FUNCTIONS AND  GRAPHS

Construction of the sine function

Definition of the sine of an acute angle:

Let A be an acute angle of a right angle triangle, remember that the sine of angle A is the quotient between the opposite side of a right angle BC and the hypotenuse AC.

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

Check that if only the length of the side AB is modified, then the side BC and the hypotenuse AC also change, however the angle A doesn´t change and the quotient BC/AC, which is the value of sine doesn´t change either.

Definition of the sine of any angle.

Let A be any angle, if we represent it with the vertex at the origin of coordinates and one side against the positive X axis OX then the sine of the angle can be obtained as the quotient between the ordinate of any point of the second side and its distance from the vertex. (The positive angles are measured in an anti clockwise direction)

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

Check that if only the distance from the point P to the origin is modified, without changing the angle, the coordinates x and y  also change, however the angle A doesn´t change and the quotient y/d, which is the value of sine, does not change either.

The sine at the goniometric circumference.

Goniometric circumference is the name given to that which has its centre at the origin of coordinates and a radius of one. Any point on the circumference is at distance 1 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 sine coincides with the ordinate of the point of intersection of the other side with the goniometric circumference.

3.- Modify the value of the angle and see how the sine of the angle is the length of the green segment. (You can use the coloured arrows or write the value of any angle)

The sine function

Construction of the sine function from the goniometric circumference.

4.- Increase the value of the angle at the goniometric circumference and watch the values of sine on the circumference and the graph y=sen(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 sine function

After a complete lap around 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 function does the sine look like for values close to zero?

Author: Juan Madrigal Muga

	Ministerio de Educación, Cultura y Deporte. Año 2000