Mathematics Education

When you are in middle school they teach you basic terminology in geometry.
Tangent lines just touch a circle in one point. From a Latin word related to English "tactile", or "to touch".
Secant line cut a circle in two points. From a Latin word related to English "sector", or "to cut".
Chords are part of the secant line. From a Latin word that meant the string of a musical instrument.
Complements and Supplements are pairs of angles that add up to 90 and 180 degrees respectively.
Hypotenuse is the longest line in a right triangle. From a Late Latin "hypotenusa", from Greek "hypoteinousa" meaning "stretching under" (the right angle)

Now suddenly in tenth grade when teaching basic trigonometry the words are mysteriously redefined as ratios. We were always taught the mnemonic SOHCAHTOA for sine=opposite/hypotenuse, cosine=adjacent/hyptenuse, and tangent=opposite/adjacent. Then Secant=1/cosine and cotangent=1/tangent. The word "trigonometry" means "the study of triangles".

I don't remember any explanation being given as to why "tangent" and "secant" were being redefined.

ETYMOLOGY LESSON The word "sine" is from Latin "sinus" which means "fold in a garment". The word was a 16th century mistake. The translator confused two different words in Arabic. Arabic jiba "chord of an arc" (from Sanskrit jya "bowstring"), was confused with jaib "fold in a garment." The word "sine" stayed with us for the last four centuries because of this mistake, but the intention was to refer to a "chord".

My question is how do you relate the geometry "line-circle" definitions with the trigonometric "right triangle ratio" definitions? When you were in school did you wonder why the words were being redefined.

The more you study mathematics, the more this pattern repeats. Concepts are taught first to just understand the concept. Then the same concept is repeated at a higher level course where more accurate (or entirely different) definitions are given. Then this may happen additional times when the concept is abstracted into something else, a theorem is stated and proved that encompasses all of the preceding.

For example, consider solving 2X = 4. You first learn how to solve this without understanding it. Then, you may learn the Fundamental Theorem of Algebra without quite understanding it. Then you may see the proof of the FTA in your upper level complex variables course. You then learn general Field theory and solving equations over more general algebraic structures. Then you may learn Galois theory in your Abstract Algebra course. In my first year graduate algebra course, I learned what "X" is. At some point, if you really like equations, you may learn Galois Cohomology and Algebraic Topology. It's like that. The instance you pointed out is one of many.

In the case of trigonometric functions, their proper definitions may be given in terms of exponential functions of a complex variable. These definitions appear to have nothing to do with circles or triangles. see this.

Quote: pacomartin

When you are in middle school they teach you basic terminology in geometry.
Tangent lines just touch a circle in one point. From a Latin word related to English "tactile", or "to touch".
Secant line cut a circle in two points. From a Latin word related to English "sector", or "to cut".
Chords are part of the secant line. From a Latin word that meant the string of a musical instrument.
Complements and Supplements are pairs of angles that add up to 90 and 180 degrees respectively.
Hypotenuse is the longest line in a right triangle. From a Late Latin "hypotenusa", from Greek "hypoteinousa" meaning "stretching under" (the right angle)

Now suddenly in tenth grade when teaching basic trigonometry the words are mysteriously redefined as ratios. We were always taught the mnemonic SOHCAHTOA for sine=opposite/hypotenuse, cosine=adjacent/hyptenuse, and tangent=opposite/adjacent. Then Secant=1/cosine and cotangent=1/tangent. The word "trigonometry" means "the study of triangles".

I don't remember any explanation being given as to why "tangent" and "secant" were being redefined.

It's not so much a redefinition as an associated concept. The tangent of an angle A (tan A) is the length of the tangent line segment between angle=0 and angle=A, where the tangent line touches the unit circle only at angle=0. The secant of the same angle A is the length of the secant line segment that intersects the center of the unit circle and the tangent line at angle A. In other words, "tangent line" and "secant line" still mean what they mean. The trig functions are concerned with the length of segments on those lines based on the angle (and normalized to the unit circle). This diagram from Wikipedia may help: