Trigonometric Identities PDF

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Trigonometric Identities - Description

Dear readers, here we are offering Trigonometric Identities PDF to all of you. Trigonometric Identities Formula and Trigonometric Identities are those trigonometric relations that are true for all the values of the angles used in them on which the used trigonometric ratios are defined. In modern mathematics, the main task of the Trigonometry Formula is to measure the sides of the triangle and to establish mutual relations between the sides and angles. If the basic formula of trigonometry is strong then there will be no problem with trigonometry-related formulas and trigonometry questions.

Therefore, to make your basics strong, all the formulas of Trigonometry have been explained in this article. Do follow all the trigonometry formulas and tricks mentioned here and also share them with your friends. Questions based on the formula of trigonometric functions are often asked in board exams or competitive exams.

Trigonometric Identities PDF Overview

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.

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
  • tan⁡(�+�)=tan⁡�+tan⁡�1–tan⁡�.tan⁡�
  • tan⁡(�–�)=tan⁡�–tan⁡�1+tan⁡�.tan⁡�

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)]/2

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