•Topic: Sinusoidal functions

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

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

•

•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”

•

•

•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.

•

•

•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?

•

•-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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