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Trigonometric Functions - Contents

Trigonometric Functions - Contents. Radian Measure Area/Circumference/Length of Arc Area of a Sector Area of a minor segment Small Angles. Trigonometric Results Trigonometric Graphs Graphical Solutions Derivatives of Trigonometric Functions Integrals of Trigonometric Functions.

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Trigonometric Functions - Contents

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  1. Trigonometric Functions - Contents • Radian Measure • Area/Circumference/Length of Arc • Area of a Sector • Area of a minor segment • Small Angles • Trigonometric Results • Trigonometric Graphs • Graphical Solutions • Derivatives of Trigonometric Functions • Integrals of Trigonometric Functions (Right Click Mouse  Pointer Options Arrow Options Visible)

  2. 1 Unit 1 Unit Radian Measure 1c One radian is the angle that a one unit arc makes with centre of a unit circle. 2π= 360o Proof Circumference = 2πr π= 180o π radians = 180o = 2π (1) = 2π  The circumference is 2π

  3. Example 1 Convert into degrees. 5π 3 5π 3 5x180 3 = π = 180o Radian Measure Conversions Example 2 Convert 45o into radians. = 300o 180o = π radians 1o = π/180 radians Example 3 Convert 1.6c into degrees. 45o = π/180 x 45 radians = 45π/180 radians π radians = 180o = π/4 radians 1 radians = 180/π deg 1.6 radians = 180/πx 1.6 deg = 91o 40’

  4. Circumference= 2r Area/Circumference/Length of Arc = D Area= r2 r Length of Arc l = r ( is in radians)

  5. 4 Circumference= 2r Area/Circumference/Length of Arc = 2 x 3 = D = 6 (exact) ≈ 18.8 cm (Approximate) Area= r2 3 cm = x 32 = 9(exact) Length of Arc ≈ 28.3 cm2 (Approximate) l = r ( is in radians) = 3 x /4 = 3/4 (exact) ≈ 2.4 cm (Approximate)

  6. Angle  Revolution =  2 =  r2 2  A = r Area of Sector Area of a Sector A = ½r2 Proof ( is in radians) Area of Sector Area of Circle Example A r2 Find the area of a sector with radius 3cm and angle /6. A = ½ r2 A = ½r2 = ½ 32x /6. = 3/4 cm2. ≈ 2. 36 cm2.

  7. a  b r Area = ½absin  Area of Minor Segment Area of Sector Area of Triangle = - Area of Sector Area of a Minor Segment A = ½r2 ( - sin ) ( is in radians) Example Find the area of a minor segment formed with radius 3cm and angle /6. Proof = ½r2 - ½ r2 sin  = ½r2( - sin ) = ½r2( - sin ) = ½ 32x (/6 - sin /6) = ½ 32x (/6 - ½) = (3/4 –3/8)cm2 ≈ 1. 98 cm2.

  8. lim Sin x = 1 x0 x0 x x lim Tan x = 1 Small Radian Angles For small angles Sin x ≈ x h Tan x ≈ x x o a=1 Cos x ≈ 1 Therefore Cos x = a/h What happens to a/h as x 0. a/h 1/1 1

  9. lim Sin x = 1 x0 x0 x x lim Tan x = 1 Small Radian Angles For small angles Sin x ≈ x h Tan x ≈ x x o a=1 Cos x ≈ 1 Therefore Sin x = o/h What happens to o/h as x 0. o/h x  0/1  0

  10. lim Sin x = 1 x0 x0 x x lim Tan x = 1 Small Radian Angles For small angles Sin x ≈ x h Tan x ≈ x x o a=1 Cos x ≈ 1 Therefore Tan x = o/a What happens to o/a as x 0. o/a 0/1  0

  11. Evaluate: lim Sin 5x x0 x lim Sin 5x lim 5(Sin 5x) = x0 x0 5x x 5 lim (Sin 5x) = x0 5x Example: Small Radian Angles = 5 x 1 = 5

  12. Trigonometric Results π/2 π-θ θ 2nd 1st S A Note: π=180o 0 π 2π T C 3rd 4th π+θ 2π-θ 3π/2

  13. Sin π/3 = √3/2 Sin π/6 = 1/2 Trigonometric Results Cos π/6 = √3/2 Cos π/3 = 1/2 60o π/3 Tan π/3 = √3 Tan π/6 = 1/√3 π/6 2 2 √3 π/3 60o π/3 60o 1 2 π/3 = 60o π/6 = 30o

  14. Trigonometric Results Sin π/4 = 1/√2 Cos π/4 = 1/√2 π/4 Tan π/4 = 1 45o √2 1 π/4 45o 1 π/4 = 45o

  15. A S T C 5x180o 4 Sin Sin is -ve in the 3rd Quadrant. = -1 √2 Find the exact value of: Trigonometric Results 5π 4 Sin= = Sin 225o = Sin (180o + 45o) = -Sin 45o

  16. A S T C 30o 2 1 Solve for 0 ≤ θ ≤ 2π Trigonometric Results Evaluate Sinθ = 1/2 Sin is +ve in the 1st & 2nd Quadrants. θ = 30o θ = π/6 θ = π - π/6 θ = 5π/6

  17. Trigonometric Graphs y = sin(x) π/2 3π/2 Geogebra

  18. Trigonometric Graphs y = cos(x) π/2 3π/2 Geogebra

  19. Trigonometric Graphs y = tan(x) π/2 3π/2 Geogebra

  20. Trigonometric Graphs y = cosec(x) π/2 3π/2 Geogebra

  21. Trigonometric Graphs y = sec(x) π/2 3π/2 Geogebra

  22. Trigonometric Graphs y = cot(x) π/2 3π/2 Geogebra

  23. y = 3sin(x) y = 4 sin(x) y = 2sin(x) y = sin(x) The aSin x Family of Curves Trigonometric Graphs Geogebra

  24. y = sin 2x y = sin 3x y = sin 4x The Sin bx Family of Curves Trigonometric Graphs y = sin x Geogebra

  25. y = sin x + 1 y = sin x y = sin x - 1 The Sin x + c Family of Curves Trigonometric Graphs Geogebra

  26. One Solution Graphical solve y = cos x and y = x Graphical Solution y = x y = cos x π/4 π/2

  27. d dx d dx d dx d dx d dx d dx (sin x) = cos x [sin f(x)] = f’(x) cos f(x) (cos x) = sin x [cos f(x)] = -f’(x) sin f(x) (tan x) = sec2 x [tan f(x)] = f’(x) sec2 f(x) Function of Function Rule Derivative Derivative of Trigonometric Functions Sin x Cos x Tan x

  28. d dx d dx d dx d dx d dx d dx (sin 4x) = 4 cos 4x [sin f(x)] = f’(x) cos f(x) [cos f(x)] = -f’(x) sin f(x) [cos (x2 - 2x + 1)] = -(2x - 2) sin (x2 -2x + 1) [tan (3x + 1)] = 3 sec2 (3x + 1) [tan f(x)] = f’(x) sec2 f(x) Derivative Examples Derivative of Trigonometric Functions Sin 4x Tan (3x + 1) Cos (x2 - 2x + 1) = 2(1 - x) sin (x2 -2x + 1)

  29. ∫ ∫ sin x dx = - cos x + C sin (ax +b) dx = - 1/a cos (ax + b) + C ∫ cos x dx = sin x + C ∫ cos (ax +b) dx = 1/a sin (ax + b) + C ∫ ∫ sec2 x dx = tan x + c sec2 (ax +b) dx = 1/a tan (ax + b) + C Function of Function Rule Integral Integrals of Trigonometric Functions Sin x Cos x Tan x

  30. ∫ ∫ sin (ax +b) dx = - 1/a cos (ax + b) + C sin (3x +2) dx = -1/3 cos (3x + 2) + C Integral of Trigonometric Functions Integration Examples

  31. ∫ sin x dx = - cos x + C π π π ∫ 2 2 2 - cos x sin x dx = 0 0 = - cos - (-cos 0) Integral of Trigonometric Functions Integration Examples = 0 – (-1) = 1

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