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Comprehensive Guide to Graphing Trigonometric Functions

Learn how to graph sine, cosine, tangent, cotangent, secant, and cosecant functions with step-by-step instructions and practical applications. Master different forms and understand amplitudes, periods, and phase shifts.

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Comprehensive Guide to Graphing Trigonometric Functions

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  1. Chapter 3Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and y = k +A cos (Bx + C) 3.4 Additional Applications 3.5 Graphing Combined Forms 3.6 Tangent, Cotangent, Secant, and Cosecant Functions Revisited

  2. 3.1 Basic Graphs • Graphs of y = sin x and y = cos x • Graphs of y = tan x and y = cot x • Graphs of y = csc x and y = sec x • Graphing with a graphing calculator

  3. y = sin x

  4. y = cos x

  5. y = tan x

  6. Other Graphs

  7. 3.2 Graphing y= k + A sin Bx and y = k + A cos Bx • Graphing y = A sin x and y = A cos x • Graphing y = sin Bx and y = cos Bx • Graphing y = A sin Bx and y = A cos Bx • Graphing y= k + A sin Bx and y = k + A cos Bx • Applications

  8. Comparing Amplitudes • Compare the graphs of y = 1/3 sin x and y = 3 sin x • The effect of A in y = A sin x is to increase or decrease the y values without affecting the x values.

  9. Comparing Periods • Compare the graphs of y = sin 2x and y = sin ½ x The graph shows the change in the period.

  10. Amplitude and Period For both y = A sin Bx and y = A cos Bx: Amplitude = |A| Period = 2p/B

  11. Vertical Shift • y = -2 + 3 cos 2x, -p x  2p • Find the period, amplitude, and phase shift and then graph

  12. Period and Frequency • For any periodic phenomenon, if P is the period and f is the frequency, P = 1/f.

  13. 3.3 Graphing y = k + A sin (Bx + C) and y = k + A cos (Bx + C) • Graphing y = A sin (Bx + C) and y = A cos (Bx + C) • Graphing y = k + A sin (Bx + C) and y = k + A cos (Bx + C) • Finding the equation for the graph of a simple harmonic motion

  14. Finding Period and Phase Shift • y = A sin (Bx + C) and y = A cos (Bx + C) • These have the same general shape as y = A sin Bx and y = A cos Bx translated horizontally. • To find the translation: x = -C/B (phase shift) and x = -C/B + 2p/B

  15. Phase shift and Period • Find the period and phase shift of y = sin(2x + p/2) • The period is p. • The phase shift is –p/4.

  16. Steps for Graphing

  17. 3.4 Additional Applications • Modeling electric current • Modeling light and other electromagnetic waves • Modeling water waves • Simple and damped harmonic motion: resonance

  18. Alternating Current Generator • I = 35 sin (40pt – 10p) (current) • Amplitude = 35 • Phase shift: 40pt = 10p t = ¼ • Frequency = 1/Period = 20 Hz • Period = 1/20

  19. Electromagnetic Waves • E = A sin 2p(vt – r/l) • t = time, r = distance from the source, l is the wavelength, v is the frequency

  20. Water Waves • y = A sin 2p(f1t – r/l) • t = time, r = distance from the source, l is the wavelength, f1 is the frequency

  21. Damped Harmonic Motion • Y = (1/t)sin (p/2)t, 1  t  8 • First, graph y = 1/t. • Then, graph y = sin(pt/2) keeping high and low points within the envelope.

  22. 3.6 Tangent, Cotangent, Secant, and Cosecant Functions Revisited • Graphing y = A tan (Bx + C) and y = cot (Bx + c) • Graphing y = A sec (Bx + C) and y = csc (Bx + c)

  23. y = tan x

  24. y = cot x

  25. y = csc x

  26. Y = sec x

  27. Graphing y = A tan (Bx + C) • Y = 3 tan (p/2(x) + p/4), -7/2  x  5/2 • Phase shift = -1/2 • Period = 2 • Asymptotes at -7/2, -3/2, ½, and 5/2

  28. Graph of y = sec x • Graph y = 5 sec (1/2(x) + p) for -7p x  3p.

  29. Graphing y = A csc(Bx + C) • Graph y = 2 csc (p/2(x) = p) for -2 < x < 10

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