Introduction to the trigonometric functions

1. Introduction to the
trigonometric functions
Goethe Schule Wetzlar
Oberstufengymnasium
Goetheschule Wetzlar • Frankfurter Straße 72 • 35578 Wetzlar
www.goetheschule-wetzlar.eu
Unsere Kooperationspartner : Alle Mittelstufenschulen der Stadt Wetzlar
und des Lahn-Dill-Kreises

2. Structure
1. Learning goal
2. GeoGebra: Change presettings
3. Task for the students
4. Possibility of differentiated instruction
5. Possibility to continue this teaching unit
GoetheSchuleWetzlar

3. Learning goal
The students should discover the most
important properties of the trigonometric
functions (sine, cosine, tangent) by exploring
their graphs.
1
GoetheSchuleWetzlar

4. 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
𝜋
2
(or 𝜋) as unit for the x-axis
2
GoetheSchuleWetzlar

5. GeoGebra: Change presettings (II)
3
GoetheSchuleWetzlar

6. GeoGebra: Change presettings (III)
4
GoetheSchuleWetzlar

7. 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.
5
GoetheSchuleWetzlar

8. 6
GoetheSchuleWetzlar
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

9. 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
7
GoetheSchuleWetzlar

10. 8
GoetheSchuleWetzlar
2𝜋 even _______
𝑥 = 𝑘 ∙ 𝜋 𝑘 ∈ ℤ0 𝑥 =
𝜋
2
+ 𝑘 ∙ 𝜋 𝑘 ∈ ℤ0 𝑥 = 𝑘 ∙ 𝜋 𝑘 ∈ ℤ0
𝑥 = 𝑘 ∙ 𝜋 𝑘 ∈ ℤ0 ℝ0 𝑥 =
3𝜋
2
+ 𝑘 ∙ 2𝜋 𝑘 ∈ ℤ0
𝜋 −1; 1 ℝ0
_______ 𝑥 = 𝑘 ∙ 𝜋 𝑘 ∈ ℤ0 𝑥 =
𝜋
2
+ 𝑘 ∙ 2𝜋 𝑘 ∈ ℤ0
odd ℝ0 ℝ0 ∖
𝜋
2
+ 𝑘 ∙ 𝜋, 𝑘 ∈ ℤ0
−1; 1 2𝜋 odd
𝑥 =
𝜋
2
+ 𝑘 ∙ 𝜋 𝑘 ∈ ℤ0 𝑥 = 𝜋 + 𝑘 ∙ 2𝜋 𝑘 ∈ ℤ0 𝑥 = 𝑘 ∙ 2𝜋 𝑘 ∈ ℤ0

11. 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:
• 𝑓 𝑥 = 𝑎 ∙ sin 𝑏 ∙ 𝑥 − 𝑐 + 𝑑
• 𝑔 𝑥 = 𝑎 ∙ cos 𝑏 ∙ 𝑥 − 𝑐 + 𝑑
• ℎ 𝑥 = 𝑎 ∙ tan 𝑏 ∙ 𝑥 − 𝑐 + 𝑑
9
GoetheSchuleWetzlar

12. 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
10
GoetheSchuleWetzlar

13. Possibility to continue this
teaching unit (III)
11
GoetheSchuleWetzlar

14. Possibility to continue this
teaching unit (IV)
12
GoetheSchuleWetzlar

15. Possibility to continue this
teaching unit (V)
• enter the following functions (only one
function at a time) in the input bar:
• 𝑓 𝑥 = 𝑎 ∙ sin 𝑏 ∙ 𝑥 − 𝑐 + 𝑑
• 𝑔 𝑥 = 𝑎 ∙ cos 𝑏 ∙ 𝑥 − 𝑐 + 𝑑
• ℎ 𝑥 = 𝑎 ∙ tan 𝑏 ∙ 𝑥 − 𝑐 + 𝑑
13
GoetheSchuleWetzlar

16. Possibility to continue this
teaching unit (VI)
• choose the following settings:
• 𝑎 = 1
• 𝑏 = 1
• 𝑐 = 0
• 𝑑 = 0
• vary each of the parameters (only one
parameter at a time) and observe how the
graph of the function changes
14
GoetheSchuleWetzlar

17. Possibility to continue this
teaching unit (VII)
15
GoetheSchuleWetzlar

18. Possibility to continue this
teaching unit (VIII)
• the students should discover the following
functions of the parameters :
• 𝑎: vertical dilation (amplitude)
• 𝑏: horizontal dilation
• 𝑐: horizontal translation (phase shift)
• 𝑑: vertical translation
16
GoetheSchuleWetzlar

19. Thank you for your
attention!!!
17
GoetheSchuleWetzlar