Beyond SOH-CAH-TOA: An Example of How History Helps Us to Understand Trigonometry
Aha! Moment
As a teacher several years ago, I recall introducing my students to trigonometric ratios using the acronym “SOH CAH TOA.” If they remembered the acronym, students could calculate the trigonometric ratios sine, cosine, and tangent given a right triangle when they were provided with some information about the measures of angles and/or side lengths. This was the way I was introduced to trigonometry, and using the acronym reminded me how to calculate the values of trigonometric ratios for acute angles in right triangles. Teaching using just the acronym, though, deprives the students of an opportunity to understand what the ratios represent. Students may be able to tell you that sine can be calculated by finding the ratio of the length of the side opposite an angle in a right triangle to the length of the hypotenuse of the right triangle.
However, the acronym does not help them understand why the ratio is defined this way. Armed only with “SOH CAH TOA,” it becomes confusing for students to find the values of trigonometric ratios for angles whose measures are 90 degrees or greater, or negative. The acronym is not a bad tool, but it is insufficient to help students understand trigonometric ratios.
Using History to Explain What Sine Measures
The Eureka Math curriculum addresses the historical context that led to the formalizing of the trigonometric ratios sine, cosine, and tangent. It describes the research of ancient astronomers who were interested in the position of stars, and in particular, the heights of the stars in the sky. The lesson outlines the research of ancient Indian astronomer-mathematician Aryabhata I, whose ideas built upon those of ancient Babylonian and Greek astronomers. He believed that the stars rotated about the Earth in circular orbits. The diagram below models the sun rotating counterclockwise about the Earth in a circular orbit relative to an observer standing on the Earth’s surface and facing north. The sun rises in the east (to the right) and then rotates counterclockwise until it sets in the west (to the left). The shaded region represents when the sun would not be visible to the observer.
To model the movement of the sun, we could measure the angle of elevation of the sun as it moves counterclockwise, as shown:
The position of the sun relative to the earth could be determined using only the distance r and the angle of elevation of the sun with the horizon (the horizontal line through point E). Since the radius is constant in our model, the perceived height of the sun is determined solely by the angle of elevation x measured in degrees.
The research of Aryabhata I provided a means by which the height of astronomical objects could be calculated.
Aryabhata I used the abbreviation jya (pronounced jhah) for half of the length of the chord joining the endpoints of the arc of a circle. The term jya was an abbreviation for a Sanskrit term representing a half-chord of a circle, and the value jya represents the height of a star for a given angle of counterclockwise rotation from the eastern horizon.
In his text Aryabhatiya, Aryabhata constructed a table of values of the jya in increments of 3 3/4 degrees, which were used to calculate the positions of astronomical objects.
From Jya to Sine
Transcribed letter-by-letter into Arabic in the 10th century, jya became jiba. In medieval writing, scribes regularly omitted vowels to save time, space, and resources, so the Arabic scribes wrote just jb. Since jiba isn’t a real word in Arabic, later readers interpreted jb as jaib, which is an Arabic word meaning cove or bay. When translated into Latin around 1150 C.E., jaib became sinus, which is the Latin word for bay. Sinus got shortened into sine in English. (Eureka Math A Story of Functions, Grade 11, Module 2, Lesson 3)
Why This Matters
While it is interesting to learn about how the term jya evolved into the term sine, the importance of the history behind sine is that students understand it represents a height. Within the context of astronomy, it represents the height of an object in the sky. In a mathematical context, it represents the vertical distance from the center of the unit circle to any point on that circle. Knowing this is extremely important when the students are required to expand their understanding of trigonometry beyond right triangles. When viewing sine as a height, it is much easier to understand when sine is positive (when a point is above the center of the circle) and when it is negative (when the point is below the center). It is easier to consider whether or not calculated values are reasonable. For instance, the sine of an angle that has a very small positive value will be close to zero, because an object in the sky would be just above the horizon line. Understanding the history helps students understand the graphs of trigonometric functions as well. For instance, it makes sense that the graph of y = sin(x) has a relative maximum at x = 90 degrees because this would represent when the sun is directly overhead at its maximum perceived height in the sky.
This illustration demonstrates that providing a historical context for mathematics can not only engage students, but help them to understand mathematical concepts in a way that is richer than providing the information without a context.
This post is authored by Sara Lack, a Eureka Math writer for A Story of Functions.
