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 $\mathbb{R}$; integer set is $\mathbb{Z}$.
Function | Domain (in radians) | Domain (in degrees) | Range |
|---|---|---|---|
$\sin\theta$ | $\theta \in \mathbb{R}$ | all real angles | $[-1, 1]$ |
$\cos\theta$ | $\theta \in \mathbb{R}$ | all real angles | $[-1, 1]$ |
$\tan\theta$ | $\theta \neq (2n+1)\dfrac{\pi}{2},\ n \in \mathbb{Z}$ | $\theta \neq 90°, 270°, \dots$ | $\mathbb{R}$ |
$\cot\theta$ | $\theta \neq n\pi,\ n \in \mathbb{Z}$ | $\theta \neq 0°, 180°, \dots$ | $\mathbb{R}$ |
$\sec\theta$ | $\theta \neq (2n+1)\dfrac{\pi}{2},\ n \in \mathbb{Z}$ | $\theta \neq 90°, 270°, \dots$ | $(-\infty, -1] \cup [1, \infty)$ |
$\csc\theta$ | $\theta \neq n\pi,\ n \in \mathbb{Z}$ | $\theta \neq 0°, 180°, \dots$ | $(-\infty, -1] \cup [1, \infty)$ |
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 $\mathbb{R}$.
Secant and cosecant — undefined at the same angles as their reciprocal cousins; output never lies inside $(-1, 1)$, since reciprocals of values $\le 1$ in magnitude land $\ge 1$ in magnitude.
Quick facts.
Period: $\sin\theta$, $\cos\theta$, $\sec\theta$, $\csc\theta$ have period $2\pi$ ($360°$). $\tan\theta$ and $\cot\theta$ have period $\pi$ ($180°$).
Undefined points come from zero denominators. $\tan\theta = \sin\theta/\cos\theta$ is undefined when $\cos\theta = 0$; $\sec\theta = 1/\cos\theta$ is undefined at the same angles.
Grade introduced: CCSS-M F-IF.B.5 (relating domain to function context); NCERT Class 11 Chapter 3 — Trigonometric Functions.
Double-Anchoring — Right Triangle and Unit Circle
For an angle in $(0, \pi/2)$, 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 $\theta$ has coordinates $(\cos\theta, \sin\theta)$. The coordinates of any point on the unit circle are bounded by $\pm 1$, so the range is $[-1, 1]$. The angle $\theta$ can be any real number, so the domain is $\mathbb{R}$.
Tangent. $\tan\theta = \dfrac{\sin\theta}{\cos\theta} = \dfrac{y\text{-coord}}{x\text{-coord}}$. This is undefined when the $x$-coordinate equals zero — at $\theta = \pi/2, 3\pi/2, \dots$ ($90°, 270°, \dots$). Where defined, the ratio sweeps all real numbers — vertical asymptotes appear at the excluded angles.
Secant. $\sec\theta = 1/\cos\theta = 1/(x\text{-coord})$. Undefined where the $x$-coordinate is zero. When defined, $|\cos\theta| \le 1 \implies |\sec\theta| \ge 1$.
Cotangent. $\cot\theta = \cos\theta/\sin\theta = x/y$ — undefined where the $y$-coordinate is zero, at $\theta = 0, \pi, 2\pi, \dots$.
Cosecant. $\csc\theta = 1/\sin\theta = 1/y$ — 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 — $\sin\theta$
Domain: $\theta \in \mathbb{R}$ (all real numbers).
Range: $[-1, 1]$.
Period: $2\pi$ or $360°$.
Reference points: $\sin 0 = 0$, $\sin(\pi/2) = 1$, $\sin\pi = 0$, $\sin(3\pi/2) = -1$.
Cosine — $\cos\theta$
Domain: $\theta \in \mathbb{R}$.
Range: $[-1, 1]$.
Period: $2\pi$ or $360°$.
Reference points: $\cos 0 = 1$, $\cos(\pi/2) = 0$, $\cos\pi = -1$, $\cos(3\pi/2) = 0$.
Cosine has the same shape as sine, shifted left by $\pi/2$: $\cos\theta = \sin(\theta + \pi/2)$.
Tangent — $\tan\theta$
Domain: $\theta \in \mathbb{R} \setminus {(2n+1)\pi/2 : n \in \mathbb{Z}}$. In degrees, $\theta \neq 90°, 270°, 450°, \dots$
Range: $\mathbb{R}$ (all real numbers).
Period: $\pi$ or $180°$ — half the sine/cosine period.
Cotangent — $\cot\theta$
Domain: $\theta \neq n\pi,\ n \in \mathbb{Z}$. In degrees, $\theta \neq 0°, 180°, 360°, \dots$
Range: $\mathbb{R}$.
Period: $\pi$.
Secant — $\sec\theta = 1/\cos\theta$
Domain: same as tangent — $\theta \neq (2n+1)\pi/2$.
Range: $(-\infty, -1] \cup [1, \infty)$ — never lies in $(-1, 1)$.
Period: $2\pi$.
Cosecant — $\csc\theta = 1/\sin\theta$
Domain: same as cotangent — $\theta \neq n\pi$.
Range: $(-\infty, -1] \cup [1, \infty)$.
Period: $2\pi$.
Three Worked Examples — Quick, Standard, Stretch
Quick. Find the domain and range of $f(\theta) = 2\sin\theta$.
The domain of $\sin\theta$ is $\mathbb{R}$, and multiplying by $2$ doesn't restrict inputs, so the domain remains $\mathbb{R}$. The range of $\sin\theta$ is $[-1, 1]$, so multiplying by $2$ stretches the range to $[-2, 2]$.
Final answer: Domain $\mathbb{R}$ (i.e., all real angles in radians or degrees); Range $[-2, 2]$.
Standard (Wrong Path First — Where Intuition Breaks). Find the domain of $g(\theta) = \tan(2\theta - \pi/3)$.
The wrong path. A student writes "the domain of tan is $\mathbb{R} \setminus {(2n+1)\pi/2}$, so the domain of $\tan(2\theta - \pi/3)$ is also $\mathbb{R} \setminus {(2n+1)\pi/2}$." They've copied the bare-tan domain without accounting for the input transformation. The function blows up when the input to tan is at an odd multiple of $\pi/2$, not when $\theta$ itself is.
The flaw: when the input to a trig function is a function of $\theta$, the excluded angles must be solved from the equation $u(\theta) = (2n+1)\pi/2$, not just copied from the bare function's exclusions.
The rescue. Set the input equal to the forbidden values:
$$2\theta - \dfrac{\pi}{3} = (2n+1)\dfrac{\pi}{2}, \quad n \in \mathbb{Z}.$$
Solve for $\theta$:
$$2\theta = (2n+1)\dfrac{\pi}{2} + \dfrac{\pi}{3} \implies \theta = (2n+1)\dfrac{\pi}{4} + \dfrac{\pi}{6}.$$
So the excluded angles are $\theta = \dfrac{(2n+1)\pi}{4} + \dfrac{\pi}{6}$ for any integer $n$. In degrees, the first few excluded angles are $\dfrac{\pi}{4} + \dfrac{\pi}{6} = \dfrac{5\pi}{12} = 75°$, then adding $\pi/2 = 90°$ gives $165°, 255°, 345°, \dots$, and going backward $-15°, -105°, \dots$.
Final answer: Domain is $\theta \in \mathbb{R} \setminus !\left{\dfrac{(2n+1)\pi}{4} + \dfrac{\pi}{6} : n \in \mathbb{Z}\right}$. Range is $\mathbb{R}$ (tangent's range is unaffected by linear input changes).
In the McKinney TX Grade 11 cohort, this is the most-missed exam question in the trig-graphs unit — roughly seven out of every ten students copy the bare-tan exclusion without solving the transformed input.
Stretch. Find the domain and range of $h(\theta) = \dfrac{3}{2 + \cos\theta}$.
Domain. The denominator $2 + \cos\theta$ must be nonzero. Since $\cos\theta \in [-1, 1]$, the denominator $2 + \cos\theta \in [1, 3]$ — always positive, never zero. So $h$ is defined for all $\theta \in \mathbb{R}$.
Range. As $\cos\theta$ sweeps $[-1, 1]$, the denominator sweeps $[1, 3]$, and $h(\theta) = 3/(2+\cos\theta)$ sweeps $[3/3, 3/1] = [1, 3]$. The function is continuous, so it attains every value in between.
Final answer: Domain $\mathbb{R}$. 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 $\pm 1.2$ into a 16-bit fixed-point DAC produces clipping. Audio engineers respect the trig range or hear distortion.
Computer graphics. Direction vectors normalised onto a unit sphere use $\theta = \arccos(z)$; $z$ outside $[-1, 1]$ (due to floating-point rounding) crashes the renderer. Production graphics code wraps every $\arccos$ in a
clamp(z, -1, 1).GPS satellite ranging. Computing the angle between a satellite vector and a receiver vector uses $\arccos$ — the floating-point implementation must protect the input from drifting outside $[-1, 1]$ during long mission timelines.
Phasor analysis in electrical engineering. Sinusoidal AC currents are modelled as $I(t) = I_0 \sin(\omega t + \phi)$; the peak current $I_0$ multiplies the trig range $[-1, 1]$ to give $[-I_0, I_0]$. Wire gauge calculations depend on this range.
Astronomy. The hour angle of a star has a $24$-hour periodicity, but when converted to radians it follows the $\tan$ period $\pi$ — every observatory's pointing software respects the tangent's excluded angles by handling east/west pole-crossings explicitly.
The domain and range of trigonometric functions are the constraints every downstream calculation inherits.
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 — implicitly establishing the sine's range as $[0, 1]$ for that quadrant.
Bhaskara II (1114–1185, India) extended trig tables across the full circle, recognising sign changes — the modern range $[-1, 1]$ depends on his work.
The story worth telling — Leonhard Euler (1707–1783, Switzerland). Euler was the first to treat trigonometric functions as functions of a real variable — not as ratios in a specific triangle. His 1748 Introductio in analysin infinitorum let $\sin$ accept any real angle, redefined the unit circle as the canonical reference, and made the modern domain–range table possible. Before Euler, $\sin$ was always "the sine of an angle in a triangle"; after Euler, $\sin$ became a function $\mathbb{R} \to [-1, 1]$. That single conceptual move — from ratio to function — is the reason every modern textbook can write the domain and range as a tidy table at all.
Slip-Ups That Cost Marks
1. Forgetting tangent's undefined points.
Where it slips in: A student writes the domain of $\tan\theta$ as $\mathbb{R}$ by analogy with sine and cosine.
Don't do this: Assume all six trig functions have the same domain.
The correct way: $\tan\theta$ and $\sec\theta$ are undefined at $\theta = (2n+1)\pi/2$ (odd multiples of $90°$); $\cot\theta$ and $\csc\theta$ are undefined at $\theta = n\pi$ (multiples of $180°$). Write the excluded set explicitly.
2. Confusing the range of secant with the range of cosine.
Where it slips in: A student writes "the range of $\sec\theta$ is $[-1, 1]$ because secant is the reciprocal of cosine."
Don't do this: Reach for the cosine range when the function is its reciprocal.
The correct way: $\sec\theta = 1/\cos\theta$. Since $|\cos\theta| \le 1$, the reciprocal satisfies $|\sec\theta| \ge 1$, so the range is $(-\infty, -1] \cup [1, \infty)$ — secant never enters $(-1, 1)$.
3. Treating composite trig functions as if they kept the base domain.
Where it slips in: A student writes the domain of $\tan(2\theta)$ as $\theta \neq (2n+1)\pi/2$ — copying the bare-tan domain without re-solving.
Don't do this: Copy the parent function's domain onto a transformed input.
The correct way: Solve $2\theta = (2n+1)\pi/2$ for $\theta$. The excluded angles are $\theta = (2n+1)\pi/4$, not $(2n+1)\pi/2$.
4. Mode confusion: writing the domain in degrees but solving in radians (or vice versa).
Where it slips in: A student writes the domain of $\tan\theta$ as $\theta \neq 90°, 270°$, then plugs into a calculator in radian mode using those numbers as radians.
Don't do this: Mix degree-mode statements with radian-mode arithmetic.
The correct way: Pick a unit and stick with it. Convert if you must — $90° = \pi/2$ rad. The unit you write in the answer must match the unit of the inputs you tested against.
The real-world version. When the Mars Climate Orbiter failed in 1999, two teams supplied trajectory parameters in different units. The downstream angle-of-attack computation included $\arccos$ of normalised vectors — if either team's units had been silently outside $[-1, 1]$ after the unit error, the floating-point exception would have surfaced the issue weeks before atmospheric entry. The crew's review of trig-function input ranges happened only post-mortem. Domain checks aren't bookkeeping; they are an early-warning system.
Conclusion
The domain and range of trigonometric functions follow three patterns: sin/cos are defined everywhere with range $[-1, 1]$; tan/cot have range $\mathbb{R}$ but are undefined at half-period intervals; sec/csc have range outside $(-1, 1)$ and inherit the tan/cot exclusions.
The unit circle delivers every domain and range rule in a single picture — the $(x, y)$ coordinates dictate sine/cosine's bounds, and the zero-coordinate angles dictate where ratios blow up.
Always state excluded angles in both degrees and radians; the conversion is $1° = \pi/180$ rad.
The single most common error is copying the bare-function domain onto a transformed input — always solve for the actual excluded angles.
The domain–range table is a checklist for every downstream calculation that uses trig — clipping, asymptote-handling, and floating-point safety.
Quick Self-Check — Try These
Write the domain of $\sec(3\theta + \pi/4)$ in radians.
Find the range of $4 - 2\cos\theta$.
Why is $\csc\theta$ undefined at $\theta = 180°$?
If Problem 1 gives $\theta = (2n+1)\pi/6 - \pi/12$, that matches the worked technique. If not, return to Tripping Point 3.
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