Introduction to the trigonometric functions

Goethe Schule WetzlarOberstufengymnasium

Goetheschule Wetzlar • Frankfurter Straße 72 • 35578 Wetzlar www.goetheschule-wetzlar.eu

Unsere Kooperationspartner : Alle Mittelstufenschulen der Stadt Wetzlar und des Lahn-Dill-Kreises

Structure1. Learning goal 2. GeoGebra: Change presettings3. Task for the students4. Possibility of differentiated instruction5. Possibility to continue this teaching unit

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Learning goal

The students should discover the most important properties of the trigonometric functions (sine, cosine, tangent) by exploring their graphs.

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GeoGebra: Change presettings (I)

• when you enter trigonometric functions, the default unit is radians

• use radians:• right-click on the graphics view and pick Graphics

• click on the tab xAxis• put a cross in the box next to Distance• choose (or ) as unit for the x-axis

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GeoGebra: Change presettings (II)

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GeoGebra: Change presettings (III)

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Task for the students

Enter the sine (cosine, tangent) function in the input bar. Discover the properties of each of these trigonometric functions by exploring their graphs and complete the following table.

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The trigonometric functions (sine, cosine, tangent) function f(x) = sin(x) g(x) = cos(x) h(x) = tan(x)

domain

codomain

symmetry

roots

maxima

minima

inflection points

period

Possibility of differentiated instruction

• solutions in disorder (see next slide) • only for students who have difficulties to

complete the task • they can cut out the solutions and glue

them on the appropriate place on the table

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2𝜋 even _______

𝑥= 𝑘∙𝜋 ሺ𝑘 ∈ℤ0ሻ 𝑥= 𝜋2 + 𝑘∙𝜋 ሺ𝑘 ∈ℤ0ሻ

𝑥= 𝑘∙𝜋 ሺ𝑘 ∈ℤ0ሻ

𝑥= 𝑘∙𝜋 ሺ𝑘 ∈ℤ0ሻ ℝ0 𝑥= 3𝜋2 + 𝑘∙2𝜋 ሺ𝑘 ∈ℤ0ሻ 𝜋 ሾ−1;1ሿ ℝ0

_______

𝑥= 𝑘∙𝜋 ሺ𝑘 ∈ℤ0ሻ 𝑥= 𝜋2 + 𝑘∙2𝜋 ሺ𝑘 ∈ℤ0ሻ odd ℝ0 ℝ0 ∖ቄ𝜋2 + 𝑘∙𝜋,𝑘 ∈ℤ0ቅ

ሾ−1;1ሿ 2𝜋 odd

𝑥= 𝜋2 + 𝑘∙𝜋 ሺ𝑘 ∈ℤ0ሻ 𝑥= 𝜋+ 𝑘∙2𝜋 ሺ𝑘 ∈ℤ0ሻ 𝑥= 𝑘∙2𝜋 ሺ𝑘 ∈ℤ0ሻ

Possibility to continue this teaching unit (I)

• the students discover the transformations of the trigonometric functions

• in general the transformations can be represented as follows:

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Possibility to continue this teaching unit (II)

• the students discover the function of the parameters a, b, c, d by using sliders

• create sliders• click on the button Slider• click on the desired position• choose the name (a, b, c, d), the interval of

the values and the increment (steps)• click on apply

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Possibility to continue this teaching unit (III)

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Possibility to continue this teaching unit (IV)

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Possibility to continue this teaching unit (V)

• enter the following functions (only one function at a time) in the input bar:

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Possibility to continue this teaching unit (VI)

• choose the following settings:

• vary each of the parameters (only one parameter at a time) and observe how the graph of the function changes

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Possibility to continue this teaching unit (VII)

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Possibility to continue this teaching unit (VIII)

• the students should discover the following functions of the parameters :• : vertical dilation (amplitude) • : horizontal dilation • : horizontal translation (phase shift)• : vertical translation

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Thank you for your attention!!!

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