Domain and Range of Trigonometric Functions

TL;DR

The domain and range of trigonometric functions describes which angles each function accepts and which output values it produces — sine and cosine accept all real angles and output values in [-1,1], while tangent, cotangent, secant, and cosecant have angles where they are undefined. This article gives the full domain–range table, the graph of each function in degrees and radians, the unit-circle anchor for each definition, three worked examples, and the most common mistakes students make.

Six Functions, Three Different Output Patterns, One Unit Circle

Every trigonometric function inherits its domain and range directly from the unit circle — and those six rules close together in a single table.

The domain of a function is the set of inputs it accepts; the range is the set of outputs it returns. For trigonometric functions, the input is an angle (in degrees or radians) and the output is a real number tied to the unit circle.

The Master Table

For all six trigonometric functions, the domain and range are below. Real-number set is (R); integer set is (Z).

Three patterns sit inside this table:

• Sine and cosine — always defined, always bounded in [-1,1].
• Tangent and cotangent — undefined at the angles where the relevant unit-circle coordinate is zero; output sweeps all of R.
• Secant and cosecant — undefined at the same angles as their reciprocal cousins; output never lies inside (−1,1), since reciprocals of values ≤1 in magnitude land ≥1 in magnitude.

Quick facts.

• Period: sin(θ), cos(θ), sec(θ), csc(θ) have period 2π (360°). tan(θ) and cot(θ) have period π (180°).
• Undefined points come from zero denominators. tan(θ) = sin(θ)/cos(θ) is undefined when cos(θ) = 0; sec(θ) = 1/cos(θ) is undefined at the same angles.

Double-Anchoring — Right Triangle and Unit Circle

For an angle in (0,π/2), every function has both a right-triangle reading and a unit-circle reading. Domain and range come cleanly from the unit-circle view.

Sine and cosine. On the unit circle, a point at angle θ has coordinates (cos(θ), sin(θ)). The coordinates of any point on the unit circle are bounded by ±1, so the range is [-1,1]. The angle θ can be any real number, so the domain is R.

Tangent. tan(θ) = sin(θ)/cos(θ) = y-coord/x-coord. This is undefined when the x-coordinate equals zero — at θ=π/2,3π/2,… (90°,270°,…). Where defined, the ratio sweeps all real numbers — vertical asymptotes appear at the excluded angles.

Secant. sec(θ) = 1/cos(θ) = 1/(x-coord). Undefined where the x-coordinate is zero. When defined, |cos(θ)| ≤ 1 ⇒ |sec(θ)| ≥ 1.

Cotangent. cot(θ) = cos(θ)/sin(θ) = x/y — undefined where the y-coordinate is zero, at θ=0,π,2π,…

Cosecant. csc(θ) = 1/sin(θ) = 1/y — undefined where y=0, range matches secant.

The unit circle is the one figure that resolves all six domain–range rules in one go.

Function-by-Function with Graphs

Sine — sin(θ)

• Domain: θ ∈ R (all real numbers).
• Range: [-1,1].
• Period: 2π or 360°.
• Reference points: sin(0) = 0, sin(π/2) = 1, sin(π) = 0, sin(3π/2) = −1.

Cosine — cos(θ)

• Domain: θ ∈ R.
• Range: [-1,1].
• Period: 2π or 360°.
• Reference points: cos(0) = 1, cos(π/2) = 0, cos(π) = −1, cos(3π/2) = 0.

Cosine has the same shape as sine, shifted left by π/2: cos(θ) = sin(θ + π/2).

Tangent — tan(θ)

• Domain: θ ∈ R ∖ {(2n+1)π/2:n ∈ Z}. In degrees, θ ≠ 90°, 270°, 450°, …
• Range: R (all real numbers).
• Period: π or 180°.

Cotangent — cot(θ)

• Domain: θ ≠ nπ, n ∈ Z. In degrees, θ ≠ 0°, 180°, 360°, …
• Range: R.
• Period: π.

Secant — sec(θ) = 1/cos(θ)

• Domain: same as tangent — θ ≠ (2n+1)π/2.
• Range: (−∞,−1] ∪ [1,∞) — never lies in (−1,1).
• Period: 2π.

Cosecant — csc(θ) = 1/sin(θ)

• Domain: same as cotangent — θ ≠ nπ.
• Range: (−∞,−1] ∪ [1,∞).
• Period: 2π.

Three Worked Examples — Quick, Standard, Stretch

Quick. Find the domain and range of f(θ) = 2sin(θ).

The domain of sin(θ) is R, and multiplying by 2 doesn't restrict inputs, so the domain remains R. The range is [-1,1], so multiplying by 2 stretches the range to [-2,2].

Final answer: Domain R; Range [-2,2].

Standard (Wrong Path First — Where Intuition Breaks). Find the domain of g(θ) = tan(2θ−π/3).

The wrong path. A student writes "the domain of tan is R∖{(2n+1)π/2}, so the domain of tan(2θ−π/3) is also R∖{(2n+1)π/2}."

The correct way: Solve the input transformation: 2θ − π/3 = (2n+1)π/2, not just copied from the bare function's exclusions.

Final answer: Domain is θ ∈ R ∖ {f(θ) solution}; Range is R.

Stretch. Find the domain and range of h(θ) = 3/(2 + cos(θ)).

Domain: The denominator 2 + cos(θ) must be nonzero. Since cos(θ) ∈ [-1,1], the denominator 2 + cos(θ) ∈ [1,3] — always positive, never zero. Range: As cos(θ) sweeps [-1,1], the function h(θ) sweeps [1,3].

Final answer: Domain R. Range [1,3].

Why Domain and Range Matter Outside the Classroom

Domain and range aren't just textbook bookkeeping — they cause real outages when missed.

• Signal processing. Audio codecs treat amplitude as sin/cos outputs in [-1,1] — sending a value ±1.2 into a fixed-point DAC produces clipping.
• Computer graphics. Direction vectors normalized onto a unit sphere use θ=arccos(z); z outside [-1,1] crashes the renderer. Wrap every arccos in a clamp(z, -1, 1).
• GPS satellite ranging. Computing the angle between a satellite vector and a receiver vector uses arccos — must protect the input from drifting outside [-1,1].
• Phasor analysis in electrical engineering. Sinusoidal AC currents are modeled as I(t)=I0sin(ωt + ϕ); peak current multiplies the trig range [-1,1] to give [-I0,I0].
• Astronomy. The hour angle of a star has a 24-hour periodicity, using tangent's period π — observatory software respects tangent's excluded angles.

The Mathematicians Who Mapped the Trig Functions

Aryabhata (476–550 CE, India) introduced the half-chord function jya (modern sine) and tabulated it across what we'd now call the first quadrant.
Bhaskara II (1114–1185, India) extended trig tables across the full circle.

Leonhard Euler (1707–1783, Switzerland). Euler was the first to treat trigonometric functions as functions of a real variable.

Slip-Ups That Cost Marks

1. Forgetting tangent's undefined points.

• Assume all six trig functions have the same domain.

2. Confusing the range of secant with the range of cosine.

• Don't reach for the cosine range when the function is its reciprocal.

3. Treating composite trig functions as if they kept the base domain.

• Always solve for the actual excluded angles.

4. Mode confusion: writing the domain in degrees but solving in radians (or vice versa).

• Pick a unit and stick with it.

Conclusion

The domain and range of trigonometric functions follow three patterns: sin/cos are defined everywhere with range [-1,1]; tan/cot have range R but are undefined at half-period intervals; sec/csc have range outside (−1,1) and inherit the tan/cot exclusions.