Counting Complete Cycles of cos(4x) on [-270°, 270°]

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2014-05-13 21:22

In collections CCGPS AA Unit 5 Lesson 2

This GeoGebra example demonstrates the graph of the cosine function g(x)=cos(4x) on the restricted domain [-270°, 270°] and asks how many complete cycles occur in that interval. It highlights periodic functions, angular frequency, period length, degree/radian conversion, domain restrictions, and graph interpretation. Since the period of cos(4x) is 90°, the 540° interval contains six complete cycles. The construction uses LaTeX and FractionText for formulas, If/Min/Max for conditional display and bounds, Sequence and TableText for counting or tabular text, and objects such as stepMax, stepper, steps, counter, toggle, button1, domdeg, domrad, fullper, fplotdeg, fplotrad, xmin, xmax, and ymin to control domain, period, plotting range, and interactive stepping. It is suitable for high-school algebra, pre-calculus trigonometry lessons, classroom demonstrations, and student exploration of cosine cycles.

GeoGebra Commands & Parameters

LaTeX(steps) → counter
LaTeX("\text{Identify the period of }\textcolor{#000099}{g(x).}") → text1
LaTeX("\text{This will be needed to calculate the number of complete cycles of }\textcolor{#000099}{g(x)}\text{ that}\cr\text{exist in the restricted domain.}") → text2
LaTeX("\text{Set the argument of the function equal to the period of the parent cosine}\cr\text{function, }\textcolor{#000099}{\cos(x)}\text{, and solve for }\textcolor{#000099}{x}.") → text3
FractionText(permult) → permulttext
LaTeX((((("\text{The period of the function }\textcolor{#000099}{g(x) = " * g) * "}\text{ is }\textcolor{#000099}{") * permulttext) * "\pi}.")) → text5
FractionText(dlrad) → dlradtext
FractionText(drrad) → drradtext
If(toggle ≟ 1, LaTeX[(("\text{The period of }\textcolor{#000099}{g(x)}\text{, converted to degrees, is }\textcolor{#000099}{" * perdeg) * "^{\circ}}.")], LaTeX[(((("\text{The domain of }\textcolor{#000099}{g(x)}\text{, converted to radians, is }\textcolor{#000099}{\left[" * dlradtext) * "\pi,") * drradtext) * "\pi\right]}.")]) → text9
If(toggle ≟ 1, LaTeX["\text{In order to determine the number of completed cycles, calculate the total}\cr \text{number of degrees within the domain of the function, and then divide this}\cr\text{number by the period measured in degrees.}"], LaTeX["\text{In order to determine the number of completed cycles, calculate the total}\cr \text{number of radians within the domain of the function, and then divide this}\cr\text{number by the period measured in radians.}"]) → text11
LaTeX("\text{Domain:}") → text12
LaTeX("\text{Period:}") → text14
LaTeX("\frac{Domain}{Period}:") → text16
If(toggle ≟ 1, LaTeX[(((((("\textcolor{#000099}{" * dr) * "^{\circ}-\left(") * dl) * "^{\circ}\right)=") * domdeg) * "^{\circ}}")], LaTeX[(((((("\textcolor{#000099}{" * drradtext) * "\pi-\left(") * dlradtext) * "\pi\right)=") * domrad) * "\pi}")]) → text13
If(toggle ≟ 1, LaTeX[(("\textcolor{#000099}{" * perdeg) * "^{\circ}}")], LaTeX[(("\textcolor{#000099}{" * permulttext) * "\pi}")]) → text15
LaTeX("\text{Verify the result by graphing.}") → text19
If(toggle ≟ 1, dl - perdeg / 4, ((dlrad - permult / 4) * π)) → xmin
If(toggle ≟ 1, dr + perdeg / 4, ((drrad + permult / 4) * π)) → xmax
Min(0 - a / 4, c - a - a / 4) → ymin
Max(0 + a / 4, c + a + a / 4) → ymax
LaTeX((((((((("\text{The number of complete cycles of }\textcolor{#000099}{g(x)=" * g) * "}\text{ on the}\cr\text{restricted domain }\textcolor{#000099}{[") * dl) * "^{\circ},") * dr) * "^{\circ}]}\text{ is }\textcolor{#000099}{") * fullper) * "}.")) → text18
If(toggle ≟ 1, Sequence[(dl + (n * perdeg), fplotdeg(dl + (n * perdeg))), n, 0, fullper, 1], Sequence[(((dlrad + (n * permult)) * π), fplotrad(((dlrad + (n * permult)) * π))), n, 0, fullper, 1]) → list1
Sequence(Vector[Element[list1, n], Element[list1, n + 1]], n, 1, Length[list1] - 1) → list2
LaTeX((((((("\textcolor{#000099}{g(x)=" * g) * "}\text{ on }\textcolor{#000099}{\left[") * dl) * "^{\circ},") * dr) * "^{\circ}\right]}")) → Problem
LaTeX("\text{Convert the period to degrees or convert the domain to radians.}") → text6
If(toggle ≟ 1, LaTeX["\text{Converting period to degrees}"], LaTeX["\text{Converting domain to radians}"]) → toggleText
LaTeX("\text{Since the domain is given in degrees and the period is found in radians, one}\cr\text{or the other must be converted so that the units match before any comparison can be made.}") → text7
LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → text20
If(toggle ≟ 1, LaTeX[(((((("\textcolor{#000099}{\text{Greatest integer less than or equal to }\left(\frac{" * domdeg) * "^{\circ}}{") * perdeg) * "^{\circ}}\right)=") * fullper) * "}")], LaTeX[(((((("\textcolor{#000099}{\text{Greatest integer less than or equal to }\left(\frac{" * domrad) * "\pi}{") * permult) * "\pi}\right)=") * fullper) * "}")]) → text17
LaTeX((((("\text{Determine the number of complete cycles }\textcolor{#000099}{g(x)}\text{ makes over the domain }\textcolor{#000099}{[" * dl) * "^{\circ}, ") * dr) * "^{\circ}]}.")) → text10
If(toggle ≟ 1, LaTeX[(((((("\textcolor{#000099}{1}\text{ radian}\textcolor{#000099}{=\frac{180^{\circ}}{\pi}}\text{, so }\textcolor{#000099}{" * permulttext) * "\pi}\text{ radians is }\textcolor{#000099}{") * permulttext) * "\pi\cdot\frac{180^{\circ}}{\pi}=") * perdeg) * "^{\circ}}.")], LaTeX[(((((((((("\textcolor{#000099}{1^{\circ}=\frac{\pi\text{ radians}}{180}}\text{, so }\cr \textcolor{#000099}{" * dl) * "^{\circ}}\text{ is }\textcolor{#000099}{") * dl) * "^{\circ}\frac{\pi}{180^{\circ}}=") * dlradtext) * "\pi}\cr \text{and}\cr \textcolor{#000099}{") * dr) * "^{\circ}\frac{\pi}{180^{\circ}}=") * drradtext) * "\pi}.")]) → text8
TableText({{LaTeX[(("\textcolor{#000099}{" * b) * "x = 2\pi}")], LaTeX["\text{Set the argument equal to the period of }\textcolor{#000099}{\cos(x)}."]}, {LaTeX[(("\textcolor{#000099}{x = \frac{2}{" * b) * "}\pi}")], LaTeX["\text{Solve for }\textcolor{#000099}{x}."]}, {LaTeX[(("\textcolor{#000099}{x = " * permulttext) * "\pi}")], LaTeX["\text{Simplify.}"]}}) → text4