So far, you have learned about the three basic trigonometric functions and their inverse functions. In this guide, you will learn about the reciprocal functions of the three trig functions. In trigonometry, the reciprocal functions play a crucial role in understanding the relationships between the different trigonometric functions. The three reciprocal functions are cosecant, secant, and cotangent, denoted by , , and respectively.
The Cosecant Function
The cosecant function, denoted by , is a trigonometric function that is defined as the reciprocal of the sine function. In other words, .
That's a strange graph! Let's break it down by exploring the main characteristics of the cosecant function. The domain of the cosecant function is all real numbers, except for , where n is an integer. These values of correspond to the vertical asymptotes of the cosecant function, which means that the function becomes infinitely large at every integer multiple of , so , , , etc. At every integer multiple of , , and since cosecant is defined as , that means we would attempt to compute , which is not possible. Therefore, we have a vertical asymptote at these values. The range of the cosecant function is .
The period of the cosecant function is , which means that the function repeats every units. This is the same period as the sine function, which makes sense because the cosecant function is defined as the reciprocal of the sine function.
In terms of the unit circle, the cosecant function can be understood as the ratio of the length of the hypotenuse to the length of the y-coordinate. The hypotenuse is always equal to 1, so can also be written as . The cosecant function is related to the sine function in that it is the reciprocal of the sine function. In other words, . This means that when the sine function is at its minimum or maximum values, the cosecant function is at its maximum or minimum values, respectively.
The Secant Function
The secant function, denoted by , is a trigonometric function that is defined as the reciprocal of the cosine function. In other words, .
The domain of the secant function is all real numbers, and the range is the set of all real numbers greater than or equal to 1. The secant function has vertical asymptotes at , where is an integer. This means that the secant function becomes infinitely large at these values of . The expression will always yield an odd integer, so the vertical asymptotes of the cosecant function are at , , , etc. As you might have already noticed, these values are where , and since we cannot divide 1 by 0, we have a vertical asymptote.
The period of the secant function is , which means that the function repeats every units. This is the same period as the cosine function, which makes sense because the secant function is defined as the reciprocal of the cosine function. In terms of the unit circle, the secant function can be understood as the ratio of the length of the hypotenuse to the length of the x-coordinate. The hypotenuse is always equal to 1, so can also be written as .
The secant function is related to the cosine function in that it is the reciprocal of the cosine function. In other words, . This means that when the cosine function is at its minimum or maximum values, the secant function is at its maximum or minimum values, respectively.
The Cotangent Function
The cotangent function, denoted by , is a trigonometric function that is defined as the reciprocal of the tangent function. In other words, .
The domain of the cotangent function is all real numbers, except for , where is an integer. These values of correspond to the vertical asymptotes of the cotangent function, which means that the function becomes infinitely large at these points. At every integer multiple of , the tangent function is 0, so trying to compute 1/0 yields a value that is undefined, and therefore, a vertical asymptote. Also, since , taking the reciprocal of the function yields . This also means that when , there is a vertical asymptote. The range of the cotangent function is .
The period of the cotangent function is , which means that the function repeats every units. This is the same period as the tangent function, which makes sense because the cotangent function is defined as the reciprocal of the tangent function. In terms of the unit circle, the cotangent function can be understood as the ratio of the x-coordinate to the y-coordinate. This means that .
The cotangent function is related to the tangent function in that it is the reciprocal of the tangent function. In other words, . This means that when the tangent function is at its minimum or maximum values, the cotangent function is at its maximum or minimum values, respectively. While the tangent function was always increasing, the cotangent function is always decreasing because we took the reciprocal of the tangent function.
CHOSHACAO: You can use the acronym CHOSHACAO to remember the reciprocal functions. CHO = Cosecant is Hypotenuse over Opposite. SHA = Secant is Hypotenuse over Adjacent. CAO = Cotangent is Adjacent over Hypotenuse. In case you didn't notice, this acronym is the reciprocal of SOHCAHTOA!
Frequently Asked Questions
What's the formula for secant cosecant and cotangent functions?
Secant, cosecant, and cotangent are the reciprocal trig functions: - sec θ = 1 / cos θ, defined where cos θ ≠ 0. Range: (-∞, -1] ∪ [1, ∞). Vertical asymptotes at cos θ = 0. Period = 2π. sec(−θ) = sec θ (even). - csc θ = 1 / sin θ, defined where sin θ ≠ 0. Range: (-∞, -1] ∪ [1, ∞). Vertical asymptotes at sin θ = 0. Period = 2π. csc(−θ) = −csc θ (odd). - cot θ = cos θ / sin θ = 1 / tan θ, defined where sin θ ≠ 0 (equivalently tan θ ≠ 0). Period = π. Vertical asymptotes where tan θ = 0 (i.e., sin θ = 0). cot(−θ) = −cot θ (odd). Its graph is decreasing between consecutive asymptotes. For AP work, remember domain exclusions (denominator ≠ 0) and typical graph features (asymptotes, |y| ≥ 1 for sec/csc, period, parity). For the Topic 3.11 study guide see (https://library.fiveable.me/ap-pre-calculus/unit-3/secant-cosecant-cotangent-functions/study-guide/nhIcN0Whx8hECmPVKlvK). For extra practice questions, check (https://library.fiveable.me/practice/ap-pre-calculus). Also keep your calculator in radian mode on the AP exam when working trig.
How do I find where sec x and csc x have vertical asymptotes?
Think “where the denominator is zero.” Sec x = 1/cos x has vertical asymptotes exactly where cos x = 0, and csc x = 1/sin x has asymptotes where sin x = 0. In radians those are: - sec x: x = π/2 + kπ, for any integer k (period 2π, asymptotes every π) - csc x: x = kπ, for any integer k (period 2π) (Also note cot x = cos x / sin x has asymptotes where sin x = 0—same x = kπ—and period π.) Why this works: at those x the reciprocal functions blow up because you’re dividing by zero, so the graph goes to ±∞. Remember ranges: sec and csc have |y| ≥ 1. For more examples and graphs tied to the AP CED, see the Topic 3.11 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/secant-cosecant-cotangent-functions/study-guide/nhIcN0Whx8hECmPVKlvK) and the Unit 3 overview (https://library.fiveable.me/ap-pre-calculus/unit-3). For extra practice, try problems at (https://library.fiveable.me/practice/ap-pre-calculus). On the AP exam, angles are in radians, so use those values when solving graph or equation questions.
I'm confused about the difference between cot x and tan x - can someone explain?
Short version: tan x = sin x / cos x, while cot x = cos x / sin x (equivalently cot x = 1 / tan x). So they’re reciprocals—where one is undefined the other is zero. Key differences you should memorize (CED keywords): - Definitions: tan x = sin x/cos x; cot x = cos x/sin x (cot is the reciprocal of tan) (3.11.A.4). - Domains: tan is undefined where cos x = 0; cot is undefined where sin x = 0 (so cot has vertical asymptotes at multiples of π, because sin = 0 there) (3.11.A.4–5). - Zeros/asymptotes: tan = 0 when sin x = 0; cot = 0 when cos x = 0. - Period & symmetry: tan and cot have period π; cot is odd (cot(−x) = −cot x) and is decreasing between consecutive asymptotes (3.11 keywords). - Graph tip: cot graphs look like reflected/shifted tan graphs because of the reciprocal relationship; expect vertical asymptotes where the denominator is 0 and monotonic decrease between them. For more practice and the topic study guide, check the Fiveable Topic 3.11 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/secant-cosecant-cotangent-functions/study-guide/nhIcN0Whx8hECmPVKlvK) and do lots of graph/identity problems from the unit page (https://library.fiveable.me/ap-pre-calculus/unit-3). Remember on the AP exam angles are in radians unless told otherwise.
When do I use csc x vs sec x in problems?
Use sec x when the problem naturally involves cosine (sec x = 1/cos x, cos x ≠ 0); use csc x when it naturally involves sine (csc x = 1/sin x, sin x ≠ 0). That’s the basic rule from the CED (3.11.A.1–A.2). Quick reminders that help you choose: - If the original expression/graph/model has cos or factors of cos in the denominator, write sec. If it has sin or factors of sin in the denominator, write csc. - Domain/asymptotes: sec has vertical asymptotes where cos x = 0 (x = π/2 + kπ); csc has asymptotes where sin x = 0 (x = kπ). - Range: both have |y| ≥ 1. Periods: sec and csc period 2π; cot is π and equals cos/sin when useful (3.11.A.4–A.5). - For solving equations, convert reciprocals to sine/cosine (or vice versa) to use reference angles and unit-circle values easily. For more examples and graphs to practice these choices, see the Topic 3.11 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/secant-cosecant-cotangent-functions/study-guide/nhIcN0Whx8hECmPVKlvK) and the Unit 3 overview (https://library.fiveable.me/ap-pre-calculus/unit-3). For extra practice problems (1000+), check (https://library.fiveable.me/practice/ap-pre-calculus).
What's the range of secant and cosecant functions and why is it weird?
Secant and cosecant are just reciprocals: sec θ = 1/cos θ and csc θ = 1/sin θ. Since sin and cos only take values between −1 and 1, their reciprocals can never land strictly between −1 and 1—any number x with |x| < 1 would require 1/x to be bigger than 1 in magnitude, which isn’t possible. So the range for both is (−∞, −1] ∪ [1, ∞). It feels “weird” because unlike sine/cosine (which are bounded between −1 and 1), sec and csc blow up to ±∞ when cos or sin = 0, creating vertical asymptotes at those x-values (domain excludes where denominator = 0). Also note: sec is even, csc is odd, sec and csc have period 2π (cot has period π)—all from the CED (Topic 3.11). For a focused review check the Topic 3.11 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/secant-cosecant-cotangent-functions/study-guide/nhIcN0Whx8hECmPVKlvK). For more practice across Unit 3, use the unit overview (https://library.fiveable.me/ap-pre-calculus/unit-3) and the 1000+ practice questions (https://library.fiveable.me/practice/ap-pre-calculus).
How do I graph cot x step by step?
Step-by-step to graph y = cot x (use radians on the AP exam): 1. Identify domain/asymptotes: cot x = cos x / sin x, so undefined where sin x = 0 → vertical asymptotes at x = kπ (…, −π, 0, π, 2π, …). Mark these. 2. Period and symmetry: period = π (CED 3.11.A.4–A.5). Cot is odd: cot(−x) = −cot x. 3. Zeros: cot x = 0 when cos x = 0 → x = π/2 + kπ. Plot zeros at π/2, 3π/2, etc. 4. Behavior between asymptotes: on each interval (kπ, (k+1)π) cot is continuous and strictly decreasing from +∞ (just right of kπ) to −∞ (just left of (k+1)π). So draw a smooth curve passing through the zero at the midpoint and dropping from +∞ to −∞. 5. Key values to plot: cot(π/4)=1, cot(3π/4)=−1; these help shape each hump. For one period 0
Why does cotangent decrease between its asymptotes but tangent increases?
Think of cot θ = cos θ / sin θ and tan θ = sin θ / cos θ. Take derivatives: d/dθ(tan θ) = sec^2 θ > 0 wherever tan is defined, so tan is increasing on each interval between its vertical asymptotes. d/dθ(cot θ) = −csc^2 θ < 0 where cot is defined, so cot is decreasing between its asymptotes. You can also see it without calculus: pick an interval between cot’s asymptotes, e.g. (0, π). On (0, π) sin θ > 0, so cot θ has the same sign and general trend as cos θ. As θ increases from 0 to π, cos θ decreases, so cot θ = cos/sin decreases. For tan on (−π/2, π/2), cos θ > 0 so tan follows sin, which increases there—hence tan increases. These facts match the CED essentials (cot = cos/sin, vertical asymptotes where sin = 0, cot decreasing between asymptotes; tan’s derivative sec^2 > 0). For the AP topic study guide, see Fiveable’s Topic 3.11 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/secant-cosecant-cotangent-functions/study-guide/nhIcN0Whx8hECmPVKlvK). For extra practice, check Fiveable’s AP Precalculus problems (https://library.fiveable.me/practice/ap-pre-calculus).
I don't understand why sec x = 1/cos x - how does that work?
Think of cosine as a ratio from right triangles or the unit circle: cos θ = adjacent/hypotenuse. The secant is defined as the reciprocal of cosine, so sec θ = hypotenuse/adjacent = 1/(adjacent/hypotenuse) = 1/cos θ. Algebraically that means sec θ · cos θ = 1, so sec θ = 1/cos θ whenever cos θ ≠ 0 (if cos θ = 0 the reciprocal is undefined, which is why sec has vertical asymptotes at those angles). This is exactly the CED reciprocal identity (3.11.A.1): sec θ is the reciprocal of cosine. Remember the consequences: domain excludes angles where cos θ = 0, range |y| ≥ 1, sec is even (sec(−θ)=sec θ), and its graph has vertical asymptotes where cosine is zero. For a quick review, see the Topic 3.11 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/secant-cosecant-cotangent-functions/study-guide/nhIcN0Whx8hECmPVKlvK). For more practice, check the AP Precalculus practice set (https://library.fiveable.me/practice/ap-pre-calculus).
What are the domain restrictions for csc x sec x and cot x?
For each reciprocal trig function, exclude inputs that make the denominator 0. - sec x = 1/cos x: domain = all real x with cos x ≠ 0. So x ≠ π/2 + kπ for any integer k. (Vertical asymptotes at those x; period 2π; sec is even.) - csc x = 1/sin x: domain = all real x with sin x ≠ 0. So x ≠ kπ for any integer k. (Vertical asymptotes at kπ; period 2π; csc is odd.) - cot x = cos x / sin x (or 1/tan x): domain = all real x with sin x ≠ 0 (equivalently tan x ≠ 0). So x ≠ kπ for any integer k. (Vertical asymptotes at kπ; cot has period π and is odd.) On the AP exam your calculator should be in radian mode and remember these domain exclusions when sketching graphs or solving equations. For a quick refresher, see the Topic 3.11 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/secant-cosecant-cotangent-functions/study-guide/nhIcN0Whx8hECmPVKlvK). For extra practice, try more problems at (https://library.fiveable.me/practice/ap-pre-calculus).
How do I solve equations with secant and cosecant functions?
Solve sec and csc equations by using their reciprocal definitions and checking domain restrictions. Steps: 1. Rewrite in sine/cosine: sec θ = a → cos θ = 1/a; csc θ = b → sin θ = 1/b. (Remember sec = 1/cos, csc = 1/sin from the CED.) 2. Check feasibility: |1/a| ≤ 1 (or |1/b| ≤ 1). If not, no real solutions. 3. Solve the trig equation for the basic angle (use radians on the AP exam). For cos θ = c, solutions are θ = ±arccos(c) + 2πk (sec/csc have period 2π). For sin θ = s, solutions are θ = arcsin(s) + 2πk or π − arcsin(s) + 2πk. Don’t forget to exclude angles where denominator would be 0 (cos θ = 0 for sec, sin θ = 0 for csc). 4. Give general solutions with k ∈ Z, then restrict to the interval asked (e.g., 0 ≤ θ < 2π). Quick example: solve sec θ = 2. Cos θ = 1/2 → θ = ±π/3 + 2πk → θ = π/3 + 2πk or θ = 5π/3 + 2πk (in [0,2π)). For more on graphs, asymptotes, periods and symmetry (sec even, csc odd), check the Topic 3.11 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/secant-cosecant-cotangent-functions/study-guide/nhIcN0Whx8hECmPVKlvK). For extra practice (1000+ problems) use the AP practice bank (https://library.fiveable.me/practice/ap-pre-calculus).
When does cot x equal zero and when is it undefined?
Cot x = cos x / sin x, so treat it like a fraction. - cot x = 0 when the numerator is 0 but the denominator isn’t. So cos x = 0 ⇒ x = π/2 + kπ (k an integer). At those angles sin x = ±1, so cot x = 0. - cot x is undefined when the denominator is 0, i.e., sin x = 0 ⇒ x = kπ (k an integer). Those x-values are vertical asymptotes of y = cot x. Remember cot has period π and is odd, and its graph is decreasing between consecutive asymptotes. On the AP exam angles are in radians unless stated otherwise—practice these exact-angle values (see the Topic 3.11 study guide: https://library.fiveable.me/ap-pre-calculus/unit-3/secant-cosecant-cotangent-functions/study-guide/nhIcN0Whx8hECmPVKlvK) and try related practice problems (https://library.fiveable.me/practice/ap-pre-calculus).
What's the difference between finding asymptotes for sec x vs csc x?
Short answer: you find asymptotes for sec x where cos x = 0 and for csc x where sin x = 0—because sec = 1/cos and csc = 1/sin, and vertical asymptotes occur where the denominator is zero (domain exclusions). Details you can use on AP problems: - Sec x: vertical asymptotes at x = π/2 + kπ (cos = 0). Period = 2π. Sec is even, so branches repeat symmetrically. - Csc x: vertical asymptotes at x = kπ (sin = 0). Period = 2π. Csc is odd, so branches are symmetric through the origin. Graph tip: look at the zeros and extrema of the original sine/cosine. For sec, asymptotes line up with cosine zeros and the secant “U” or upside-down “U” branches sit around cosine extrema (|y| ≥ 1). For csc, do the same but using sine’s zeros/extrema. Want more practice or a quick refresher on these exact points and graphs? Check the Topic 3.11 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/secant-cosecant-cotangent-functions/study-guide/nhIcN0Whx8hECmPVKlvK) and try problems at (https://library.fiveable.me/practice/ap-pre-calculus).
How do I know if a trig function problem wants me to use reciprocal functions?
Look for three quick clues that tell you to use reciprocals (sec, csc, cot): 1. The expression or equation literally uses a reciprocal (sec θ, csc θ, cot θ) or shows 1/cos, 1/sin, or cos/sin. If you see sec, csc, or cot, you’re in Topic 3.11 territory (they’re defined where the corresponding denominator ≠ 0). 2. The algebra or simplification pushes you there—e.g., you get something like 1/cos θ = 4 or (cos θ)/(sin θ) = 2. Convert to the reciprocal identity (sec θ = 1/cos θ, cot θ = cos θ/sin θ) to solve or to find domain exclusions (vertical asymptotes where cos θ = 0 for sec, sin θ = 0 for csc/cot). 3. Graph or range clues: if a graph has vertical asymptotes where sin or cos = 0 or outputs only |y| ≥ 1, that’s a reciprocal graph (sec/csc). Cotangent has period π and is decreasing between asymptotes. Strategy: rewrite reciprocals into sines/cosines (or vice versa) to use unit-circle values, check domain exclusions, and use the CED facts: sec even, csc/cot odd; sec/csc period 2π, cot period π. For the topic study guide and example problems, see the Fiveable Topic 3.11 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/secant-cosecant-cotangent-functions/study-guide/nhIcN0Whx8hECmPVKlvK) and practice problems (https://library.fiveable.me/practice/ap-pre-calculus). Remember: on the AP exam, angles are in radians unless told otherwise and watch domain restrictions when solving equations.
Can someone explain why the range of sec x is (-∞, -1] ∪ [1, ∞) and not all real numbers?
Think of sec x as 1/cos x. Cos x always lies between -1 and 1, so |cos x| ≤ 1 for every x where cos x exists. If 0 < |cos x| ≤ 1, then 1/|cos x| ≥ 1, so |sec x| = |1/cos x| ≥ 1. That means sec x can never be a number whose absolute value is less than 1. The smallest possible magnitudes occur when cos x = ±1, giving sec x = ±1. When cos x = 0, sec x is undefined (vertical asymptotes). So the only real outputs are ≤ −1 or ≥ 1, i.e. (−∞, −1] ∪ [1, ∞). This matches the CED Essential Knowledge (sec is reciprocal of cos; range |y| ≥ 1 and vertical asymptotes where cos = 0). For a short topic review, check the Fiveable study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/secant-cosecant-cotangent-functions/study-guide/nhIcN0Whx8hECmPVKlvK) and practice problems (https://library.fiveable.me/practice/ap-pre-calculus).
I keep mixing up which reciprocal goes with which trig function - any memory tricks?
Nice—a few quick, reliable tricks that'll stick: - Flip rule (core): sec = 1/cos, csc = 1/sin, cot = 1/tan (and cot = cos/sin). Say it out loud: “sec over cos, csc over sin, cot over tan.” - Unit-circle picture (best mental model): cos = x-coordinate, sin = y. Their reciprocals are 1/x and 1/y. So if you imagine x or y = 0, you immediately know which reciprocal is undefined: cos = 0 → sec undefined; sin = 0 → csc undefined; tan = 0 → cot undefined. - Quick mnemonic: pair the first letters—sec & cos both have a “c” sound (sec ↔ cos), csc & sin both start with “s” (csc ↔ sin), cot & tan are both “t”-family reciprocals (cot ↔ tan). - Spot checks: sec and cos are even functions; csc and cot are odd. Range for sec/csc: |y| ≥ 1. Periods: sec/csc 2π, cot π. If you want practice problems and a short study guide that lines up with the CED, check the Topic 3.11 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/secant-cosecant-cotangent-functions/study-guide/nhIcN0Whx8hECmPVKlvK) and try problems at (https://library.fiveable.me/practice/ap-pre-calculus).