SOLVING TRIGONOMETRIC EQUATIONS FOR ANGLES

Solved Problems

Problem 1 :

Solve the following equation :

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

Problem 2 :

Solve the following equation :

sin θ + √3 cos θ  =  1

Solution :

r = √a² + b²

a = 1, b = √3, then r = √1² + √3²  =  2

Divide the given equation by "r", we get

(1/2) sin θ + √3/2 cos θ = 1/2

sin(π/6) sin θ + cos(π/6) cos θ = 1/2

cos (π/6 - θ)  =  cos^-1(1/2)

cos (θ-π/6)  =  cos^-1(1/2)

θ  =  2nπ ± π/3

 θ - π/6  =  2nπ ± π/3

 θ  =  2nπ ± π/3 + π/6

Problem 3 :

Solve the following equation :

cot θ + cosecθ  =  √3

Solution :

cot θ + cosecθ  = √3

(cos θ /sin θ) + (1/sin θ)  =  √3

(cos θ + 1)/sin θ  =  √3

cos θ + 1 = √3 sin θ

√3 sin θ - cos θ  = 1  ----(10)

a = √3, b = -1

r =  √a² + b²  =  √(√3)² + (-1)²  =  √4  =  2

Divide (1) by 2, we get

(√3/2) sin θ - (1/2) cos θ  = (1/2)

 (1/2) cos θ - (√3/2) sin θ = (-1/2)

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

cos ((π/3) + θ)  =  cos 2π/3

θ  =  2nπ ± a

(π/3) + θ  =  2nπ ± 2π/3

θ  =  2nπ ± (2π/3) - (π/3)

Problem 4 :

Solve the following equation :

tan θ + tan (θ + π/3) + tan (θ + 2π/3)  =  √3

Solution :

tan (θ + π/3)  =  (tan θ + tan π/3) / (1 - tan θ tan π/3)

tan (θ + π/3)  =  (tan θ + √3) / (1 - √3 tan θ)  ------(1)

tan (θ + 2π/3)  =  (tan θ + tan 2π/3) / (1 - tan θ tan 2π/3)

tan (θ + 2π/3)  =  (tan θ - √3) / (1 + √3 tan θ) ------(2)

(1) + (2) 

  =  (tan θ+√3) / (1-√3 tan θ) + (tan θ-√3) / (1+√3tan θ)

  =  [(tan θ+√3)(1+√3tan θ) + (1-√3 tan θ) (tan θ-√3)]/(1 - 3tan² θ)

  =  8 tan θ / (1 - 3 tan² θ)

tan θ + (1) + (2)  =  √3

 tan θ + (8 tan θ / (1 - 3 tan² θ))  =  √3

 (tan θ - 3 tan³ θ + 8 tan θ )/(1 - 3 tan² θ)  =  √3

 (9 tan θ - 3 tan³ θ)/(1 - 3 tan² θ)  =  √3

3 (3 tan θ - tan³ θ)/(1 - 3 tan² θ)  =  √3

3 tan 3θ  =  √3

tan 3θ  =  √3/3

tan 3θ  =  1/√3

For tan θ, we have θ = nπ + a

3θ = nπ + π/6

θ = nπ/3 + π/18

Problem 5 :

Solve the following equation :

cos 2θ = (√5 + 1)/4

Solution :

cos 2θ = (√5 + 1)/4

cos 2θ  =  cos 36

θ = 2nπ ± a

2θ = 2nπ ± 36

θ = nπ ± 18

θ = nπ ± π/10

Problem 6 :

Solve the following equation :

2 cos² x − 7 cos x + 3 = 0

Solution :

2 cos² x − 7 cos x + 3 = 0

Let t = cos x

2 t² − 7t + 3 = 0

(2t - 1)(t - 3)  = 0

θ = 2nπ ± a

a = π/3

θ = 2nπ ± π/3

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