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Trigonometry Formulas & Trigonometric Identities

Trigonometry Formulas & Identities

Basic Trigonometric Function Formulas

There are basically 6 ratios used for finding the elements in Trigonometry. They are called trigonometric functions. The six trigonometric functions are sine, cosine, secant, co-secant, tangent and co-tangent.

By using a right-angled triangle as a reference, the trigonometric functions and identities are derived:

  • 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

The Reciprocal Identities are given as:

  • 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

Trigonometry 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
csc 2 √2 2/√3 1 -1
sec 1 2/√3 √2 2 -1 1

Periodicity Identities (in Radians)

These formulas are used to shift the angles by π/2, π, 2π, etc. They are also 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

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]

Trigonometric Identities

There are various identities in trigonometry which are used to solve many trigonometric problems. Using these trigonometric identities or formulas, complex trigonometric questions can be solved quickly. Let us see all the fundamental trigonometric identities here.

Reciprocal Trigonometric Identities

The reciprocal trigonometric identities are:

  • Sin θ = 1/Csc θ or Csc θ = 1/Sin θ
  • Cos θ = 1/Sec θ or Sec θ = 1/Cos θ
  • Tan θ = 1/Cot θ or Cot θ = 1/Tan θ

Pythagorean Trigonometric Identities

There are three Pythagorean trigonometric identities in trigonometry that are based on the right-triangle theorem or Pythagoras theorem.

  • sina + cosa = 1
  • 1+tan2 a  = sec2 a
  • coseca = 1 + cota

Ratio Trigonometric Identities

The trigonometric ratio identities are:

  • Tan θ = Sin θ/Cos θ
  • Cot θ = Cos θ/Sin θ

Trigonometric Identities of Opposite Angles

The list of opposite angle trigonometric identities are:

  • Sin (-θ) = – Sin θ
  • Cos (-θ) = Cos θ
  • Tan (-θ) = – Tan θ
  • Cot (-θ) = – Cot θ
  • Sec (-θ) = Sec θ
  • Csc (-θ) = -Csc θ

Trigonometric Identities of Complementary Angles

In geometry, two angles are complementary if their sum is equal to 90 degrees. Similarly, when we can learn here the trigonometric identities for complementary angles.

  • Sin (90 – θ) = Cos θ
  • Cos (90 – θ) = Sin θ
  • Tan (90 – θ) = Cot θ
  • Cot ( 90 – θ) = Tan θ
  • Sec (90 – θ) = Csc θ
  • Csc (90 – θ) = Sec θ

Trigonometric Identities of Supplementary Angles

Two angles are supplementary if their sum is equal to 90 degrees. Similarly, when we can learn here the trigonometric identities for supplementary angles.

  • sin (180°- θ) = sinθ
  • cos (180°- θ) = -cos θ
  • cosec (180°- θ) = cosec θ
  • sec (180°- θ)= -sec θ
  • tan (180°- θ) = -tan θ
  • cot (180°- θ) = -cot θ

Sum and Difference of Angles Trigonometric Identities

Consider two angles , α and β, the trigonometric sum and difference identities are as follows:

  • sin(α+β)=sin(α).cos(β)+cos(α).sin(β)
  • sin(α–β)=sinα.cosβ–cosα.sinβ
  • cos(α+β)=cosα.cosβ–sinα.sinβ
  • cos(α–β)=cosα.cosβ+sinα.sinβ

Double Angle Trigonometric Identities

If the angles are doubled, then the trigonometric identities for sin, cos and tan are:

  • sin 2θ = 2 sinθ cosθ
  • cos 2θ = cos2θ – sinθ = 2 cos2θ – 1 = 1 – 2sin2 θ
  • tan 2θ = (2tanθ)/(1 – tan2θ)

Half Angle Identities

If the angles are halved, then the trigonometric identities for sin, cos and tan are:

  • sin (θ/2) = ±√[(1 – cosθ)/2]
  • cos (θ/2) = ±√(1 + cosθ)/2
  • tan (θ/2) = ±√[(1 – cosθ)(1 + cosθ)]

Product-Sum Trigonometric Identities

The product-sum trigonometric identities change the sum or difference of sines or cosines into a product of sines and cosines.

  • Sin A + Sin B = 2 Sin(A+B)/2 . Cos(A-B)/2
  • Cos A + Cos B = 2 Cos(A+B)/2 . Cos(A-B)/2
  • Sin A – Sin B = 2 Cos(A+B)/2 . Sin(A-B)/2
  • Cos A – Cos B = -2 Sin(A+B)/2 . Sin(A-B)/2

Trigonometric Identities of Products

These identities are:

  • Sin A. Sin B = [Cos (A – B) – Cos (A + B)]/2
  • Sin A. Cos B = [Sin (A + B) – Sin (A – B)]/2
  • Cos A. Cos B = [Cos (A + B) – Cos (A – B)]