The term trigonometry may sound foreboding and a topic that is not included in an elementary school curriculum, but the foundational concepts of trigonometry are taught to even very young children. Trigonometry is basically the field of mathematics that describes the relationships between the angles and sides of triangles, or triangle measurement. Trigonometry is critical in the fields of navigation and surveying. It is important to engage elementary age students in understanding the basic concepts of trigonometry and this can be done effectively through creating games such as: Trigonometry Outdoors, Trigonometry Indoor Search Game and Triangle Card Match.

The purpose of the trigonometry outdoors search game is to have students observe their environment in a different way than usual and to see how the outdoor environment is filled with different kinds of triangles. Prior to going outdoors to look for triangles in the environment, talk with students about the field of trigonometry and how it was founded in the 15th century to help people find their way to places and to survey the earth to make maps and to locate land boundaries. Show them examples of acute triangles, right triangles and equilateral triangles. Take students outside and have them find samples of these triangles in the environment. The student who finds examples of all three kinds of triangles is the winner.

The purpose of the trigonometry indoor search game is similar to the outdoor game but the search is completed indoors and students keep a journal of the triangles they find, their location and the purpose of the triangle. This part of the game teaches aspects of the scientific method such as observation, documentation (journal) and developing hypotheses (purpose). Before starting the game, show students examples of acute triangles, right triangles and equilateral triangles and talk about some of the uses of triangles--building bridges, roofs, stairs etc. Ask students to find as many triangles as they can for a week and keep their journal up-to-date. At the end of the week, the student with the most notations wins.

The triangle card match game helps students learn about the different kinds of triangles by the angles of the sides. Use 5 by 7 inch note cards and draw one of each of these types of triangles on a card: scalene, isosceles, equilateral, acute scalene, right scalene, obtuse scalene, right isosceles, obtuse isosceles, obtuse isosceles and equilateral. You may want to have multiple examples of each type of triangle so the game lasts longer. Write the name of each kind of triangle on a card too. To play the match game, place each triangle card face up and put the cards with the names of the triangle face down in a pile. Have students draw the name of a triangle from the pile and find the triangle it goes with. The student with the most matches wins.

-Mary Johnson-Gerard

Trigonometry is a study of math whose origins date back to the ancient Egyptians. The principles of trigonometry deal mostly with the sides, angles and functions of triangles. The most common triangle that is used in trigonometry is the right triangle, which is the basis for the Pythagorean Theorem, in which the square of both sides of a right triangle are equal to the square of its longest side or hypotenuse.

Greeks and Trigonometry

In the second century B.C., Hipparchus derived a trigonometric table measuring chord lengths of a circle having a fixed radius. Hipparchus built the values in increasing degrees, beginning with 71 and ending with 180, incrementing in units of 71 degrees. In the second century A.D., Ptolemy defined Hipparchus' value for the radius as 60 and created a table of chords incrementing one degree, from 0 degrees to 180 degrees. This table of chords also showed how to find unknown parts of triangles from given parts.

In the sixth century, India based its trigonometry on the sine function, which was the length of the side opposite the angle in a right triangle of a specific hypotenuse instead of a ratio. The Indians used various values for the hypotenuse. They built sine tables from these functions and later introduced a cosine function and tables.

Abu Abdallah Muhammad Ibn Jabir al-Battani, who lived from 858 to 929, formally introduced the cosine function, as he built on the work of the Indians and Greeks. Muslim mathematicians also introduced the polar triangle for spherical triangles, sine and tangent tables created in 1/60th-of-a-degree steps. Nasir ad-Din at-Tusi, who lived from 1201 to 1274, wrote a book separating plane and spherical trigonometry into its own field of study, called the Book of the Transversal Figure. Muslim mathematicians revived the long-dead tangent function, invented by the Chinese but lost, and added the co-tangent, co-secant, and secant functions.

Georges Joachim defined trigonometric functions as ratios instead of lengths of lines during the 13th century. French mathematician François Viète, who lived from 1540 to 1653, introduced the polar triangle into spherical trigonometry and published two books, Canon Mathematicus, and Universalium Inspectionum Liber Singularis, in 1579. These two books were mathematical tables in which the values for sine are computed to 10 to the negative eighth power . In the 17th century, John Napier, a Scottish mathematician, invented logarithms, memory tricks to remember the 10 laws of how to solve spherical triangles; he also came up with what are now called Napier's analogies to help mathematicians solve oblique spherical triangles. In the 18th century, Leonhard Euler defined trigonometric functions in terms of complex numbers showing how basic laws of trigonometry were the consequences of arithmetic of complex numbers.

Early forms of trigonometry appeared in Chinese mathematics in the sixth century, but major advances in trigonometry did not happen until the 12th and 13th centuries, even though astronometrical calculations and calendar science demanded it. Shen Kuo, who lived from 1031 to 1095, used trigonometry to solve problems of chords and arcs. Guo Shoujing worked on arcs and circles, which formed the foundation of spherical trigonometry during the 12th and 13th centuries. Most of China's mathematics were lost after the Yuan Dynasty took root in 1271 until the 19th century