You may have learned about trigonometric functions such as sine and cosine as being defined by the ratios of sides of a triangle (SOHCAHTOA), or in terms of points and lines related to the unit circle. For me, it didn’t totally click until I started to think visually about inscribing a triangle inside of a circle.
For example, you can think about the sine function as measuring the distance from the x-axis of a point on the unit circle at a particular angle. The sign (+/-) of that value indicates if the point lies above or below the axis. Similarly the cosine can be thought of as measuring the distance from the y-axis of that same point.
It is useful to note that the cosine of an angle is the same as the sine of the complement of the angle. In other words, it is the same operation as sine, just with respect to the y-axis instead of the x-axis.
The word sine originally came from the latin sinus, meaning “bay” or “inlet”. However, it had a long path to get there. The earliest known reference to the sine function is from Aryabhata the Elder, who used both ardha-jya (half-chord) and jya (chord) to mean sine in Aryabhatiya, a Sanskrit text finished in 499 CE.
Jya, meaning chord, became jiba in Arabic, and was abbreviated as just jb. When the term was translated to latin in the twelfth century, jb was incorrectly read as jaib (meaning “bay” or “inlet”), and thus translated as sinus.
The sine function has a direct connection to chords on a circle. Pick two points on the unit circle,
then the length of the line connecting the two points is exactly twice the value of the sine of half the
anlge between them. That is, chord_length(θ) / 2 = sin(θ / 2).
Tangent comes from the latin tangere, the verb meaning “to touch”. A line tangent to a circle intersects it at exactly one point. From this, a geometric construction of the tangent function makes a lot of sense: take the line tangent from a point on the unit circle and calculate the distance along that line from the point of intersection with the circle, to the point of intersection with the x-axis. Similarly, the distance from that same point on the unit circle to the y-axis is the value of the cotangent function.
The origin of secant can be traced to latin as well. It comes from the latin word secare, meaning “to cut”. By definition a secant line on a circle is any line that intersects it in two places, you can think of this line as cutting the circle in two pieces. The secant function is the distance from the origin to the point where the tangent line intercepts the x-axis. Note that if this secant line is extended, it cuts the unit circle neatly in half. Again, the cosecant can be thought of as being the same as the same function with respect to the y-axis.
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