Unit 3 · Topic 3.11

3.11 The Secant, Cosecant, and Cotangent Functions

Secant, cosecant and cotangent are the reciprocals of cosine, sine and tangent. Wherever the original function is 0, the reciprocal has a vertical asymptote, and wherever the original is ±1, the reciprocal touches it.

Key terms

secant
cosecant
cotangent
reciprocal
vertical asymptote

Definitions

Secant: sec θ = 1/cos θ, defined where cos θ ≠ 0.

Cosecant: csc θ = 1/sin θ, defined where sin θ ≠ 0.

Cotangent: cot θ = 1/tan θ = cos θ / sin θ, defined where sin θ ≠ 0. The form cos θ / sin θ is the safer one, because it also works where tan θ is undefined: cot(π/2) = 0/1 = 0.

Secant and cosecant graphs

Taking a reciprocal turns small outputs into large ones. As cos θ approaches 0, sec θ grows without bound, so the graph of sec θ has vertical asymptotes wherever cos θ = 0, at θ = π/2 + kπ.

Where cos θ = 1, sec θ = 1; where cos θ = −1, sec θ = −1. Between asymptotes, the secant graph forms U-shaped branches: opening up from the points where cosine has its maximums, and opening down from the points where cosine has its minimums.

Cosecant does the same with sine. Its vertical asymptotes are where sin θ = 0, at θ = kπ, and its branches touch the sine graph at sine's maximums and minimums.

Because |cos θ| ≤ 1 and |sin θ| ≤ 1, their reciprocals are never between −1 and 1. Both sec θ and csc θ have range (−∞, −1] ∪ [1, ∞). Both have period 2π, like cosine and sine.

Reading the reciprocal from the original

Because these are reciprocals, sec θ always has the same sign as cos θ, and csc θ the same sign as sin θ. If cos θ = 0.2, then sec θ = 5; if sin θ = −0.5, then csc θ = −2.

On an interval where cos θ keeps one sign, sec θ moves the opposite way: where cosine increases, secant decreases, and the reverse. Small values of cosine become large values of secant.

Cotangent graph

cot θ has vertical asymptotes where sin θ = 0 (θ = kπ) and zeros where cos θ = 0 (θ = π/2 + kπ).

Its period is π, like tangent. But between consecutive asymptotes, cotangent is always decreasing, falling from ∞ to −∞, while tangent is always increasing.

Function	Asymptotes	Period	Range
sec θ	θ = π/2 + kπ	2π	(−∞, −1] ∪ [1, ∞)
csc θ	θ = kπ	2π	(−∞, −1] ∪ [1, ∞)
cot θ	θ = kπ	π	All real numbers

Worked examples

Try each one yourself first, then open the solution.

Example 1
Exact reciprocal values
Find sec(π/3), csc(7π/6), cot(3π/4) and sec(π/2), if they exist.
Show the solutionHide the solution
1. Step 1: cos(π/3) = 1/2, so sec(π/3) = 2.
2. Step 2: sin(7π/6) = −1/2, so csc(7π/6) = −2.
3. Step 3: cot(3π/4) = cos(3π/4)/sin(3π/4) = (−√2/2)/(√2/2) = −1.
4. Step 4: cos(π/2) = 0, so sec(π/2) is undefined.
Answer: sec(π/3) = 2, csc(7π/6) = −2, cot(3π/4) = −1, sec(π/2) is undefined.
Example 2
Describing the cosecant graph
Describe the graph of y = csc θ on 0 < θ < 2π.
Show the solutionHide the solution
1. Step 1: Asymptotes where sin θ = 0: θ = 0, π and 2π.
2. Step 2: On (0, π), sin θ is positive with maximum 1 at π/2. So csc θ is positive, opens upward, and has a relative minimum at (π/2, 1).
3. Step 3: On (π, 2π), sin θ is negative with minimum −1 at 3π/2. So csc θ is negative, opens downward, and has a relative maximum at (3π/2, −1).
Answer: Asymptotes at θ = 0, π, 2π; an upward branch with relative minimum (π/2, 1) and a downward branch with relative maximum (3π/2, −1).
Example 3
Trap: cot θ where tan θ is undefined
Find cot(π/2) and cot(π).
Show the solutionHide the solution
1. Step 1: Using cot θ = 1/tan θ fails at π/2, because tan(π/2) is undefined. That doesn't mean cot(π/2) is undefined.
2. Step 2: Use cot θ = cos θ / sin θ: cot(π/2) = 0/1 = 0.
3. Step 3: cot(π) = cos π / sin π = −1/0, which is undefined. θ = π is a vertical asymptote of cotangent.
Answer: cot(π/2) = 0; cot(π) is undefined.

Common mistakes

Confusing reciprocal functions with inverse functions. sec θ = 1/cos θ, but arccos is the inverse of cosine.
Putting secant's asymptotes where sin θ = 0. Secant's come from cosine's zeros.
Thinking cotangent increases like tangent. Each branch of cotangent decreases.

On the exam

Expect exact values and questions about where these graphs have asymptotes, often by matching the reciprocal's graph to the original.
They also appear in identities, such as 1 + tan² θ = sec² θ, in topic 3.12.

Connected topics

Videos

AP Precalculus – 3.11 Cosecant, Secant, Cotangent Functions
The AlgebrosWatch on YouTube (opens in a new tab)
Secant, Cosecant, and Cotangent in Under 3 mins (AP Precalculus Unit 3 Topic 3.11)
Maximum InsightWatch on YouTube (opens in a new tab)
Secant (sec), cosecant (csc) and cotangent (cot) example | Trigonometry | Khan Academy
Khan AcademyWatch on YouTube (opens in a new tab)
AP Precalculus Notes (Topic 3.11) The Secant, Cosecant, and Cotangent Functions
Mr. SindelWatch on YouTube (opens in a new tab)
The OTHER Trigonometric Values EXPLAINED [AP Precalculus Topic 3.11]
Michael Porinchak - AP Statistics & AP PrecalculusWatch on YouTube (opens in a new tab)