Angle between 2 lines

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  • Angle between 2 lines

    1. 1. Angle between 2 LinesPreliminary Extension MathematicsDate: Tuesday 10th May 2011
    2. 2. Angle between 2 lines y line l1 has gradient m1 l2 l1 line l2 has gradient m2 ∴ m1 = tan α and m2 = tan β θ α β x 0
    3. 3. Angle between 2 lines y line l1 has gradient m1 l2 l1 line l2 has gradient m2 ∴ m1 = tan α and m2 = tan β θ α β x and α +θ =β (Why?) 0
    4. 4. Angle between 2 lines y line l1 has gradient m1 l2 l1 line l2 has gradient m2 ∴ m1 = tan α and m2 = tan β θ and α +θ =β α β x (Exterior angle of V) 0
    5. 5. Angle between 2 lines y So l2 l1 θ = β −α θ α β x 0
    6. 6. Angle between 2 lines y So l2 l1 θ = β −α ∴ tan θ = tan(β − α ) θ α β x 0
    7. 7. Angle between 2 lines y So l2 l1 θ = β −α ∴ tan θ = tan(β − α ) θ tan β − tan α = α β 1 + tan β tan α x 0 You will learn this formula later
    8. 8. Angle between 2 lines y So l2 l1 θ = β −α ∴ tan θ = tan(β − α ) θ tan β − tan α = α β 1 + tan β tan α x 0 m1 − m2 = 1 + m1m2 Why?
    9. 9. Angle between 2 lines y So l2 l1 θ = β −α ∴ tan θ = tan(β − α ) θ tan β − tan α = α β 1 + tan β tan α x 0 m1 − m2 = 1 + m1m2 When tan θ is positive, θ is acute. When tan θ is negative, θ is obtuse.
    10. 10. Angle between 2 lines y Thus for two lines of gradient l2 l1 m1 and m2 the acute angle between them is given by θ m1 − m2 tan θ = α β x 1 + m1m2 0 Note that m1m2 ≠ −1 what does this mean?
    11. 11. Angle between 2 lines y Thus for two lines of gradient l2 l1 m1 and m2 the acute angle between them is given by θ m1 − m2 tan θ = α β x 1 + m1m2 0 Note that m1m2 ≠ −1 the formula does not work for perpendicular lines
    12. 12. Example 1Find the acute angle between y = 2x + 1 and y = −3x − 2(to nearest degree)
    13. 13. Example 1Find the acute angle between y = 2x + 1 and y = −3x − 2(to nearest degree) ∴ m1 = 2 and m2 = −3
    14. 14. Example 1Find the acute angle between y = 2x + 1 and y = −3x − 2(to nearest degree) ∴ m1 = 2 and m2 = −3 m1 − m2 tan θ = 1 + m1m2
    15. 15. Example 1Find the acute angle between y = 2x + 1 and y = −3x − 2(to nearest degree) ∴ m1 = 2 and m2 = −3 m1 − m2 tan θ = 1 + m1m2 2+3 ∴ tan θ = 1− 6
    16. 16. Example 1Find the acute angle between y = 2x + 1 and y = −3x − 2(to nearest degree) ∴ m1 = 2 and m2 = −3 m1 − m2 tan θ = 1 + m1m2 2+3 ∴ tan θ = 1− 6 ∴ tan θ = −1
    17. 17. Example 1Find the acute angle between y = 2x + 1 and y = −3x − 2(to nearest degree) ∴ m1 = 2 and m2 = −3 m1 − m2 tan θ = 1 + m1m2 2+3 ∴ tan θ = 1− 6 ∴ tan θ = −1 ∴ tan θ = 1
    18. 18. Example 1Find the acute angle between y = 2x + 1 and y = −3x − 2(to nearest degree) ∴ m1 = 2 and m2 = −3 m1 − m2 tan θ = 1 + m1m2 2+3 ∴ tan θ = 1− 6 ∴ tan θ = −1 ∴ tan θ = 1 → θ = 45°
    19. 19. Example 2Find the acute angle between 3x − 2y + 7 = 0 and(to nearest degree) 2y + 4x − 3 = 0
    20. 20. Example 2Find the acute angle between 3x − 2y + 7 = 0 and(to nearest degree) 2y + 4x − 3 = 0∴2y = 3x + 7
    21. 21. Example 2Find the acute angle between 3x − 2y + 7 = 0 and(to nearest degree) 2y + 4x − 3 = 0∴2y = 3x + 7 3 7∴y = x + 2 2
    22. 22. Example 2Find the acute angle between 3x − 2y + 7 = 0 and(to nearest degree) 2y + 4x − 3 = 0∴2y = 3x + 7 3 7∴y = x + 2 2 3∴ m1 = 2
    23. 23. Example 2Find the acute angle between 3x − 2y + 7 = 0 and(to nearest degree) 2y + 4x − 3 = 0∴2y = 3x + 7 similarly ∴2y = −4x + 3 3 7∴y = x + 3 2 2 ∴ y = −2x + 2 3∴ m1 = ∴ m2 = −2 2
    24. 24. Example 2Find the acute angle between 3x − 2y + 7 = 0 and(to nearest degree) 2y + 4x − 3 = 0applying the formula m1 − m2 tan θ = 3 1 + m1m2 m1 = m2 = −2 2
    25. 25. Example 2Find the acute angle between 3x − 2y + 7 = 0 and(to nearest degree) 2y + 4x − 3 = 0applying the formula m1 − m2 tan θ = 3 1 + m1m2 m1 = m2 = −2 2 3 +2 ∴ tan θ = 2 1− 3
    26. 26. Example 2Find the acute angle between 3x − 2y + 7 = 0 and(to nearest degree) 2y + 4x − 3 = 0applying the formula m1 − m2 tan θ = 3 1 + m1m2 m1 = m2 = −2 2 3 +2 ∴ tan θ = 2 1− 3 −7 ∴ tan θ = 4
    27. 27. Example 2Find the acute angle between 3x − 2y + 7 = 0 and(to nearest degree) 2y + 4x − 3 = 0applying the formula m1 − m2 tan θ = 3 1 + m1m2 m1 = m2 = −2 2 3 +2 ∴ tan θ = 2 1− 3 −7 7 ∴ tan θ = ∴ tan θ = → θ = 60° 4 4
    28. 28. Example 3 - by thinking anddrawing....Find the acute angle between y= x+3 and y = −3x + 5(to nearest degree)
    29. 29. Example 3 - by thinking anddrawing....Find the acute angle between y= x+3 and y = −3x + 5(to nearest degree) y = −3x + 5 y y= x+3 θ α β x 0
    30. 30. Example 3 - by thinking anddrawing....Find the acute angle between y= x+3 and y = −3x + 5(to nearest degree) m1 = 1 → α = 45° y = −3x + 5 y y= x+3 θ α β x 0
    31. 31. Example 3 - by thinking anddrawing....Find the acute angle between y= x+3 and y = −3x + 5(to nearest degree) m1 = 1 → α = 45° y = −3x + 5 y y= x+3 m2 = −3 → β = 108° θ α β x 0
    32. 32. Example 3 - by thinking anddrawing....Find the acute angle between y= x+3 and y = −3x + 5(to nearest degree) m1 = 1 → α = 45° y = −3x + 5 y y= x+3 m2 = −3 → β = 108° But α +θ =β θ α β x 0
    33. 33. Example 3 - by thinking anddrawing....Find the acute angle between y= x+3 and y = −3x + 5(to nearest degree) m1 = 1 → α = 45° y = −3x + 5 y y= x+3 m2 = −3 → β = 108° But α +θ =β θ ∴θ = 63° α β x 0

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