Intuition of Trigonometric Functions

The Etymology of Sine

The English word “sine” originates from Hindu works which used the Sanskrit word jya-ardha meaning “half chord”. This was often shortened to jya or jiva. Later, the Arabs translated the word phonetically as jiba, which is meaningless in Arabic. Since, the Arabic language uses accents instead of characters to represent vowels, over time the consonants jb were mistaken for the word jaib, meaning bosom or breast. When Arabic works were eventually translated into Latin, jaib was translated into the equivalent Latin word sinus. Finally, this Latin word was anglicized into the word sine, which we now use in English.

Relationship to the Unit Circle

I’m not sure how helpful these diagrams are in terms of a greater understanding of trigonometry, but they at least help explain where the rest of the trigonometric functions get their names.

These relationships can be easily proved using similar triangles and the definitions of the functions. I will leave that for the reader to work out.

Where the Trigonometric Functions Get Their Names

First, I need to give some definitions.

A tangent intersects the circumference of a circle at one point. A chord is a line segment whose endpoints both lie on the circumference of the circle. A secant is the extension of a chord.

Now, hopefully, the names of the rest of the trigonometric functions are clear. “Sine” and “Cosine” are both half-chords of the circle (which is ultimately the origin of their confused etymology). “Secant” and “Cosecant” are both half-secants of the circle. Finally, “Tangent” and “Cotangent” are both half-tangents of the circle.

Also notice that \(\{ \sin \theta, \sec \theta, \tan \theta \}\) and \(\{\cos \theta, \csc \theta, \cot \theta \}\) are duals of each other. You will see in the identities of later posts, there is always a symmetry between these functions, which is a direct result of the symmetry of their geometry.

Graphs

I provide the graphs of the trigonometric functions. It’s nice to keep these in mind when interpreting the consequences of trig identities.

Notice the asymptotes in $\sec$, $\csc$, $\tan$, and $\cot$. See if you can determine which values of $\theta$ cause these asymptotes.

What are Trigonometric Functions Fundamentally?

Eventually, we are going to see that all trigonometric functions can be expressed solely in terms of $\sin$. Each of the six functions are just syntactic sugar for a singular deeper truth, which is the relationship of a circle’s radius, arc length, and corresponding chord length.