AP PRECALCULUS — UNIT 3B · TRIGONOMETRIC & POLAR FUNCTIONS
3.11 — The Secant, Cosecant, and Cotangent Functions
Notes — The Reciprocal Trig Functions
💡 Learning Objectives (3.11.A)
By the end of this lesson you will be able to:
- Define csc(θ), sec(θ), cot(θ) as reciprocals of sin, cos, tan respectively
- Identify domains, ranges, periods, and asymptotes of each reciprocal function
- Sketch y = sec(x), y = csc(x), and y = cot(x) over multiple periods
- Evaluate reciprocal trig functions at common angles
1. Definitions
💡 The Three Reciprocal Functions
- csc(θ) = 1 / sin(θ) (undefined where sin θ = 0)
- sec(θ) = 1 / cos(θ) (undefined where cos θ = 0)
- cot(θ) = cos(θ) / sin(θ) = 1 / tan(θ) (undefined where sin θ = 0)
⚠️ Common mistake
Be careful: csc is the reciprocal of SIN (not cos), even though it starts with the letter c. The mnemonic ‘they pair OPPOSITELY’ helps: cosecant ↔ sine, secant ↔ cosine.
2. Domains and Ranges
Function | Domain (excludes…) | Range | Period |
|---|---|---|---|
y = csc x | x = kπ | (−∞,−1] ∪ [1, ∞) | 2π |
y = sec x | x = π/2 + kπ | (−∞,−1] ∪ [1, ∞) | 2π |
y = cot x | x = kπ | all reals | π |
Notice: csc and sec each have RANGE excluding (−1, 1) — they never produce values inside that gap. cot covers all real numbers, just like tan.
3. Asymptotes and Zeros
Vertical asymptotes occur wherever the reciprocal would divide by zero:
- y = csc(x): asymptotes at x = kπ (where sin x = 0)
- y = sec(x): asymptotes at x = π/2 + kπ (where cos x = 0)
- y = cot(x): asymptotes at x = kπ (where sin x = 0)
Zeros: csc and sec NEVER equal zero (their numerators are 1). cot(x) = 0 wherever cos(x) = 0, i.e. at x = π/2 + kπ.
4. Key Values
θ | 0 | π/6 | π/4 | π/3 | π/2 | π | 3π/2 |
|---|---|---|---|---|---|---|---|
sin | 0 | 1/2 | √2/2 | √3/2 | 1 | 0 | −1 |
cos | 1 | √3/2 | √2/2 | 1/2 | 0 | −1 | 0 |
csc | undef | 2 | √2 | 2√3/3 | 1 | undef | −1 |
sec | 1 | 2√3/3 | √2 | 2 | undef | −1 | undef |
cot | undef | √3 | 1 | √3/3 | 0 | undef | 0 |
5. Sketching the Graphs
Each reciprocal graph mirrors a U-shape between asymptotes:
- y = csc(x) sits above y = sin(x), bouncing off y = 1 at peaks and asymptoting where sin = 0
- y = sec(x) sits above (or below, in the negative regions) y = cos(x), bouncing off y = 1 at the cosine peaks
- y = cot(x) is similar to tan(x) but reflected and shifted; it DECREASES on every (kπ, (k+1)π)
📝 A handy sketching trick
Draw the parent sine (or cosine, or tan) lightly first, then mark its zeros — those become the new asymptotes. Where the parent has a peak (max or min), the reciprocal touches y = 1 or y = −1 from outside.
6. Symmetry
- sec is EVEN: sec(−θ) = sec(θ) (since cos is even)
- csc is ODD: csc(−θ) = −csc(θ) (since sin is odd)
- cot is ODD: cot(−θ) = −cot(θ)
7. Identities Worth Knowing
Three identities follow directly from the Pythagorean identity sin² + cos² = 1:
- Divide by cos²: tan²(θ) + 1 = sec²(θ)
- Divide by sin²: 1 + cot²(θ) = csc²(θ)
Together with sin² + cos² = 1, these are the THREE PYTHAGOREAN IDENTITIES of trigonometry.
8. Summary
- csc, sec, cot are reciprocals of sin, cos, tan respectively
- csc and sec have range (−∞, −1] ∪ [1, ∞); cot covers all reals
- Periods: csc and sec have period 2π; cot has period π
- Asymptotes appear wherever the parent function equals zero
- Three Pythagorean identities: sin² + cos² = 1, tan² + 1 = sec², 1 + cot² = csc²