4.5 Graphs of Sine and Cosine Functions

*Sketch sine and cosine graphs*Use amplitude and period*Sketch translations of sine and cosine graphs

Key Points

» One period = the intercepts, the maximum points, and the minimum points

Amplitude and Period of Sine and Cosine Curves

» Amplitude: of y = a sinx and y = a cosx represent half the distance between the maximum and minimum values of the function and is given by ˃ Amplitude = IaI

» Period: Let be be a positive real number. The period of y = a sinbx and y = a cosbx is given by ˃ Period = (2π)/b

Graphing By Hand

» Sketch the graph y = 2sinx by hand on the interval [-π, 4π]

Graphing Using Period

» Sketch the graph of y = cos (x/2) by hand over the interval [-4π, 4π]

Translations of Sine and Cosine

» The constant c in the general equations ˃ y = a sin(bx – c) and y = a cos(bx – c)

create horizontal translations of the basic sine and cosine curves.

» One cycle of the period starts at bx – c = 0 and ends at bx – c = 2π

» The number c/b is called a phase shift

Horizontal Translation

» Sketch the graph y = ½ sin (x – π/3)

Vertical Translations

» Sketch the graph y = 2 + 3 cos2x

Writing an Equation

» Find the amplitude, period, and phase shift for the sine function whose graph is shown. Then write the equation of the graph.

» For a person at rest, the velocity v (in liters per second) of air flow during a respiratory cycle (the time from the beginning of one breath to the beginning of the next) is given by where t is the time (in seconds) .

(inhalation occurs when v> 0 and exhalation occurs when v < 0 )a) Graph on the calculatorb) Find the time for one full respiratory cyclec) Find the number of cycles per minuted) The model is for a person at rest. How might the

model change for a person who is exercising?

3sin85.

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» A company that produces snowboards, which are seasonal products, forecast monthly sales for 1 year to be where S is the sales in thousands of units and t is the time in months, with t = 1 corresponding to January.

a) Graph the function for a 1 year periodb) What months have maximum sales and which

months have minimum sales.

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tS

Mathematical Modeling

» Throughout the day, the depth of the water at the end of a dock in Bangor, Washington varies with the tides. The table shows the depths (in feet) at various times during the morning.

» Use a trigonometric function to model this data.» A boat needs at least 10 feet of water to moor at

the dock. During what times in the evening can it safely dock?

Time 12am 2 am 4 am 6 am 8 am 10am 12pm

Depth, y 3.1 7.8 11.3 10.9 6.6 1.7 .9