Trigonometric Identities

 

What are Trigonometric Identities?

Trigonometric Identities are the equalities that involve trigonometry functions and holds true for all the values of variables given in the equation. 

There are various distinct trigonometric identities involving the side length as well as the angle of a triangle. The trigonometric identities hold true only for the right-angle triangle.

All the trigonometric identities are based on the six trigonometric ratios. They are sine, cosine, tangent, cosecant, secant, and cotangent. All these trigonometric ratios are defined using the sides of the right triangle, such as an adjacent side, opposite side, and hypotenuse side. All the fundamental trigonometric identities are derived from the six trigonometric ratios.

List of 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.

  • sin2 a + cos2 a = 1
  • 1+tan2 a  = sec2 a
  • cosec2 a = 1 + cot2 a

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 – sin2 θ = 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)]/2


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