Trigonometric Formulas

Trigonometric Formulas

We have learned trigonometric formulae from fifth standard from the basic concepts. And we know that all the trigonometric formulae are based on the sin, cosine, tan and cot angles.

In this article we learn about basic to all relations and identities of the trigonometric functions.

In a right-angled triangle ABC, if ϴ is the angle made by sides AB and AC, and angle ABC is the right angle then we can say that AB is the opposite side, BC is the adjacent side and AC is the hypotenuse.

Now, we can define basic formulae as follows:

  • sinϴ = opposite side/ hypotenuse = AB/ AC
  • cosϴ = adjacent side/ hypotenuse = BC/AC
  • tanϴ = opposite side/ adjacent side= AB/BC
  • cotϴ = adjacent side/ opposite side= BC/AB
  • secϴ= hypotenuse/ adjacent side= AC/BC = 1/cosϴ
  • cosecϴ= hypotenuse/ opposite side= AC/AB= 1/sinϴ

Fundamental identities:

  • Sin2ϴ + cos2ϴ = 1
  • 1 + tan2ϴ = sec2ϴ
  • 1 + cot2ϴ = cosec2ϴ

Some basic formula:

  • Sin(-ϴ) = -sin
  • Cos(-ϴ) = +cosϴ
  • tan(-ϴ) = -tanϴ
  • Cot(-ϴ) = -Cotϴ
  • Sec(-ϴ) = +Secϴ
  • Cosec(-ϴ) = -Cosecϴ

 

  • Cos(A + B) = CosA CosB – SinA SinB
  • Cos(A – B) = CosACosB + SinASinB
  • Sin(A + B) = SinA CoSB + CosA SinB
  • Sin(A – B) = SinA CosB – CosA SinB
  • tan(A + B) = (tanA + tanB)/ (1 – tanA tanB)
  • tan(A – B) = (tanA – tanB)/ (1+ tanA tanB)

 

  • Sin2ϴ= 2Sinϴ Cosϴ
  • Cos2ϴ = Cos2ϴ – Sin2ϴ

            = 1 – 2Sin2ϴ

            = 2Cos2ϴ – 1

 

  • tan(2ϴ) = 2tanϴ/ (1 – tan2ϴ)
  • Sin(2ϴ) = 2tanϴ/ (1 + tan2ϴ)
  • Cos(2ϴ) = (1 – tan2ϴ) / (1 + tan2ϴ)

 

  • Sin(3ϴ) = 3Sinϴ – 4Sin3ϴ
  • Cos(3ϴ) = 4Cos3ϴ – 3Cosϴ
  • tan(3ϴ) = (3tanϴ – tan3ϴ)/ (1 – 3tan2ϴ)

 

  • 2SinA CosB = Sin(A + B) + Sin(A – B)
  • 2CosA SinB = Sin(A + B) – Sin(A – B)
  • 2CosA CosB = Cos(A + B) + Cos(A – B)
  • 2SinA SinB = Cos(A – B) – Cos(A + B)

 

  • Cos2ϴ = (1 + Cos2ϴ)/ 2
  • Sin2ϴ = (1 – Cos2ϴ)/ 2

 

  • SinA + SinB = 2Sin(A+B/2) Cos(A – B/2)
  • SinA – SinB = 2Cos(A + B/2) Sin(A – B/2)
  • CosA + CosB = 2Cos(A + B/2) Cos(A – B/2)
  • CosA – CosB = -2Sin(A + B/2) Sin(A – B/2)

= 2Sin(A + B/2) Sin(B – A/2)

 

In any triangle ABC,

  • Sin (A + B) = SinC
  • Sin (B + C) = SinA
  • Sin (C + A) = SinB

And

  • Cos (A + B) = -CosC
  • Cos (B + C) = – CosA
  • Cos (C + A) = -CosB

 

  • Cos (π/2 – ϴ) = Sinϴ
  • Sin (π/2 – ϴ) = Cosϴ

 

Hyperbolic functions of Sin and Cosine:

  • Sin (iϴ) = i*Sin(hϴ)
  • Cos (iϴ) = Cos(hϴ)
  • Sin(iϴ) = (e – e-iϴ)/ 2i
  • Cos (iϴ) = (e + e-iϴ) / 2
  • Cosh2ϴ – Sinh2ϴ = 1

Updated: September 1, 2021 — 8:42 pm

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