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TRIGONOMETRY

D.EDWARDS. TRIGONOMETRY. Pythagoras’ theorem. Square on Hypotenuse. =. +. Square on Leg 1. Square on Leg 2. Sine Cosine Tangent. Trigonometric functions. Prove:. Trigonometric identities. Special Angles. Trigonometric functions. proof. proof. proof. ?. ?. ?. ?. ?. ?. ?. ?.

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TRIGONOMETRY

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  1. D.EDWARDS TRIGONOMETRY

  2. DEdwards Pythagoras’ theorem Square on Hypotenuse = + Square on Leg 1 Square on Leg 2

  3. Sine • Cosine • Tangent Trigonometric functions

  4. Prove: • . • . Trigonometric identities

  5. Special Angles DEdwards Trigonometric functions proof proof proof ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

  6. Sin 30° = • Cos 30° = • Tan 30° = • Sin 60° = • Cos 60° = • Tan 60° = Trigonometric functions 60° 30° 2 2 60° 60° 1 2 table

  7. Sin 45° = • Cos 45° = • Tan 45° = Trigonometric functions 45° 1 1 table

  8. Trigonometric Graphs are PERIODIC i.e. they repeat themselves after a “cycle” is complete GRAPHS OF TRIGONOMETRIC FUNCTIONS

  9. GRAPHS OF TRIGONOMETRIC FUNCTIONS y-intercept = 0 x-intercepts = 0 °, ±180°, ±360°

  10. GRAPHS OF TRIGONOMETRIC FUNCTIONS y-intercept = 1 x-intercepts = ±90°, ±270°

  11. GRAPHS OF TRIGONOMETRIC FUNCTIONS

  12. Complementary Relationships • Sin x = Cos ( 90 - x) • Cos x = Sin ( 90 - x) • Tan x = Trigonometric relationships

  13. Supplementary Relationships • Sin x = Sin ( 180 - x) • Cos x = - Cos ( 180 - x) • Tan x = - Tan ( 180 - x) Trigonometric relationships

  14. GRAPHS OF TRIGONOMETRIC FUNCTIONS The function has ASYMPTOTES at these points

  15. TRIGONOMETRIC FUNCTIONS on the cartesian plane (0,1) THE UNIT CIRCLE Circle on the Cartesian plane with radius of 1 unit

  16. TRIGONOMETRIC FUNCTIONS on the cartesian plane (0,1) THE UNIT CIRCLE Terminal Side: Other side forming desired angle Initial Side: Positive side of x axis

  17. TRIGONOMETRIC FUNCTIONS on the cartesian plane (0,1) THE UNIT CIRCLE: 1st Quadrant Terminal Side Theta θ: Angle between Terminal side & Initial side Reference Angle α: Acute Angle between Terminal Side & x-axis Initial Side

  18. TRIGONOMETRIC FUNCTIONS on the cartesian plane (0,1) THE UNIT CIRCLE: 2nd Quadrant Terminal Side Theta , θ Reference Angle α Initial Side

  19. TRIGONOMETRIC FUNCTIONS on the cartesian plane (0,1) THE UNIT CIRCLE: 3rd Quadrant Theta , θ Reference Angle α Initial Side Terminal Side

  20. TRIGONOMETRIC FUNCTIONS on the cartesian plane (0,1) THE UNIT CIRCLE: 4th Quadrant Theta , θ Reference Angle α Initial Side Terminal Side

  21. TRIGONOMETRIC FUNCTIONS on the cartesian plane (0,1) THE UNIT CIRCLE Coordinates of points on the unit circle show (cosθ , sin θ) When the terminal side is drawn through that point

  22. TRIGONOMETRIC FUNCTIONS on the cartesian plane (0,1) THE UNIT CIRCLE (cosθ , sin θ) P(a , b) x <- 1 (radius) -> <- b -> O θ ° X <- a ->

  23. TRIGONOMETRIC FUNCTIONS on the cartesian plane (0,1) THE UNIT CIRCLE Terminal Side Coordinates of points on the unit circle show (cosθ , sin θ) When the terminal side is drawn through that point x θ=90 ° cos90 °=0 sin90 ° =1 Hence point on circle is (0,1)

  24. TRIGONOMETRIC FUNCTIONS on the cartesian plane (0,1) THE UNIT CIRCLE Coordinates of points on the unit circle show (cosθ , sin θ) When the terminal side is drawn through that point θ=180 ° Cos180° = -1 Sin 180° = 0 Hence point on circle is(-1,0) x Terminal Side

  25. DEdwards TRIGONOMETRIC FUNCTIONS on the cartesian plane (0,1) THE UNIT CIRCLE Coordinates of points on the unit circle show (cosθ , sin θ) When the terminal side is drawn through that point θ=270 ° Cos 270° = 0 Sin 270° = -1 Hence point on circle is (0, -1) x Terminal Side

  26. TRIGONOMETRIC FUNCTIONS on the cartesian plane (0,1) θ=30 ° Cos 30 °= 0.866=0.9(1dp) sin30 ° = 0.5 Hence point on circle is (0.9 , 0.5) THE UNIT CIRCLE (cosθ , sin θ) (0.9 , 0.5) x θ=30 °

  27. TRIGONOMETRIC FUNCTIONS on the cartesian plane (0,1) A ll are Positive THE 1st QUADRANT 0 <θ≤ 90 For all points (x,y) x is positive & y is positive So, all points (cos θ,sin θ ) Cos θ :positive Sin θ : positive Tan θ = sin θ /cos θ=positive

  28. TRIGONOMETRIC FUNCTIONS on the cartesian plane S (0,1) A INE (only) is Positive ll are Positive THE 2nd QUADRANT 90 ° <θ≤ 180° For all points (x,y) x is negative & y is positive So, all points (cos θ,sin θ ) Cos θ :negative Sin θ : positive Tan θ = sin θ /cos θ=negative

  29. TRIGONOMETRIC FUNCTIONS on the cartesian plane S (0,1) A INE (only) is Positive ll are Positive THE 3rd QUADRANT 180 ° <θ≤ 270° For all points (x,y) x is negative & y is negative So, all points (cos θ,sin θ ) Cos θ :negative Sin θ : negative Tan θ = sin θ /cos θ=positive T AN (only) is Positive

  30. TRIGONOMETRIC FUNCTIONS on the cartesian plane S (0,1) A INE (only) is Positive ll are Positive THE 4th QUADRANT 270 ° <θ≤ 360° For all points (x,y) x is positive & y is negative So, all points (cos θ,sin θ ) Cos θ :positive Sin θ : negative Tan θ = sin θ /cos θ=negative C T OS (only) is Positive AN (only) is Positive

  31. TRIGONOMETRIC FUNCTIONS on the cartesian plane S (0,1) A INE (only) is Positive ll are Positive C T OS (only) is Positive AN (only) is Positive

  32. “CAST” Relationships • Sin x = Sin (180 - x) = -Sin (180 + x) = Sin ( 360 - x) • Cos x = -Cos (180 - x) = -Cos (180+ x) = Cos (360 - x) • Tan x = -Tan (180 - x) = Tan (180+ x) = -Tan(360 - x) Trigonometric relationships

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