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Chapter 6 Periodic Functions 6.1 The Sine and Cosine Functions

Chapter 6 Periodic Functions 6.1 The Sine and Cosine Functions 6.2 Circular Functions and their Graphs 6.3 Sinusoidal Models 6.4 Inverse Circular (Trigonometric) Functions. Average Daily High Temperatures in New York City. H(t) = 24sin(0.0172(t-120)) + 62.

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Chapter 6 Periodic Functions 6.1 The Sine and Cosine Functions

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  1. Chapter 6Periodic Functions 6.1 The Sine and Cosine Functions 6.2 Circular Functions and their Graphs 6.3 Sinusoidal Models 6.4 Inverse Circular (Trigonometric) Functions

  2. Average Daily High Temperatures in New York City H(t) = 24sin(0.0172(t-120)) + 62 On what days was the average daily high temperature equal to 50 degrees? 50 = 24sin(0.0172(t-120)) + 62

  3. Inverse Trigonometric Functions solve sin(t) = 1 solve sin(t) = 1/2 solve sin(t) = -√2/2

  4. Inverse Trigonometric Functions solve sin(t) = 1/3 t = sin-1(1/3)ort = arcsin(1/3) t = .34calculator solve sin(t) = -1/3 t = 3.14 + .34 = 3.44t = 6.28 - .34 = 5.94unit circle symmetry t = 3.14 - .34 = 2.80unit circle symmetry t = 3.44 + 2kπ t = 5.94 + 2kπ t = .34 + 2kπ t = 2.80 + 2kπ

  5. Inverse Sine Function y = sin-1(x) or y = arcsin(x) means sin(y) = x and π/2 ≤ y ≤ π/2 arc between –π/2 and π/2 number between -1 and 1 sin-1(x) sin-1(0) sin-1(-1/2) otherwise calculator sin-1(√3/2) sin-1(1) sin-1(-√2/2)

  6. Inverse Sine Function y = sin-1(x) or y = arcsin(x) means sin(y) = x and π/2 ≤ y ≤ π/2 y=sin(x) y=arcsin(x) y=Sin(x)

  7. Inverse Cosine Function y = cos-1(x) or y = arccos(x) means cos(y) = x and 0 ≤ y ≤ π y=cos(x) y=arccos(x) y=Cos(x)

  8. Inverse Tangent Function y = tan-1(x) or y = arctan(x) means tan(y) = x and π/2 < y < π/2 y=tan(x) y=arctan(x) y=Tan(x)

  9. More Practice Page330: #53,#55,#63, #65

  10. More Practice Page330: #63 (check)

  11. More Practice Page330: #65 (check) 1+4*sin(Pi*(t+2.5)/5) = 0

  12. 50 = 24sin(0.0172(t-120)) + 62 7π/6 = 0.0172(t-120) (1/0.0172)*(7π/6) = t-120 -12 = 24sin(0.0172(t-120)) (1/0.0172)*(7π/6) + 120 = t -12/24 = sin(0.0172(t-120)) 333 = t -1/2 = sin(0.0172(t-120)) arcsin(-1/2) = 0.0172(t-120)

  13. 50 = 24sin(0.0172(t-120)) + 62 -π/6 = 0.0172(t-120) (1/0.0172)*(-π/6) = t-120 -12 = 24sin(0.0172(t-120)) (1/0.0172)*(-π/6) + 120 = t -12/24 = sin(0.0172(t-120)) 90 = t -1/2 = sin(0.0172(t-120)) arcsin(-1/2) = 0.0172(t-120)

  14. Homework Page330: #53-70 TURN IN: #54, 56, 58, 66

  15. FINAL EXAM - REVIEW • TRIG FUNCTIONS: • Basic evaluations (π/6, π/4, π/3, π/2, π … ) • Given info to formula to graph. • Picture to info to formula • Given sin(t), determine cos(t), tan(t), cot(t), sec(t) and csc(t). • Use inverse trig ideas to solve a trig equation

  16. FINAL EXAM - REVIEW • EXAM 2 • Find a shifted exponential formula • Determine instantaneous rate of change for logistic model • 3) Find and interpret vertex in context. • Use logarithms to solve an exponential equation. • Find polynomial formula from zeroes and y intercept.

  17. FINAL EXAM - REVIEW • EXAM 1 • Identify formulas to match graphs. • Determine an inverse function. • 3) Determine a linear model in context. • Determine a basic exponential model in context. • Determine a linear model from two points.

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