The reciprocal trig functions

Each new function is a pointwise reciprocal of an old one:

1. Secant: $\sec\theta = 1/\cos\theta$. Since cos(π/3) = 1/2, sec(π/3) = 2.
2. Cosecant: $\csc\theta = 1/\sin\theta$. Since sin(π/6) = 1/2, csc(π/6) = 2.
3. Cotangent: $\cot\theta = \cos\theta/\sin\theta$, the reciprocal of tangent wherever both are defined. cot(π/4) = 1; cot(π/6) = (√3/2)/(1/2) = √3.

The recipe acts on the value, not the angle. To evaluate sec(π/3), first get cos(π/3) = 1/2, then flip the 1/2 into 2. Nothing is done to the π/3 itself. Signs come along for free: the reciprocal of a negative number is negative, so each reciprocal function keeps its parent's sign in every quadrant.

A reciprocal explodes where its parent is zero: 1 divided by a number near 0 is enormous. So:

1. csc has vertical asymptotes where sin θ = 0: at θ = 0, π, 2π, every multiple of π.
2. sec has them where cos θ = 0: at θ = π/2, 3π/2, every odd multiple of π/2.
3. cot shares csc's asymptotes (multiples of π), because its denominator is also sin θ.

The classic wrong answer puts csc's asymptotes at π/2 and 3π/2, the peaks of sine. Exactly backwards: at a peak, sin θ = ±1, and the reciprocal of ±1 is just ±1. The peaks are where the reciprocal curve touches its parent, the calmest points on the graph. The explosions happen over the parent's x-intercepts, where there is a zero to divide by.

Because |sin θ| ≤ 1 and |cos θ| ≤ 1, their reciprocals satisfy |csc θ| ≥ 1 and |sec θ| ≥ 1. The range of both is $(-\infty, -1] \cup [1, \infty)$: everything except the open band between −1 and 1. There is no angle whose secant is 0.4, and none whose cosecant is 0.

That is why the graphs look like stacks of U-shapes: each branch sits on a peak of the parent wave (touching at height 1 or −1) and flees to ±∞ as it approaches the neighboring asymptotes. Small parent values near the zero crossings become huge reciprocal values; the parent's modest hills become the reciprocal's resting points. Cotangent is the exception: as a ratio rather than a pure reciprocal, its range is all real numbers, its period is π, and every branch falls from +∞ to −∞.

The notation collision of the unit: $\csc\theta$ and $\sin^{-1}\theta$ are different functions. Cosecant flips the output value: sin(π/6) = 1/2, so csc(π/6) = 2. Arcsine (written sin⁻¹) runs the function backwards: sin⁻¹(1/2) = π/6, an angle. One answers "one over the sine," the other answers "which angle has this sine."

The same goes for sec versus cos⁻¹ and cot versus tan⁻¹. A quick self-test: sec θ · cos θ = 1 wherever both are defined, because that is what reciprocal means. No such product rule holds for cos⁻¹. When the −1 sits on a trig function name, it means inverse function; when you want the reciprocal, say sec, csc, or cot, or write 1/cos θ explicitly. The exam leans on this distinction, and so do the scenarios below.