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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). Comparison of Sine and Cosine Graphs. The connection between sine and cosine.

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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. Comparison of Sine and Cosine Graphs

  3. The connection between sine and cosine We know sin(t) is the q coordinate. We know cos(t) is the p coordinate. We know p2 + q2 = 1. Trig Identity[sin(t)]2 + [cos(t)]2 = 1 sin2(t) + cos2(t) = 1 CYU 6.5/298

  4. Rate of Change Find the rate of change of f(t) = sin(t) at t = π = -0.9999833335

  5. other “nice” arcst = π/4 p2 + q2 = 1 and p = q p2 + p2 = 1 2p2 = 1 p2 = 1/2 p = +/- √(1/2) = +/- √2/2 sin (π/4) = √2/2 = 0.707107 cos(π/4) = √2/2 = 0.707107

  6. other “nice” arcst = π/6 triangle is equilateral each side measures 1 p2 + (1/2)2 = 12 p2 = 3/4 p = +/- √(3/4) = +/- √3/2 sin (π/6) = 1/2 = 0.50 cos(π/6) = √3/2 = 0.86603

  7. other “nice” arcst = π/3 triangle is equilateral each side measures 1 (1/2)2 + (q)2 = 12 q2 = 3/4 q = +/- √(3/4) = +/- √3/2 sin (π/3) = √3/2 = 0.86603 cos(π/3) = 1/2 = 0.50

  8. Other Circular (Trigonometric) Functionspage 301 tangent cotangent secant cosecant CYU 6.7/304 and #15,#17, #19

  9. Graphs of Other Circular (Trigonometric) Functions Function has a numerator and denominator since y intercept: f(0) = tan(0) = sin(0)/cos(0) = 0/1 = 0. x intercept(s): sin(t) = 0 for t = 0, π, -π, 2π, -2π, 3π… VA: cos(t) = 0 for t = π/2, -π/2, 3π/2, -3π/2, 5π/2… HA: not possible with an infinite number of VAs

  10. Domain: all reals except π/2,-π/2, 3π/2, -3π/2, 5π/2 Range: all reals

  11. Graphs of Other Circular (Trigonometric) Functions Function has a numerator and denominator since y intercept: f(0) = __________ x intercept(s): cos(t) = 0 for t = ______________ VA: sin(t) = 0 for t = ___________________ HA: not possible with an infinite number of VAs

  12. Graphs of Other Circular (Trigonometric) Functions Function has a numerator and denominator since y intercept: f(0) = __________ x intercept(s): none since numerator never equals 0 VA: cos(t) = 0 for t = ___________________ HA: not possible with an infinite number of VAs

  13. Graphs of Other Circular (Trigonometric) Functions Function has a numerator and denominator since y intercept: f(0) = __________ x intercept(s): none since numerator never equals 0 VA: sin(t) = 0 for t = ___________________ HA: not possible with an infinite number of VAs

  14. Work carefully through pages 285 through 304. Page: 326 #1-24, 25*, 26*, 27 – work through powerpoint slides on cot(t), sec(t), csc(t) TURN IN: #18,#20,#22, 27

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