solving both cosine and sine equations using the graphing calculator

•Topic: Sinusoidal functions

Aim: How can we apply what we know about solving cosine equations to solving sine equations? 

•Do Now

•Identify the period, range and amplitude of y = 3 cos Θ   

•Think about a plan

•In Buenos Aires, Argentina, the average monthly temperature is highest in January and lowest in July, ranging from 83° F to 57° F.

•Write a cosine function that models the change in temperature according to the month of the year.

•Step 1: Find amplitude = ½ (83 – 57) = 13

•Step 2: find the period = 12 months

•Step 3: Average temperature = ½ (140) = 70°F

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•Write the Cosine function

•Y = a cos b t

•Where t = # of months since January

•January is where t = 0

•Y = 13 cos b t

•Period = 12

•12 = 2π/b

•b = π/6 (can you explain why?)

•Cosine equation is y = 13 cos π/6 t

•Lets graph the cosine equation over a two year period

•Y = 13 cos π/6 t

•Use the graphing calculator

•Window:

•Xmin = 0

•Xmax = 24

•Xscl = 2

•Ymin = -13

•Ymax = 13

•Yscl = 2

•Y1 = 13 cos πt/6 then press “graph”

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•What is the temperature in August of the first year?

•Use equation y = 13 cos πt / 6

•Where t = 7 (why is t = 7?)

•Solve y = 13 cos 7π/6

•Method 1: Use graphing calculator

2nd calc, value, x = 7, enter

Y = -11.25833

Method 2: type in 13 cos (7π/6) = -11.25833

Solution --- temp in August is 58.75° F.

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•Analysis of temperature in August

•Recall that the average temp is  70°F

•Since y = -11.25 which is the change in temp from the average, the temp in august is 70 – 11.25 = 58.75°F.

•In class assessment

•Problem #36: solving a sine equation using a cosine equation

•Complete problems 10, 11, 18, 19, 21, 27 and 30 on the handout.

•Page 857 circled problems

•All questions should be answered on separate sheet of paper with your name clearly displayed

•Hand in before the bell rings

•Exit slip

•What are the solutions to the following equation in the interval from 0 to 2π?

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•-2 cos Θ = -1.2

•Answer to the DNA question

•Because one cos curve is drawn from 0 to 24 (x-axis), the period of the curve is 24.

•Using the formula y = a cos bx, we can find a = 1 (amplitude) and b = π/12 (because period = 2π/b, so 24 = 2π/b, making b = π/12.)

•Therefore, y = 1cos (π/12)x or simply y = cos (π/12)x .

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