Chapter 3 Graphing 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

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

y = sin x

y = cos x

y = tan x

Other Graphs

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

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.

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

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

Vertical Shift y = cos 2x, -   x  2  Find the period, amplitude, and phase shift and then graph

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

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

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 + 2  /B

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

Steps for Graphing

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

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

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

Water Waves y = A sin 2p(f 1 t – r/ ) t = time, r = distance from the source, is the wavelength, f 1 is the frequency

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

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)

y = tan x

y = cot x

y = csc x

Y = sec x

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

Graph of y = sec x Graph y = 5 sec (1/2(x) +  for -7   x  3 .

Graphing y = A csc(Bx + C) Graph y = 2 csc (  /2(x) =  ) for -2 < x < 10