Given cosθ=4/5 in Quadrant I, find cotθ – GeoGebra Demo

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2014-05-13 21:01

In collections CCGPS AA Unit 5 Lesson 1

This example demonstrates how to compute cotθ when cosθ = 4/5 and θ lies in the first quadrant, using trigonometric identities. Core concepts include the definitions of trigonometric functions, quadrant sign rules, the relationship between cosine and cotangent, and the Pythagorean theorem on the unit circle. Construction highlights: LaTeX for formula rendering; If and Element for step control; Segment, Circle, Angle to draw the reference triangle and angle marks; objects stepMax, stepper, steps, counter drive a step‑by‑step animation; numer, denom, funcs, funcnum1, third store intermediate values; A, B, C, quadrant define vertices and quadrant; button1, text3, text5 provide interactive buttons and annotations; θ, θ_ref, scale, ratio manage scaling and coordinate bounds. Suitable for high‑school trigonometry instruction, self‑study review, or classroom demonstration to visualize how quadrant affects trigonometric signs.

GeoGebra Commands & Parameters

LaTeX(steps) → counter
If(quadrant ≟ "I", {(Element[third, funcnum1], numer), (numer, Element[third, funcnum1]), (denom, numer), (Element[third, funcnum1], denom), (denom, Element[third, funcnum1]), (numer, denom)}, If[quadrant ≟ "II", {((-Element[third, funcnum1]), numer), ((-numer), Element[third, funcnum1]), ((-denom), numer), ((-Element[third, funcnum1]), denom), ((-denom), Element[third, funcnum1]), ((-numer), denom)}, If[quadrant ≟ "III", {((-Element[third, funcnum1]), (-numer)), ((-numer), (-Element[third, funcnum1])), ((-denom), (-numer)), ((-Element[third, funcnum1]), (-denom)), ((-denom), (-Element[third, funcnum1])), ((-numer), (-denom))}, {(Element[third, funcnum1], (-numer)), (numer, (-Element[third, funcnum1])), (denom, (-numer)), (Element[third, funcnum1], (-denom)), (denom, (-Element[third, funcnum1])), (numer, (-denom))}]]) → Boptions
Element(Boptions, funcnum1) → B
If(quadrant ≟ "I", {"", "", "", "", "", ""}, If[quadrant ≟ "II", {"", "-", "-", "", "-", "-"}, If[quadrant ≟ "III", {"-", "-", "", "-", "-", ""}, {"-", "", "-", "-", "", "-"}]]) → signOptions
Segment(A, C) → a
Segment(A, B) → b
Segment(C, B) → c
Circle(A, B) → d
Element(denomLabelPtOpt, funcnum1) → denomLabelPt
Element(numerLabelPtOpt, funcnum1) → numerLabelPt
LaTeX(numer) → numerLabel
LaTeX(denom) → denomLabel
Angle((Length[b], 0), A, B) → θ
Angle(C, A, B) → θ_{ref}
If(funcnum1 ≟ 1, LaTeX["\sin(\theta)=\frac{\text{length of opposite side}}{\text{length of hypotenuse}}"], If[funcnum1 ≟ 2, LaTeX["\cos(\theta)=\frac{\text{length of adjacent side}}{\text{length of hypotenuse}}"], If[funcnum1 ≟ 3, LaTeX["\tan(\theta)=\frac{\text{length of opposite side}}{\text{length of adjacent side}}"], If[funcnum1 ≟ 4, LaTeX["\csc(\theta)=\frac{\text{length of hypotenuse}}{\text{length of opposite side}}"], If[funcnum1 ≟ 5, LaTeX["\sec(\theta)=\frac{\text{length of hypotenuse}}{\text{length of adjacent side}}"], LaTeX["\cot(\theta)=\frac{\text{length of adjacent side}}{\text{length of opposite side}}"]]]]]) → text3
If(funcnum1 ≟ 1, LaTeX[((((("c^2=a^2+b^2 \cr a^2=c^2-b^2 \cr a=\sqrt{c^2-b^2} \cr a=\sqrt{" * denom) * "^2-") * numer) * "^2}=") * Element[third, 1])], If[funcnum1 ≟ 2, LaTeX[((((("c^2=a^2+b^2 \cr b^2=c^2-a^2 \cr b=\sqrt{c^2-a^2} \cr b=\sqrt{" * denom) * "^2-") * numer) * "^2}=") * Element[third, 2])], If[funcnum1 ≟ 3, LaTeX[((((("c^2=a^2+b^2 \cr c=\sqrt{a^2+b^2} \cr c=\sqrt{" * denom) * "^2+") * numer) * "^2}=") * Element[third, 3])], If[funcnum1 ≟ 4, LaTeX[((((("c^2=a^2+b^2 \cr a^2=c^2-b^2 \cr a=\sqrt{c^2-b^2} \cr a=\sqrt{" * numer) * "^2-") * denom) * "^2}=") * Element[third, 4])], If[funcnum1 ≟ 5, LaTeX[((((("c^2=a^2+b^2 \cr b^2=c^2-b^2 \cr b=\sqrt{c^2-a^2} \cr b=\sqrt{" * numer) * "^2-") * denom) * "^2}=") * Element[third, 5])], LaTeX[((((("c^2=a^2+b^2 \cr c=\sqrt{a^2+b^2} \cr c=\sqrt{" * numer) * "^2+") * denom) * "^2}=") * Element[third, 6])]]]]]) → text5
Element(thirdLabelPtOpt, funcnum1) → thirdLabelPt
LaTeX(Element[third, funcnum1]) → thirdLabel
LaTeX((("\text{The length of the third side is " * Element[third, funcnum1]) * " units.}")) → text6
Element(adjacentOptions, funcnum1) → adj
Element(oppositeOptions, funcnum1) → opp
Element(hypotenuseOptions, funcnum1) → hyp
LaTeX(((Element[funcs, funcnum2] * "(\theta_{ref})=") * Element[ansOpts, funcnum2])) → text10
If(funcnum2 ≟ 1, LaTeX["\sin(\theta)=\frac{\text{length of opposite side}}{\text{length of hypotenuse}}"], If[funcnum2 ≟ 2, LaTeX["\cos(\theta)=\frac{\text{length of adjacent side}}{\text{length of hypotenuse}}"], If[funcnum2 ≟ 3, LaTeX["\tan(\theta)=\frac{\text{length of opposite side}}{\text{length of adjacent side}}"], If[funcnum2 ≟ 4, LaTeX["\csc(\theta)=\frac{\text{length of hypotenuse}}{\text{length of opposite side}}"], If[funcnum2 ≟ 5, LaTeX["\sec(\theta)=\frac{\text{length of hypotenuse}}{\text{length of adjacent side}}"], LaTeX["\cot(\theta_{ref})=\frac{\text{length of adjacent side}}{\text{length of opposite side}}"]]]]]) → text9
LaTeX((("\text{Find }" * Element[funcs, funcnum2]) * "(\theta).")) → text7
LaTeX("\text{Move the slider to solve the problem.}") → text0
LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → text12
LaTeX("\text{The lengths of two sides of the reference triangle are given by the magnitudes}\cr\text{of the numerator and denominator of the value of the trigonometric function.}") → text2
LaTeX("\text{Use the Pythagorean Theorem to find the length of the third side.}") → text4
LaTeX((("\text{Use the values from the triangle to determine }" * Element[funcs, funcnum2]) * "(\theta_{ref}).")) → text8
LaTeX((((((((((("\text{Since }\theta\text{ lies in Quadrant }" * quadrant) * "\text{, }") * Element[funcs, funcnum2]) * "(\theta)=") * Element[signOptions, funcnum2]) * Element[funcs, funcnum2]) * "(\theta_{ref})=") * Element[signOptions, funcnum2]) * Element[ansOpts, funcnum2]) * ".")) → text11
LaTeX((((((((((((("\text{In Quadrant " * quadrant) * ", if }") * Element[funcs, funcnum1]) * "(\theta)=") * Element[signOptions, funcnum1]) * "\frac{") * numer) * "}{") * denom) * "}\text{ what is }") * Element[funcs, funcnum2]) * "(\theta)?")) → Problem
LaTeX("\text{Sketch angle }θ\text{ (measured counterclockwise from the positive x-axis),}\cr\text{the associated triangle, and the reference angle }\theta_{ref}\text{ (the acute angle}\cr\text{the terminal side makes with the x-axis).}") → text1