FIND ALL SOLUTIONS OF THE EQUATION EXPRESS THE SOLUTIONS IN RADIANS

About "Find All Solutions of the Equation Express the Solutions in Radians"

Find All Solutions of the Equation Express the Solutions in Radians

The equations containing trigonometric functions of unknown angles are known as trigonometric equations. A solution of trigonometric equation is the value of unknown angle that satisfies the equation.

General Solution :

The solution of a trigonometric equation giving all the admissible values obtained with the help of periodicity of a trigonometric function is called the general solution of the equation.

Trigonometric equation

sin θ = 0

cos θ = 0

tan θ = 0

sin θ = sinα, where

α ∈ [−π/2, π/2]

cos θ = cos α, where α ∈ [0,π]

tan θ = tanα, where

α ∈ (−π/2, π/2)

General solution

θ = nπ; n ∈ Z

θ = (2n + 1) π/2; n ∈ Z

θ = nπ; n ∈ Z


θ = nπ + (−1)n α, n ∈ Z

θ = 2nπ ± α, n ∈ Z


θ = nπ + α, n ∈ Z

Find All Solutions of the Equation Express the Solutions in Radians - Examples

Question 1 :

Solve the following equations:

(v) sin 2θ − cos 2θ − sin θ + cos θ = 0

Solution :

sin 2θ − cos 2θ − sin θ + cos θ  =  0

sin 2θ − sin θ + cos θ − cos 2θ  =  0

Let us use the formula for sin C - sin D and cos C - cos D

 sin C - sin D  =  2 cos (C + D)/2 sin (C - D)/2

cos C - cos D  =  2 sin (C + D)/2 sin (C - D)/2

sin 2θ − sin θ  =  2 cos 3θ/2 sin θ/2  -----(1)

cos θ − cos 2θ  =  2 sin 3θ/2 sin θ/2  -----(2)

(1) + (2)

   =  2 cos 3θ/2 sin θ/2 + 2 sin 3θ/2 sin θ/2

  =  2 sin θ/2 [cos 3θ/2 + sin 3θ/2]  

sin θ/2  =  0 

 sin θ/2  =  0

θ/2  =  nπ

θ  =  2nπ

cos 3θ/2 + sin 3θ/2  =  0

cos 3θ/2  =  - sin 3θ/2 

sin 3θ/2/cos 3θ/2  =  -1

tan 3θ/2  =  -1

a = - π/4

 θ  =  nπ + a

3θ/2  =  nπ - π/4

3θ  =  2nπ - π/2

θ  =  2nπ/3 - π/6

Hence the solution is {2nπ, 2nπ/3 - π/6}.

(vi)  sin θ + cos θ = 2

Solution :

sin θ + cos θ = 2

Divide by 2 on both sides

(1/2) sin θ + (1/2) cos θ = 1

cos (π/4) cos θ - sin (π/4) sin θ = cos 0

cos (π/4 - θ)  =  cos 0

θ = 2nπ + a

θ = 2nπ + π/4

θ = (8n + 1)π/4

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