chapter 3 graphing trigonometric functions 3.1 basic graphics 3.2 graphing y = k + a sin bx and y =...

Chapter 3Graphing Trigonometric Functions3.1 Basic Graphics3.2 Graphing y = k + A sin Bx

and y = k +A cos Bx3.3 Graphing y = k + A sin (Bx + C)

and y = k +A cos (Bx + C)3.4 Additional Applications3.5 Graphing Combined Forms3.6 Tangent, Cotangent, Secant, and Cosecant Functions Revisited

3.1 Basic Graphs

Graphs of y = sin x and y = cos xGraphs of y = tan x and y = cot xGraphs of y = csc x and y = sec xGraphing 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 xGraphing y = sin Bx and y = cos BxGraphing y = A sin Bx and y = A cos BxGraphing y= k + A sin Bx

and y = k + A cos BxApplications

Comparing Amplitudes

Compare the graphs of y = 1/3 sin x

and y = 3 sin xThe 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 = -2 + 3 cos 2x, - x 2Find 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. 4

22

02

2

x

x

x

Steps for Graphing

3.4 Additional Applications

Modeling electric currentModeling light and other electromagnetic

wavesModeling water wavesSimple and damped harmonic motion:

resonance

Alternating Current Generator

I = 35 sin (40t – 10) (current) Amplitude = 35 Phase shift:

40t = 10t = 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(f1t – r/)

t = time, r = distance from the source, is the wavelength, f1 is the frequency

Damped Harmonic Motion

Y = (1/t)sin (/2)t, 1 t 8First, 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 5/2Phase shift = -1/2Period = 2Asymptotes 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