Trigonometric Equations

An equation involving one or more trigonometrical ratios of unknown angle is called a trigonometric equation

e.g. cos2x – 4 sinx = 1

It is to be noted that a trigonometrical identity is satisfied for every value of the unknown angle whereas trigonometric equation is satisfied only for some values (finite or infinite) of unknown angle.

e.g. sin2x + cos2x = 1 is a trigonometrical identity as it is satisfied for every value of x ∈ R. 

Solution of a Trigonometric Equation:

A value of the unknown angle which satisfies the given equation is called a solution of the equation

e.g. sinθ = 1/2 ⇒ θ = π/6. 

General Solution:

Since trigonometrical functions are periodic functions, solutions of trigonometric equations can be generalized with the help of the periodicity of the trigonometrical functions.

The solution consisting of all possible solutions of a trigonometric equation is called its general solution.

We use the following formulae for solving the trigonometric equations:

  • sinθ =  0         ⇒  θ =  nπ ,
  • cosθ = 0         ⇒  θ = (2n+1),
  • tanθ = 0         ⇒   θ =  nπ ,
  • sinθ = sinα       ⇒ θ =  nπ + (– 1)nα ,  where α ∈ [– π/2 ,  π/2]
  • cosθ = cosα ⇒ θ = 2nπ  ±  α , where α ∈ [ 0 , π]
  • tanθ = tanα      ⇒  θ = nπ + α ,   where α ∈ (– π/2, π/2)
  • sin2θ = sin2α , cos2θ =  cos2α ,  tan2 θ = tan2α ⇒ θ = nπ ± α ,
  • sinθ = 1 ⇒  θ = (4n + 1)π/2 ,
  • sinθ = –1 ⇒ θ = (4n – 1)π/2 ,
  • cosθ = 1    ⇒  θ = 2nπ ,
  • cosθ = – 1 ⇒  θ = (2n + 1)π ,
  • sinθ = sinα  and  cosθ =  cosα ⇒  θ = 2nπ + α .

Note:

  • Everywhere in this chapter n is taken as an integer, If not stated otherwise.
  • The general solution should be given unless the solution is required in a specified interval or range.
  • a is taken as the principal value of the angle. Numerically least angle is called the principal value.

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