Domain and Range of Trig Functions

The range of sine and cosine is [-1, 1] for the parent function, but amplitude and vertical shift change that immediately. The domain and range of trig functions depend on which function you are using and whether you need the inverse. For sine and cosine, domain is all real numbers. For tangent, domain excludes odd multiples of π/2 because the function has vertical asymptotes there. For inverse trig functions, the domain is the original range and the range is a restricted principal interval. OpenStax Precalculus 2e covers this in chapters 7 and 8. The most common mistake is assuming the range of sine is always [-1, 1] without checking for transformations.

Apply the Amplitude and Shift

The range of sine function for f(x) = sin(x) is [-1, 1]. When you apply an amplitude a, the range becomes [-|a|, |a|]. A vertical shift d moves it to [d - |a|, d + |a|]. For example, f(x) = 3 sin(x) + 2 has range [-1, 5] because the amplitude is 3 and the shift is 2. The failure case: a student writes the range as [-3, 3] and forgets the vertical shift. Always add the shift after multiplying by amplitude. For f(x) = a sin(bx + c) + d, the range is [d - |a|, d + |a|]. The b and c affect period and phase shift, not range. This rule works for cosine identically because cosine has the same parent range.

Exclude Every Asymptote

The domain of tan x is all real numbers except x = π/2 + kπ, where k is any integer. At these x-values, the cosine in the denominator is zero, creating a vertical asymptote. The tangent function is undefined there. A student often memorises only the first asymptote at π/2 and forgets the odd multiples. For sec x, the domain is the same as tan x: exclude π/2 + kπ. For csc x, exclude kπ (where sine is zero). For cot x, also exclude kπ. Write the domain in interval notation as a union of intervals between asymptotes. For example, the domain of tan x is ... ∪ (-π/2, π/2) ∪ (π/2, 3π/2) ∪ ... . The failure case: a student writes "all reals except π/2" and misses the infinite set of exclusions. Check by testing x = 3π/2; the function is undefined there too.

Memorise the Three Pairs

Inverse trig functions require a restricted domain on the original function so that the inverse is also a function. The domain of arcsin is [-1, 1]; its range is [-π/2, π/2]. The domain of arccos is [-1, 1]; its range is [0, π]. The domain of arctan is all real numbers; its range is (-π/2, π/2). These are the principal ranges, and you must stay within them when evaluating inverses. OpenStax Precalculus 2e chapter 8 gives these values. The failure case: a student tries arcsin(2) and expects a real answer, but 2 is outside the domain [-1, 1]. Another failure: a student writes the range of arcsin as [-π, π] instead of the restricted interval. Memorise each pair: the domain of the inverse equals the range of the original, and the range of the inverse is the restricted domain of the original.

Stay Inside the Principal Interval

The range of arcsin is [-π/2, π/2]. This is the set of all possible output angles from the inverse sine function. For any x in [-1, 1], arcsin(x) returns an angle whose sine is x, and that angle is always between -π/2 and π/2 inclusive. For example, arcsin(0) = 0, arcsin(1) = π/2, arcsin(-1) = -π/2. The failure case: a student computes arcsin(1/2) and gives 5π/6 because sin(5π/6) = 1/2, but 5π/6 is outside the range of arcsin. The correct answer is π/6. Always check that your answer falls in the principal range. For arccos, the range is [0, π], so the same input 1/2 gives π/3. For arctan, the range is (-π/2, π/2), open at both ends because asymptotes are not included.

Identify the Asymptotes, Not Holes

Tangent and secant are undefined where cosine is zero: x = π/2 + kπ. Cosecant and cotangent are undefined where sine is zero: x = kπ. These are vertical asymptotes, not holes, because the numerator does not cancel the zero. For sec x = 1/cos x, anywhere cos x = 0 creates a vertical asymptote. For csc x = 1/sin x, anywhere sin x = 0 does the same. The failure case: a student treats a hole the same as an asymptote. A hole requires a factor that cancels in numerator and denominator. In these trig functions, no cancellation occurs because the numerator is 1. Write the domain as all real numbers except these values. For tan x, the domain is {x | x ≠ π/2 + kπ, k ∈ ℤ}. For csc x, it is {x | x ≠ kπ, k ∈ ℤ}.

Swap the Pairs Correctly

To create inverse trig functions, you restrict the domain of the original trig function so it is one-to-one. For sine, the restricted domain is [-π/2, π/2]; the range remains [-1, 1]. The inverse, arcsin, swaps these: domain [-1, 1], range [-π/2, π/2]. For cosine, the restricted domain is [0, π]; the range is [-1, 1]. Arccos has domain [-1, 1], range [0, π]. For tangent, the restricted domain is (-π/2, π/2); the range is all real numbers. Arctan has domain all real numbers, range (-π/2, π/2). OpenStax Precalculus 2e section 7.4 provides these restrictions. The failure case: a student uses the original full domain for the inverse and gets a relation, not a function. Always check that the inverse's domain matches the original's restricted range.

Apply the Rules Step by Step

Example 1: Find the domain and range of f(x) = 4 sin(3x - π) + 2. The amplitude is 4, vertical shift is 2. Range: [2 - 4, 2 + 4] = [-2, 6]. Domain: all real numbers because sine accepts any real input. The phase shift and period do not affect domain or range. Example 2: Find the domain of f(x) = tan(2x). The argument is 2x, so the asymptotes occur where 2x = π/2 + kπ, meaning x = π/4 + kπ/2. Domain: all real numbers except x = π/4 + kπ/2, k ∈ ℤ. Range: all real numbers. Example 3: Evaluate arcsin(-√3/2). The input is in [-1, 1], so the domain is satisfied. The angle in [-π/2, π/2] with sine -√3/2 is -π/3. Answer: -π/3. The failure case: a student answers 4π/3 because sin(4π/3) = -√3/2, but 4π/3 is outside the range of arcsin.

Check Each Range and Domain

The six basic trigonometric functions and their domain and range, based on OpenStax Precalculus 2e chapter 7: sine: domain (-∞, ∞), range [-1, 1]; cosine: domain (-∞, ∞), range [-1, 1]; tangent: domain x ≠ π/2 + kπ, range (-∞, ∞); cotangent: domain x ≠ kπ, range (-∞, ∞); secant: domain x ≠ π/2 + kπ, range (-∞, -1] ∪ [1, ∞); cosecant: domain x ≠ kπ, range (-∞, -1] ∪ [1, ∞). The range for secant and cosecant excludes the interval (-1, 1) because the reciprocal of a number between -1 and 1 is greater than 1 in absolute value or less than -1. The failure case: a student writes the range of secant as all real numbers, missing that it cannot output values between -1 and 1.

Know the Endpoints

For inverse trig functions, from OpenStax Precalculus 2e chapter 8: arcsin: domain [-1, 1], range [-π/2, π/2]; arccos: domain [-1, 1], range [0, π]; arctan: domain (-∞, ∞), range (-π/2, π/2). The range for arctan is open at both ends because the asymptotes at y = ±π/2 are not reached. The failure case: a student includes π/2 in the range of arctan, but tan x never reaches an infinite output, so arctan cannot output an angle that gives an infinite tangent. For arcsin and arccos, the endpoints are included because sin(±π/2) and cos(0) and cos(π) are defined and reachable.

Precalculus and trigonometry students who need to find the domain and range of trig and inverse trig functions will use these definitions directly. The rules for restricted domains and principal ranges are essential for solving equations and verifying calculator results. Students who cannot evaluate sine and cosine at common angles should first master function notation from OpenStax College Algebra 2e, Section 3.1. Anyone looking for calculus-level continuity analysis or epsilon-delta proofs needs a calculus textbook or Paul's Online Math Notes. The single thing that most often goes wrong: students treat the inverse trig range as the entire set of angles where the trig function equals the input, instead of the restricted principal interval. Always check that your answer lies in the correct range.