TRIGONOMETRY I

Measurement of angles

Trigonometry is a branch of mathematics which studies the relationships between the sides and the angles of triangles.  The Babilonians and Egyptians (more than 3000 years ago) were the first to use the angles of a triangle and trigonometric ratios.  They were used to make advances in agriculture and for the construction of pyramids.  After that, the study of trigonometry moved to Greece, and from there to India and Arabia where it was used in Astronomy.  From Arabia trigonometry spread throughout Europe, where it became independent of Astronomy.       

The measurement of an angle.

Two straight lines form an angle, called the initial and final sides of the angle. The point of interserction is called the vertex of the angle.  Let us assume the vertex of the angle is on the origin of the coordinate system and the initial side on the positive x axis.   Positive angles are measured in the clockwise direction and the negative angles in the opposite or anticlockwise direction.

To obtain the representation of a definite precise determined angle, measured in degrees, in the following  Descartes applet, it is enough to write it directly in the text field corresponding to the degrees.  The Init button returns to the original situation.

Draw the following angles: 60º, 90º, 120º, 135º, 30º, -60º, 270º, 10º, -120º, -90º, -180º, 45º, -45, 360º, 420º, 43.57º, 133.45º, -23.1º.

The sexagesimal system

The degree is the most common unit of measurement for angles and arcs of a circle.  It is divided into 60 minutes, and each minute is divided into 60 seconds.  Degrees are usually indicated with the symbol °, minutes with ' and seconds with ", as in 21°43'25", which is read as: 21 degrees 43 minutes and 25 seconds.  The measurement of angles in degrees is widely used in engineering and in physical science, mainly in astronomy, navigation and topography.

Draw the following angles: 23.57º, 123.017º, -45.33º y 203.209º. Use the Windows calculator to represent the following angles, converting them firstly into degrees: 34º23'56", 126º35', 375º25'30" Check with Descartes applet above that the calculation is correct.

Complementary and supplementary angles

Complementary angles are those which add up to 90º.  In the Descartes applet below, the complementary angle AOB (red) can be seen in the colour magenta. 

Find the complementary angles of 23º, 45º, 30º 60º, 35.75, 32º11'27"

Supplementary angles are those which add up to 180º.  In the Descartes applet below, the supplementary angle can be seen in green colour.

Find the supplementary angles of: 60º, 90º, 120º, 135º, 30º, 10º, 45º, 43.57º, 133.45º.

The measurement of an angle in radians

Another form of measuring angles is the radian, which is defined as the central angle whose corresponding arc is equal to the radius.  In circumferences of different radii the angle corresponding to a radian is the same, for that reason it is valid as a measurement of an angle.  

The Descartes applet below draws the corresponding arcs to each angle in two circumferences of radii 2 and 3.  

Represent the angles 30º, 60º, 90º, 120º, 180º, -60º, -90º, -180ºy 360º. Divide the values of the arcs by their radius and see if the result is what appears as the value of each angle in radians.

Equivalences between radians and degrees

The length of the circumference of radius r is 2pr if we divide its length by the radius we will obtain the radians of an angle of 360º, for that reason the relationship between both is: 360º=2p radians. When an angle is expressed in radians it is not necessary to specify the units.

Write on your notebook how many radians are one degree.  Calculate approximately the value in degrees of a radian.  First use the arrows and then add decimals until exactly one radian appears.  Find a formula to convert degrees into radians and another to convert radians into degrees.  Express, according to the number pi, the values in radians of the angles: 180º, 360º, 90º, 45º, 30º, 60º, 135º and 120º.

Author: Miguel García Reyes

	Ministerio de Educación, Cultura y Deporte. Año 2000