The ScienceMath- project: Introduction of Trigonometric Function
Idea: Marina Rugelj, St. Stanislav Institute Ljubljana, Slovenia
Teaching Material
(Links: see first page of Internet-Version)
 Slides: Trigonometric Function (power-point)
 Applets: For viewing: please download “Ruler and Compass CaR” from
internet, free download
Next pages: Lesson Plan
Below: Worksheet: Trigonometric Function (worksheet to copy)
This project has been funded with support from the European Commission. This publication reflects the views only of the
author(s), and the Commission cannot be held responsible for any use which may be made of the information contained therein.
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The ScienceMath- project: Introduction of Trigonometric Function
Idea: Marina Rugelj, St. Stanislav Institute Ljubljana, Slovenia
Proposal for a Lesson Plan:
Introduction of Trigonometric Functions
Content
Circulation and
oscillation introduction
Teacher
Teacher shows to students
some objects from everyday life: wheel, bicycle,
children's swing, clock …
Pupils
Slide 2
T: How are they moving?
Describe their movements!
T: Could you find some
other objects which oscillate
or circulate?
T: Is there relation between
these two motions:
circulation and oscillation?
Experiment:
Take a gramophone and
put a stick on it. Behind the
gramophone place a simple
pendulum. The stick
circulates on the
gramophone and the weight
of the pendulum oscillates.
Make a projection of the
stick and the weight on the
wall. Observe the shadows
of the stick and the weight.
What can you see?
T: Yes, but only if we
attended for the same
amplitudes and frequencies.
Graph of sin motion
P: Some of them circulate
and some of them oscillate.
Worksheet
Slide 3
P: Windmill, gramophone,
swing door…
Slide 4
Slide 5
P: … that the shadow of
the stick overlaps with the
shadow of the weight.
Applet 1
Worksheet
Ps sketch the experiment.
T: Take a point T which
circulates around the unit
circle. Put the center of the
circle in the starting point of
the coordinate system and
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The ScienceMath- project: Introduction of Trigonometric Function
Idea: Marina Rugelj, St. Stanislav Institute Ljubljana, Slovenia
observe the projection of
the point T on y–axis. It
goes up and down.
Could you draw a graph of
the position of projected
point in dependence of
time?
Definition of sin
function
T: We know that ration
between hypotenuse and
cathetus in similar rightangled triangles is always
the same:
Applet 2
Ps try to draw a sin curve.
Slide 6
a / c = a’ / c’ = a’’ / c’’
It depends on the angle x,
it is a functions of the angle
x and we named this
function sinus:
f ( x)  a  sin x .
c
If hypotenuse c = 1 than
sin x = a.
We can draw this angle x in
the unit circle. We also get
the right-angled triangle.
The length of the
hypotenuse is equal 1 ( c =
1), therefore
sin x 
a a
  a.
c 1
Slide 7
Slide 8
Worksheet
Ps draw the sin value for
different angels in unit
circle.
And a is exactly the ycoordinate of the point T on
the circle.
Transformations
of sin - function:
f (t )  A sin t  c
f (t )  A sin t
T: A is amplitude.
In the case of oscillation
A means a displacement
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The ScienceMath- project: Introduction of Trigonometric Function
Idea: Marina Rugelj, St. Stanislav Institute Ljubljana, Slovenia
from the rest position.
In the case of circulation
A means bigness of a
radium.
Could you draw two
different graphs for two
different parameters A, e.g.
A = 1 and A = 2?
Slide 9
Worksheet
Applets 3
Ps try to draw the graphs
of two different amplitudes.
f (t )  sin t
T: Parameter w we name
frequency.
In the case of oscillation w
means number of beats per
second.
In the case of circulation w
means angular velocity.
Could you draw two
different graphs of two
different parameters w.
Slide 10
Worksheet
Ps try to draw the graphs
of two different
frequencies.
Slide 11
Worksheet
f (t )  sin t  c
Ps try to draw the graphs
of the function
f (t )  sin t  2 .
T: Try to draw the graph of
this function if c = 3!
Practical task
There is a windmill and
point A on the propeller as
you can see on the picture.
The function which
describes the height of the
point A is equal:
Slide 12
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The ScienceMath- project: Introduction of Trigonometric Function
Idea: Marina Rugelj, St. Stanislav Institute Ljubljana, Slovenia
h(t )  3.5 sin
2
t  7.5
12
Questions:
1. How high is rod?
2. How big is radius of the
propeller?
3. In what time does the
point come around?
Check the answers:
1. The rod is 7,5 m
high.
2. The radius of the
propeller is 3,5 m.
3. The point need 12
secunds.
Worksheet
Applet 4
Ps can work in pairs and
try to answer these
questions.
Next pages: worksheet
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The ScienceMath- project: Introduction of Trigonometric Function
Idea: Marina Rugelj, St. Stanislav Institute Ljubljana, Slovenia
Worksheet: INTRODUCTION OF TRIGONOMETRIC FUNCTIONS
1. Find and put down some objects from every-day life which oscillate or circulate!
2. Is there any relation between these two motions: circulation and oscillation? Make a
sketch of the experiment!
3. Draw a sin curve!
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The ScienceMath- project: Introduction of Trigonometric Function
Idea: Marina Rugelj, St. Stanislav Institute Ljubljana, Slovenia
4. Draw the value of sin x for four angels x!
90o < x < 180o
270o < x < 360o
180o < x < 270o
0o < x < 90o
5. Draw the graphs of two functions in the same coordinate system:
f ( x)  sin x
f ( x)  2 sin x
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The ScienceMath- project: Introduction of Trigonometric Function
Idea: Marina Rugelj, St. Stanislav Institute Ljubljana, Slovenia
6. Draw the graphs of two functions in the same coordinate system:
7. Draw the graphs of two functions in the same c. system:
f ( x )  sin x
f ( x )  sin 2 x
f ( x)  sin x
f ( x)  sin x  2
y
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The ScienceMath- project: Introduction of Trigonometric Function
Idea: Marina Rugelj, St. Stanislav Institute Ljubljana, Slovenia
8. There is a windmill and point A on the propeller as you can see on the picture.
The function which describes the height of the point A is equal:
A
h(t )  3.5 sin
2
t  7.5
12
a. How high is the rod?
b. How big is radius of the propeller?
c. In what time does the point come around?
9.
Sketch the graph!
h
t
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