The Six Ratios as Lengths

The names in this subject look arbitrary. They are not. Every one of the six functions is the length of a specific line segment in one drawing, and the names describe what those segments do.

Start with the two already known. Put the radius at angle θ on the unit circle. Drop a vertical line from the point on the rim to the x axis. That segment has length sin θ. Now measure along the x axis from the origin to the foot of that vertical. That segment has length cos θ.

This is the half chord from page 1. The full chord joins two points on the rim; the half chord runs from the rim to the midpoint of the chord, and on the unit circle it is exactly the sine.

There is a detail about signs that the picture cannot show. A drawn length is always positive. The functions are not: sin θ is negative below the x axis and cos θ is negative to the left of the origin. Read the drawn segments as magnitudes and take the sign from which side of the origin the segment lies on. Page 5 is entirely about getting those signs right.

Keep the same picture. The extended radius now leaves the circle and carries on to the tangent line, so it cuts straight through the circle on its way. A line that cuts through a circle is a secant line A straight line that cuts a curve, meeting it at two or more points. For a circle it is exactly two, because a straight line cannot cross a circle three times. The extended radius of the unit circle is a secant line, and the secant function is the distance measured along it from the origin to the tangent line., from the Latin secare, to cut.

Measure along that cutting line from the origin all the way out to where it meets the tangent line. That distance is sec θ, the secant The reciprocal of cosine, sec θ = 1 / cos θ. On the unit circle it is the distance from the origin along the extended radius out to the vertical tangent line at (1, 0), a line that cuts the circle, which is where the name comes from (Latin secare, to cut). Period 2π, never between -1 and 1, undefined wherever cosine is zero. Note it pairs with cosine, not sine. of the angle.

The relationship to the other lengths comes out of the same big triangle. That triangle has an adjacent side of exactly 1, an opposite side of tan θ, and a hypotenuse running from the origin to the intersection. Pythagoras on it gives 1² + tan²θ = sec²θ, which is one of the three identities page 6 derives algebraically. Here it is a picture.

Secant is never shorter than the radius. The shortest way from the origin to that tangent line is straight along the x axis, a distance of 1. So the magnitude of sec θ is at least 1, always, and its magnitude is exactly 1 only when the radius points along the x axis. At 0 degrees that gives sec θ = 1 and at 180 degrees it gives sec θ = minus 1, because the distance is then measured backwards along the line. That is why the graph of secant on page 4 has a hole in the middle: it never takes a value strictly between minus 1 and 1.

The words tangent and secant were given these meanings by the Danish mathematician Thomas Fincke in his Geometria rotundi of 1583.

Now do the tangent construction again against the other tangent line.

Draw the horizontal line through (0, 1), the topmost point of the circle. It touches the circle at that one point. Extend the radius until it meets this line. The segment from (0, 1) across to the meeting point has length cot θ, the cotangent The reciprocal of tangent, cot θ = 1 / tan θ = cos θ / sin θ. On the unit circle it is the length cut off on the horizontal tangent line at (0, 1) by the extended radius. Period π, range all real numbers, undefined wherever sine is zero, and it equals tan(90° − θ).. The distance from the origin out to that meeting point is csc θ, the cosecant The reciprocal of sine, csc θ = 1 / sin θ. On the unit circle it is the distance from the origin along the extended radius out to the horizontal tangent line at (0, 1). Period 2π, never between -1 and 1, undefined wherever sine is zero. Despite the "co" prefix it pairs with sine, not cosine, which is the most common confusion in the subject..

That is the entire difference. Tangent and secant are measured against the vertical tangent line at the right of the circle; cotangent and cosecant are measured against the horizontal tangent line at the top. Same two constructions, rotated a quarter turn.

This is what the "co" prefix means, and it means something precise. Two angles are complementary The angle that brings a given angle up to 90 degrees, so the complement of θ is 90° − θ. The two acute angles of a right triangle are complements of each other, which is why swapping which one you measure from swaps the opposite and adjacent sides. if they add up to 90 degrees. The cofunction The function of the complementary angle. cos θ = sin(90° − θ), cot θ = tan(90° − θ) and csc θ = sec(90° − θ), which is what the "co" prefix in cosine, cotangent and cosecant records. Geometrically it is the same construction measured from the other acute angle of the same right triangle. of an angle is the function of its complement. So cos θ = sin(90 degrees minus θ), and equally cot θ = tan(90 degrees minus θ) and csc θ = sec(90 degrees minus θ).

The reason goes back to page 1. In a right triangle the two acute angles are complementary, and swapping which one you call theta swaps the opposite and adjacent sides. So the sine of one angle is the cosine of the other, using exactly the same two side lengths in the same fraction. The "co" functions are not new functions. They are the old ones measured from the other acute corner.

Check one value. sin(30 degrees) is 0.5 exactly, and cos(60 degrees) is 0.5 exactly, and 30 plus 60 is 90.

Divide sine by cosine. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and dividing one by the other cancels the hypotenuse. What is left is opposite over adjacent, which is the tangent.

tan θ = sin θ / cos θ

cot θ = cos θ / sin θ

These are called the quotient identities, and they hold for every angle where the bottom is not 0. Combined with the reciprocal identities, every one of the six functions can now be written using only sine and cosine. Six functions, two of them genuinely independent, and even those two are linked by cos²θ + sin²θ = 1.

The quotient form agrees with the picture. Tangent blows up at 90 degrees because cos(90 degrees) is 0 and the fraction divides by zero. The geometry said the same thing in different words: the radius becomes parallel to the tangent line and never meets it. These are two descriptions of one fact.

On the unit circle there is a third reading. The point is at (cos θ, sin θ), so sin θ over cos θ is y over x, which is the slope of the radius. Tangent of an angle is the gradient of a line at that angle to the horizontal. That is the version used in graphics and in surveying, and it is why a 45 degree slope has a gradient of exactly 1, because tan(45 degrees) = 1.