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Periodic Functions And Applications III

Periodic Functions And Applications III. Significance of the constants A,B,C and D on the graphs of y = A sin(Bx+C) + D, y = A cos(Bx+C) + D Application of periodic functions Solution of simple trig equations within a specified domain Derivatives of functions involving sin x and cos x

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Periodic Functions And Applications III

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  1. Periodic Functions And Applications III Significance of the constants A,B,C and D on the graphs of y = A sin(Bx+C) + D, y = A cos(Bx+C) + D Application of periodic functions Solution of simple trig equations within a specified domain Derivatives of functions involving sin x and cos x Applications of the derivatives of sin x and cos x in life-related situations

  2. Periodic Functions And Applications III

  3. REVISION FM Page 118 Ex 5.8 New Q Page 351 Ex 10.1 No. 1-12 (parts a & b only), leave out no.10

  4. Solving Trig(onometric) Equations • Model Find all values of x (to the nearest minute) where 0< x <360 for which • (a) sin x = 0.5 • (b) tan x = -1

  5. sin is positive  angle is in Q1 or Q2 (a) sin x = 0.5 x = 30 or x = 180 - 30 = 30 or 150 30 30 Value of sin x is 0.5  30 off x-axis

  6. tan is negative  angle is in Q2 or Q4 (b) tan x = -1 x = 180 - 45 or x = 360 - 45 = 135 or 315 45 45° Value of tan x is -1  45 off x-axis

  7. New Q Ex 10.3 Page 363 2,6 FM Page 119 Ex 5.91 (orally)

  8. General Solution of a Trig Function So there appears to be more than one solution cos θ = 0.643 θ = cos-1 (0.643) θ ≈ 50° But cos 310° = 0.643 also So, how many solutions are there?

  9. y=0.643 cos curve cos θ = 0.634 θ = 50° or θ = 310° or θ = 50° + 360° or θ = 310° + 360° or θ = 50° + 2 x 360° or θ = 310° -360° or θ = 50° + 3 x 360° or θ = 310° - 2 x 360° or θ = 50° - 360° or θ = 310° - 3 x 360° or θ = 50° - 2 x 360° etc θ = 50° + 360° x n θ = 310° + 360° x n

  10. θ = 50° + 360n θ = 310° + 360°n The General Solution forcos θ = 0.643 For all integer values of n

  11. Model : Find all values of x (to the nearest minute) for which (a) sin x = 0.5

  12. sin is positive  angle is in Q1 or Q2 (a) sin x = 0.5 x = 30 or x = 180 - 30 = 30 or 150  general solution is x = 30 + n x 360 or x = 150 + n x 360 30 30 Value of sin x is 0.5  30 off x-axis

  13. Model Find all values of x (to the nearest minute) where 0 ≤ x ≤ 360 for which (a) sin2x = 0.25 (b) tan 3x = -1

  14. sin is pos or neg  angle is in Q1,Q2,Q3 or Q4 (a) sin2x = 0.25 sin x = ± 0.5 x = 30 or x = 150 or x = 210 or x = 330 30 30 30 30 Value of sin x is 0.5  30 off x-axis

  15. tan is negative  angle is in Q2 or Q4 (b) tan 3x = -1 3x = 135° + 360n or 3x = 315 + 360n x = 45 + 120n or x = 105 + 120n  45, 165, 285, 105, 225, 345 45 45° Value of tan x is -1  45 off 3x-axis

  16. New Q Page 371 Ex 10.5 1, 2, 4,10, 14 FM Page 304 Ex 14.161 -23 with GC

  17. Derivatives of functions involving sin x and cos x • Derivatives of functions involving sin x and cos x

  18. Derivative of sin x and cos x y = sin x  dy = cos x dx y = cos x  dy = -sin x dx

  19. Model : Find the derivative of(a) sin 2x(b) sin32x(c) sin2x cos3x(d) sin(π-3x) • do some examples on Graphmatica

  20. Model : Find the derivative of(a) sin 2x(b) sin32x(c) sin2x cos3x(d) sin(π-3x) ________________________________________ (a) y = sin 2x = sin u where u = 2x dy= cos u du= 2 du dx dy=dy . du dx du dx = 2 cos u = 2 cos 2x

  21. Model : Find the derivative of(a) sin 2x(b) sin32x(c) sin2x cos3x(d) sin(π-3x) ________________________________________ (b) y = sin32x ( = (sin 2x)3 ) = u3 where u = sin 2x dy= 3u2du= 2 cos 2x du dx dy=dy . du dx du dx = 3u2 . 2 cos 2x = 6 sin22x cos 2x

  22. Model : Find the derivative of(a) sin 2x(b) sin32x(c) sin2x cos3x(d) sin(π-3x) ________________________________________ (c) y = sin2x cos3x = uv where u = sin2x and v = cos3x du= 2 sinx cosx dv = -3 sin3x dx dx dy= u dv + v du dx dx dx = -3 sin3x sin2x + cos3x  2 sin x cos x = -3 sin3x sin2x + 2 cos3x sin x cos x

  23. Model : Find the derivative of(a) sin 2x(b) sin32x(c) sin2x cos3x(d) sin(π-3x) ________________________________________ (d) y = sin (π-3x) = sin 3x dy= 3 cos 3x dx

  24. NEWQ P50 2.4No. 1(a,b,d,f,h,j), 3 – 8 (all)FM Page 447 Ex 19.51,4,5

  25. Model : Find the gradient of the curve y = sin 2x at the point where x = π/3

  26. Model : Find the gradient of the curve y = sin 2x at the point where x = π/3

  27. Trig functions and motion • Consider the motion of an object on the end of a spring dropped from a height of 1m above the equilibrium point which takes 2π seconds to return to the starting point. 1m 0m -1m s = cos t v = -sin t a = -cos t

  28. Model: The motion of an object oscillates such that its displacement, s, from a fixed point is given by s = 4cos3t where s is in meters and t is in seconds.(a) How far from the fixed point is the object at the start?(b) How long does it take for the object to return to its starting point?(c) Find the object’s velocity (i) at the start (ii) as it passes the fixed point (iii) after 2 sec(d) Find its acceleration as it passes the fixed point

  29. Model: The motion of an object oscillates such that its displacement, s, from a fixed point is given by s = 4cos3t where s is in meters and t is in seconds.(a) How far from the fixed point is the object at the start? • At the start, t = 0 • When t = 0, s = 4 cos(3x0) • = 4 cos0 • = 4 x 1 • = 4 • i.e. at the start, object is 4 metres from the fixed point.

  30. Model: The motion of an object oscillates such that its displacement, s, from a fixed point is given by s = 4cos3t where s is in meters and t is in seconds. (b)How long does it take for the object to return to its starting point? Returns to starting point  s = 4  4 cos3t = 4  cos3t = 1  3t = 2nπ  t = 2nπ/3  t = 2π/3, 4π/3, 6π/3, … i.e. first returns to starting point after 2π/3 secs

  31. Model: The motion of an object oscillates such that its displacement, s, from a fixed point is given by s = 4cos3t where s is in meters and t is in seconds.(c) Find the object’s velocity (i) at the start(ii) as it passes the fixed point (iii) after 2 sec s = 4 cos3t  v = -12 sin3t (i) At the start When t = 0, v = -12 sin 0 = 0

  32. Model: The motion of an object oscillates such that its displacement, s, from a fixed point is given by s = 4cos3t where s is in meters and t is in seconds.(c) Find the object’s velocity (i) at the start (ii) as it passes the fixed point(iii) after 2 sec • (ii) at what time does it pass the fixed point  s = 0 •  4 cos 3t = 0 • cos 3t = 0 • 3t = π/2, … • t = π/6, … • When t = π/6, v = -12 sin 3π/6 • = -12

  33. Model: The motion of an object oscillates such that its displacement, s, from a fixed point is given by s = 4cos3t where s is in meters and t is in seconds.(c) Find the object’s velocity (i) at the start(ii) as it passes the fixed point (iii) after 2 sec • (iii) After 2 sec • v = -12 sin 3x2 • = -12 sin 6 • = 3.35

  34. Model: The motion of an object oscillates such that its displacement, s, from a fixed point is given by s = 4cos3t where s is in meters and t is in seconds.(d) Find its acceleration as it passes the fixed point • s = 4 cos3t • v = -12 sin3t • a = -36 cos3t It passes the fixed point when t = π/6, a = -36 cos 3π/6 = 0

  35. NEWQ P59 Ex 2.6No. 1 – 4, 7

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