Trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) relate angles to side ratios; in AP Calculus, the CED guarantees they are continuous at every point in their domains (EK LIM-2.B.2) and often require rewriting with identities to evaluate limits (EK LIM-1.E.1).
Trigonometric functions relate angles in a right triangle to ratios of side lengths. The six functions are sine (sin), cosine (cos), tangent (tan), and their reciprocals cosecant (csc), secant (sec), and cotangent (cot). Once you extend them to the unit circle, they become periodic functions defined for (almost) all real numbers, and that's the version AP Calculus cares about.
In Unit 1, trig functions show up in two specific jobs. First, they're on the CED's short list of function families that are continuous at every point in their domains (EK LIM-2.B.2), right alongside polynomial, rational, power, exponential, and logarithmic functions. Sine and cosine are continuous everywhere; tangent and secant are continuous everywhere except where cosine equals zero, because that's where they're undefined. Second, evaluating limits often means rewriting a trig expression in an alternate form using identities (EK LIM-1.E.1), the same way you'd factor a polynomial or multiply by a conjugate. Swapping tan(x) for sin(x)/cos(x) can turn an indeterminate mess into something you can just plug into.
This term lives in Unit 1 (Limits and Continuity), supporting two learning objectives. Under AP Calc 1.12.A, you determine intervals where a function is continuous, and EK LIM-2.B.2 hands you a freebie. If a function is trigonometric, it's continuous wherever it's defined, so the only real work is finding the domain. Under AP Calc 1.6.A, you evaluate limits using equivalent expressions or the squeeze theorem, and trig identities are one of the three named rewriting tools in EK LIM-1.E.1. The classic squeeze theorem result, that sin(x)/x approaches 1 as x approaches 0, is the poster child here. And this is just the entry point. Trig functions thread through the entire course, from derivatives to integrals, so getting their continuity and domains locked down in Unit 1 pays off everywhere.
Keep studying AP Calculus Unit 1
Visual cheatsheet
view galleryUnit circle (Unit 1)
The unit circle is what upgrades trig from triangle geometry to calculus-ready functions. It defines sin and cos for every real number, which is exactly why those two are continuous on all of (-∞, ∞).
Periodicity (Unit 1)
Trig functions repeat forever, so their behavior on one period tells you everything. That's also why limits of things like sin(x) as x goes to infinity don't exist; the function never settles on one value.
Inverse trigonometric functions (Unit 1)
Functions like arcsin and arctan undo trig functions, but only after restricting the domain so each output is unique. They're separate functions with their own (much smaller) domains, not the same thing as csc, sec, and cot.
Exponential functions (Unit 1)
Exponential, logarithmic, and trigonometric functions all sit on the same EK LIM-2.B.2 list of families continuous on their domains. MCQs love asking you to identify which function types carry this guarantee.
Multiple-choice questions test trig functions in two main flavors. One asks you to confirm continuity over an interval, like deciding whether h(x) = cos(2x) is continuous on the closed interval [0, π]. The move is to recognize cosine is continuous everywhere, so any interval works. Another asks which function types are guaranteed continuous on their domains, where trigonometric is one of the correct families straight from EK LIM-2.B.2. In limit problems, expect to rewrite trig expressions using identities (an "alternate form") before evaluating, or to apply the squeeze theorem. No released FRQ uses the phrase "trigonometric functions" as a vocabulary term, but trig expressions appear constantly in FRQ functions, so fluency with domains, identities, and continuity is assumed rather than tested directly.
Reciprocal is not inverse. Cosecant is 1/sin(x), a reciprocal trig function that's still in the original six. Arcsine (sin⁻¹) is the inverse function that undoes sine, answering "what angle gives this value?" The notation sin⁻¹(x) means arcsin(x), never 1/sin(x). They also have very different domains: csc(x) is undefined wherever sin(x) = 0, while arcsin(x) only accepts inputs from -1 to 1. Mixing these up wrecks both continuity and limit problems.
The six trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent, and the last three are reciprocals of the first three.
By EK LIM-2.B.2, trigonometric functions are continuous at every point in their domains, so confirming continuity reduces to finding where the function is defined.
Sine and cosine are continuous on all real numbers, while tangent and secant break wherever cos(x) = 0, and cotangent and cosecant break wherever sin(x) = 0.
EK LIM-1.E.1 lists alternate forms of trig functions as a core limit technique, so rewrite with identities like tan(x) = sin(x)/cos(x) before evaluating.
The squeeze theorem (LO 1.6.A) is the standard tool for trig limits that identities can't crack, most famously sin(x)/x approaching 1 as x approaches 0.
Continuity of trig functions in Unit 1 is the foundation for everything later, since you can't differentiate or integrate sin and cos until you trust their limit behavior.
They're the six functions sin, cos, tan, csc, sec, and cot, which relate angles to side ratios. In AP Calc Unit 1, the key facts are that they're continuous on their domains (EK LIM-2.B.2) and that limits involving them often require rewriting with identities or applying the squeeze theorem.
No. Sine and cosine are continuous on all real numbers, but tan, sec, cot, and csc have breaks wherever their denominators hit zero. Tangent, for example, is undefined at x = π/2 + nπ. The accurate statement is that trig functions are continuous on all points in their domains.
Yes. Cosine is continuous for all real numbers, and compressing it horizontally to cos(2x) doesn't change that, so it's continuous at every point of [0, π]. That's a classic Topic 1.12 question.
csc(x) is the reciprocal, 1/sin(x), while sin⁻¹(x) means arcsin(x), the inverse function that returns an angle. They are completely different functions with different domains, and confusing the ⁻¹ notation with a reciprocal is one of the most common AP Calc errors.
Two reasons from the CED. EK LIM-1.E.1 says you may need alternate forms of trig functions (identities) to evaluate a limit, and LO 1.6.A includes the squeeze theorem, which is how you prove results like the limit of sin(x)/x equaling 1 as x approaches 0.