SOLVING TRIGONOMETRIC EQUATIONS

Solve the equation on the interval 0 ≤ θ < 2π.

Problem 1 :

cos x = 0

Solution :

cos x = 0

x = cos^-1(0)

For which angle of cosine, we get 0.

cos (π/2) = 0

General solution for,

x = (2n + 1) π/2

if n = 0,

x = (2(0) + 1) π/2

x = π/2

n = 1,

x = (2(1) + 1) π/2

x = 3π/2

Solution are {π/2 and 3π/2}.

Problem 2 :

sin x = -1

Solution :

sin x = -1  

x = sin^-1 (-1)

For which angle of sin, we get -1.

sin (3π/2) = -1

x = 3π/2

So, solution is 3π/2.

Problem 3 :

tan x = -1

Solution :

tan x = -1

x = tan^-1 (-1)

α = -π/4

General solution for,

x = nπ + α

x = nπ - π/4

if n = 1,

x = (1)π - π/4

x = 3π/4

n = 2,

x = (2)π - π/4

x = 7π/4

So, the solution are {3π/4 and 7π/4}.

Problem 4 :

2 cos x - √3 = 0

Solution :

2 cos x - √3 = 0

cos x = √3/2

x = cos^-1(√3/2)

α = π/6

General solution for,

x = 2nπ ± α

x = 2nπ ± π/6

If n = 1,

x = 2(1)π + π/6

x = 13π/6

x does not belongs to the interval 0, 2π.

x ∉ (0, 2π)

n = 1,

x = 2(1)π - π/6

x = 11π/6

So, the solution are {π/6 and 11π/6}.

Problem 5 :

2 sin x + √2 = 0

Solution :

2 sin x + √2 = 0

sin x = -√2/2

sin x = -1/√2

-1/√2 < 0

sin x ϵ (-π/2 ,0)

α = -π/4

General solutions for,

x = nπ + (-1)ⁿ α

x = nπ + (-1)ⁿ (-π/4)

If n = 1,

  x = (1)π + (-1)¹ (-π/4)

= π + π/4

x = 5π/4

n = 2,

  x = (2)π + (-1)² (-π/4)

= 2π - π/4

x = 7π/4

Solutions are {5π/4 and 7π/4}.

Problem 6 :

2 sin x - 1 = 0

Solution :

2 sin x - 1 = 0

sin x = 1/2

x = sin^-1 (1/2)

α = π/6

General solutions for,

x = nπ + (-1)ⁿ α

x = nπ + (-1)ⁿ π/6

If n = 1,

x = (1)π + (-1)¹ π/6

x = π - π/6

x = 5π/6

Solutions are {π/6 and 5π/6}.

Problem 7 :

cos θ - 1 = 0

Solution :

cos θ = 1

θ = cos^-1 (1)

θ = 0