Trigonometrikus egyenletek

Jelölés: (sin α)² = sin²α

Egy nagyon fontos trigonometrikus összefüggés:

sin²α + cos²α = 1

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A következő feladatok mindegyikében: k ∈ ℤ

1. sin²α = 1/2

   α₁ = 45° + k⋅360°          α₂ = 135° + k⋅360°          α₃ = 225° + k⋅360°          α₄ = 315° + k⋅360°

   
   

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2. sin²x = 3/4

   x₁ = π/3 + kπ (60°)          x₂ = 2π/3 + kπ (120°)

   
   

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3. cos²x = 3/4

   x₁ = π/6 + kπ (30°)          x₂ = 5π/6 + kπ (330°)

   
   

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4. cos²x = 1/2

   x₁ = π/4 + kπ (45°)          x₂ = 3π/4 + kπ (135°)

   
   

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5. 2⋅cos²α + cos α − = 0

   α₁ = 60° + k⋅360°          α₂ = −60° + k⋅360°, azaz α₂ = 300° + k⋅360°          α₃ = 180° + k⋅360°
   radiánban: α₁ = π/3 + 2kπ          α₂ = 5π/3 + 2kπ          α₃ = π + 2kπ

   
   

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6. sin x = −1

   x = 3π/2 + 2kπ (270°)

   
   

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7. sin²x = 2⋅sin x + 3

   x = 3π/2 + 2kπ (270°)

   
   

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8. tg α + cos α = 0

   α₁ ≈ 38,3° + k⋅360°          α₂ ≈ 218,3° + k⋅360°

   
   

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9. 2 − 7sinα = 2cos²α + 4

   α₁ = 210° + k⋅360°          α₂ = 330° + k⋅360°

   
   

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10. 0,25⋅sin²x + 1 = (5 − cos²x)/4

    x ∈ ℝ

    
    

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11. ctg x − tg x = 2√3

    x₁ = π/12 + kπ (15°)          x₂ = 19/12π + k⋅π (285°)

    
    

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12. √3cos x = sin x

    x = π/3 + kπ

    
    

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13. sin x / (1 − cos x) = 1 + cos x

    x₁ = π/2 + 2kπ           x₂ = π + 2kπ

    
    

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14. cos x / tg x = 3/2

    x₁ = π/6 + 2kπ           x₂ = 5π/6 + 2kπ

    
    

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15. cos²x − sin x = 1

    x₁ = π + kπ     x₂ = 3π/2 + 2kπ

    
    

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16. 2⋅sin x = tg x

    x₁ = π + kπ     x₂ = π/3 + 2kπ     x₃ = 5π/3 + 2kπ

    
    

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17. sin²x − 1,5⋅sin x = −0,5

    x₁ = π/2 + 2kπ     x₂ = π/6 + 2kπ     x₃ = 5π/6 + 2kπ

    
    

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18. cos²x + (7cos x)/2 = 2

    x₁ = π/3 + 2kπ     x₂ = 5π/3 + 2kπ

    
    

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19. sin²x − sin x = −2

    x = { }

    
    

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