ACT® · Math: Functions — Trigonometric Functions
Reciprocal and Quotient Identities
Secant, cosecant, and cotangent are just reciprocals of cosine, sine, and tangent. Sec(x) = 1/cos(x), csc(x) = 1/sin(x), cot(x) = 1/tan(x) = cos(x)/sin(x).
Trig only truly needs sine and cosine. Everything else is built from them for convenience, because some ratios show up so often that giving them their own name saves work. Tangent is sin(x)/cos(x), the rise-over-run slope idea. The other three are its mirror images: secant is 1/cos(x), cosecant is 1/sin(x), cotangent is 1/tan(x), which also equals cos(x)/sin(x). Flip the fraction, get the new name.
The mechanism is pure arithmetic, not a new idea. Cosine is a ratio of sides, adjacent over hypotenuse say. Secant takes that same ratio and turns it upside down: hypotenuse over adjacent. No angles are being undone, nothing is inverted in the function sense. This is why sec(x)cot(x) simplifies so cleanly: write both as fractions of sine and cosine, cancel what matches, and you are left with a single reciprocal function.
The trap is confusing a reciprocal with an inverse function. cot(x) is 1/tan(x), a number you get by dividing. tan−1(x), sometimes written arctan, is a completely different machine: it takes a ratio and hands back an angle. Pressing the wrong key swaps a division problem for an angle-finding one. Once these six ratios are straight, they feed directly into the identities built on the unit circle and into solving equations where swapping a reciprocal in makes a messy expression collapse.
Key terms
- Secant
- The reciprocal of cosine: sec(x) = 1/cos(x).
- Cosecant
- The reciprocal of sine: csc(x) = 1/sin(x).
- Cotangent
- The reciprocal of tangent: cot(x) = 1/tan(x) = cos(x)/sin(x).
- Inverse function (tan−1)
- A function that takes a ratio and returns the angle that produced it, not a reciprocal.
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End of topic.