FUNCTIONS AND GRAPHS

Construction of the cosine function

Definition of the cosine of an acute angle:

Let A be an acute angle of a right angled triangle, remember that the cosine of the angle A is the quotient between the adjoining side of the right angle AB and the hypotenuse AC.

1.- Modify the value of angle A and see how the value of the cosine changes. (You can press the coloured arrows or write the value of the angle between 0º and 90º)

Verify that if you only modify the length of the side of the right angle AB then the side BC and the hypotenuse AC also change, however the angle A doesn´t change and neither does the quotient AB/AC either, which is the value of the cosine.       

Definition of the cosine of any angle.

Let A be any angle, if we represent it with the vertex at the origin of coordinates and one side on the positive X axis OX, then the cosine of the angle can be obtained as the quotient between the abscissa of any point of the second side and its distance from the vertex.

2.- Modify the value of the angle A and see how the value of the cosine changes. Try for positive values, negative values, values greater than 360º, etc. (You can press the coloured arrows or write the value of any angle)

Check that if you only modify the distance from the point P to the origin, without changing the angle, then the coordinates x and y also change, however the angle A doesn´t change and the quotient x/d, which is the value of the cosine, doesn´t change either.

The cosine of the goneometric circumference.

Goniometric circumference is the name given to that which has its centre at the origin of coodinates and a radius of one. Any point on the circumference is 1 unit away from the origin, therefore, if we represent the angle with the vertex at the origin of coordinates and one side on the positive X axis OX, the value of the cosine coincides with the abscissa of the point of intersection of the other side of the angle with the goniometric circumference. 

3.- Modify the value of the angle and notice that the cosine of the angle is the length of the blue segment. (You can use the coloured arrows or write the value of any angle)

The cosine function

Construction of the cosine function from the goniometric circumference.

4.- Increase the value of the angle at the goniometric circumference and watch the values of the cosine on the circumference and the graph y=cos(x), where x is the angle measured in radians. (You can use the coloured arrows or write the value of the angle between 0º and 360º)

The graph of the cosine function

After a complete lap of the goniometric circumference the values of sine start to repeat themselves again.  For that reason it is said that this function is periodic, of period 2*pi.

5.- Use the change of scale, increasing and decreasing it. (You can use the coloured arrows or write the value of any angle)

What graph does that of the cosine look like?

Author: Juan Madrigal Muga

	Ministerio de Educación, Cultura y Deporte. Año 2000