos x)]
  • or, tan (x/2) = ±√[(1 - cos x)(1 - cos x)/(1 + cos x)(1 - cos x)]
  • tan (x/2) = ±√[(1 - cos x)2/(1 - cos2x)]
  • ⇒ tan (x/2) = (1 - cos x)/sin x


Trigonometry Formulas Involving Double Angle Identities

The double of the angle x is presented through the below few trigonometry formulas.


  • 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) = sec2 x/(2 - sec2 x)
  • cosec (2x) = (sec x • cosec x)/2

Trigonometry Formulas Involving Triple Angle Identities

The triple of the angle x is presented through the below few trigonometry formulas.


  • sin 3x = 3sin x - 4sin3x
  • cos 3x = 4cos3x - 3cos x
  • tan 3x = [3tanx - tan3x]/[1 - 3tan2x]

Trigonometry Formulas - Sum and Product Identities

Trigonometric formulas for sum or product identities are used to represent the sum of any two trigonometric functions in their product form, or vice-versa.


Trigonometry Formulas Involving Product Identities

  • sinx⋅cosy = [sin(x + y) + sin(x − y)]/2
  • cosx⋅cosy = [cos(x + y) + cos(x − y)]/2
  • sinx⋅siny = [cos(x − y) − cos(x + y)]/2

Trigonometry Formulas Involving Sum to Product Identities

The combination of two acute angles A and B can be presented through the trigonometric ratios, in the below trigonometry formulas.


  • sinx + siny = 2[sin((x + y)/2)cos((x − y)/2)]
  • sinx − siny = 2[cos((x + y)/2)sin((x − y)/2)]
  • cosx + cosy = 2[cos((x + y)/2)cos((x − y)/2)]
  • cosx − cosy = −2[sin((x + y)/2)sin((x − y)/2)]

Inverse Trigonometry Formulas

Using the inverse trigonometry formulas, trigonometric ratios are inverted to create the inverse trigonometric functions, like, sin θ = x and θ = sin −1x. Here x can have values in whole numbers, decimals, fractions, and exponents.


  • sin-1 (-x) = -sin-1 x
  • cos-1 (-x) = π - cos-1 x
  • tan-1 (-x) = -tan-1 x
  • cosec-1 (-x) = -cosec-1 x
  • sec-1 (-x) = π - sec-1 x
  • cot-1 (-x) = π - cot-1 x

Trigonometry Formulas Involving Sine and Cosine Laws

Sine Law: The sine law and the cosine law give a relationship between the sides and angles of a triangle. The sine law gives the ratio of the sides and the angle opposite to the side. As an example, the ratio is taken for the side 'a' and its opposite angle 'A'.


  • (sin A)/a = (sin B)/b = (sin C)/c


Cosine Law: The cosine law helps to find the length of aside, for the given lengths of the other two sides and the included angle. As an example the length 'a' can be found with the help of the other two sides 'b' and 'c' and their included angle 'A'.


  • a2 = b2 + c2 - 2bc cosA
  • b2 = a2 + c2 - 2ac cosB
  • c2 = a2 + b2 - 2ab cosC

where, a, b, c are the lengths of the sides of the triangle, and A, B, C are the angles of the triangle.

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