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Trigonometric Properties



Key Terms
Periodicity
Even and Odd Symmetry
Boundedness
Zeros
Continuity and Discontinuities
Monotonicity on Principal Intervals
Extrema
Identity of the Six Functions: Period, Parity, Range
Critical-Point Landscape: Where Each Function Vanishes, Explodes, and Turns Around



The Structural Behavior of Sine, Cosine, and Their Companions

Every trigonometric function carries a set of structural features — periodicity, symmetry, boundedness, monotonicity, continuity — that dictate how it behaves across the entire real line. These are not isolated facts to memorize but interconnected consequences of the unit circle geometry. Periodicity follows from the circle's finite circumference. Even/odd symmetry follows from how coordinates respond to reflecting an angle across the xx-axis. Boundedness follows from the constraint x2+y2=1x^2 + y^2 = 1. Monotonicity — where each function increases or decreases — determines the intervals on which inverse trigonometric functions can be defined.

These properties govern every computation downstream. Periodicity is the reason trigonometric equations produce infinitely many solutions. Boundedness is the reason sin⁡(x)=2\sin(x) = 2 has no solution at all. Even/odd symmetry simplifies expressions involving negative angles and underlies several fundamental identities. The locations of zeros and asymptotes shape the graphs and constrain the domains of composite expressions. Understanding the properties as a coherent system — rather than as a list — is what separates mechanical calculation from genuine fluency with the trigonometric functions.

Key Terms

Core Concept

Periodic Function— a function repeating at regular intervals
Unit Circle— the geometric source of all trigonometric properties

The Six Functions

Sine— yy-coordinate; odd, bounded, period 2π2\pi
Cosine— xx-coordinate; even, bounded, period 2π2\pi
Tangent— yx\frac{y}{x}; odd, unbounded, period π\pi
Cosecant— 1sin⁡θ\frac{1}{\sin\theta}; odd, unbounded below ∣1∣|1|
Secant— 1cos⁡θ\frac{1}{\cos\theta}; even, unbounded below ∣1∣|1|
Cotangent— cos⁡θsin⁡θ\frac{\cos\theta}{\sin\theta}; odd, unbounded, period π\pi

Formulas Used

Even-Odd Identities

Sine Odd Identity— sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin\theta
Cosine Even Identity— cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos\theta
Tangent Odd Identity— tan⁡(−θ)=−tan⁡θ\tan(-\theta) = -\tan\theta

Periodicity Identities

Sine Periodicity— sin⁡(θ+2π)=sin⁡θ\sin(\theta + 2\pi) = \sin\theta
Cosine Periodicity— cos⁡(θ+2π)=cos⁡θ\cos(\theta + 2\pi) = \cos\theta
Tangent Periodicity— tan⁡(θ+π)=tan⁡θ\tan(\theta + \pi) = \tan\theta

See All Trigonometry Definitions →


Periodicity

A function ff is periodic if there exists a positive number TT such that f(θ+T)=f(θ)f(\theta + T) = f(\theta) for every θ\theta in the domain. The smallest such TT is the fundamental period.

Sine and cosine have period 2π2\pi:

sin⁡(θ+2π)=sin⁡(θ),cos⁡(θ+2π)=cos⁡(θ)\sin(\theta + 2\pi) = \sin(\theta), \quad \cos(\theta + 2\pi) = \cos(\theta)


This is a geometric fact: after a full rotation of 2π2\pi radians around the unit circle, the point (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta) returns to its starting position. The coordinates — and therefore the function values — are unchanged. Adding any integer multiple of 2π2\pi has no effect: sin⁡(θ+2nπ)=sin⁡(θ)\sin(\theta + 2n\pi) = \sin(\theta) for all integers nn.

Cosecant and secant inherit the periods of their reciprocals. Since csc⁡θ=1sin⁡θ\csc\theta = \frac{1}{\sin\theta} and sine has period 2π2\pi, cosecant also has period 2π2\pi. The same applies to secant via cosine.

Tangent and cotangent have the shorter period π\pi:

tan⁡(θ+π)=tan⁡(θ),cot⁡(θ+π)=cot⁡(θ)\tan(\theta + \pi) = \tan(\theta), \quad \cot(\theta + \pi) = \cot(\theta)


The algebraic reason: tan⁡(θ+π)=sin⁡(θ+π)cos⁡(θ+π)=−sin⁡θ−cos⁡θ=sin⁡θcos⁡θ=tan⁡θ\tan(\theta + \pi) = \frac{\sin(\theta + \pi)}{\cos(\theta + \pi)} = \frac{-\sin\theta}{-\cos\theta} = \frac{\sin\theta}{\cos\theta} = \tan\theta. A half rotation negates both coordinates, but the ratio is unaffected. Geometrically, the point diametrically opposite (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta) on the unit circle has both coordinates negated, producing the same value of yx\frac{y}{x}.

The practical consequence for equations: if θ0\theta_0 is a solution to a trigonometric equation, then θ0+2nπ\theta_0 + 2n\pi (for sine, cosine, cosecant, secant) or θ0+nπ\theta_0 + n\pi (for tangent, cotangent) are also solutions. This generates the infinite families that characterize general solutions of trigonometric equations.

The practical consequence for graphs: every trigonometric graph is fully determined by one period. The rest is repetition.

Even and Odd Symmetry

A function is even if f(−θ)=f(θ)f(-\theta) = f(\theta) for all θ\theta in its domain — its graph is symmetric about the yy-axis. A function is odd if f(−θ)=−f(θ)f(-\theta) = -f(\theta) — its graph is symmetric about the origin.

Among the six trigonometric functions, only two are even:

cos⁡(−θ)=cos⁡(θ),sec⁡(−θ)=sec⁡(θ)\cos(-\theta) = \cos(\theta), \quad \sec(-\theta) = \sec(\theta)


The remaining four are odd:

sin⁡(−θ)=−sin⁡(θ),tan⁡(−θ)=−tan⁡(θ)\sin(-\theta) = -\sin(\theta), \quad \tan(-\theta) = -\tan(\theta)

csc⁡(−θ)=−csc⁡(θ),cot⁡(−θ)=−cot⁡(θ)\csc(-\theta) = -\csc(\theta), \quad \cot(-\theta) = -\cot(\theta)


The geometric explanation is clean. On the unit circle, the angle −θ-\theta corresponds to reflecting θ\theta across the xx-axis. This reflection negates the yy-coordinate but preserves the xx-coordinate. Since cosine is the xx-coordinate, it is unchanged — even. Since sine is the yy-coordinate, it is negated — odd. Tangent, being yx\frac{y}{x}, is the ratio of an odd quantity to an even one, which is odd. The reciprocal functions inherit the parity of their parent functions: the reciprocal of an even function is even, and the reciprocal of an odd function is odd.

These symmetries are classified as the even/odd identities and are used constantly in simplification. When a negative angle appears in an expression — say, sin⁡(−3x)\sin(-3x) — the odd identity converts it immediately to −sin⁡(3x)-\sin(3x), eliminating the negative argument. Similarly, cos⁡(−α)=cos⁡(α)\cos(-\alpha) = \cos(\alpha) allows the sign to be dropped outright.

On the graphs, even symmetry (cosine, secant) produces a mirror image across the yy-axis. Odd symmetry (sine, tangent, cosecant, cotangent) produces a 180°180° rotational symmetry about the origin.

Boundedness

    A function is bounded if its output never exceeds some fixed value in absolute terms. Among the six trigonometric functions, only sine and cosine are bounded:

    −1≤sin⁡(θ)≤1,−1≤cos⁡(θ)≤1-1 \leq \sin(\theta) \leq 1, \quad -1 \leq \cos(\theta) \leq 1


    This follows directly from the unit circle. The coordinates of any point on a circle of radius 1 satisfy x2+y2=1x^2 + y^2 = 1, which forces ∣x∣≤1|x| \leq 1 and ∣y∣≤1|y| \leq 1. No amount of rotation will ever produce a sine or cosine value outside [−1,1][-1, 1]. The bounds are achieved: sin⁡θ=1\sin\theta = 1 at θ=π2\theta = \frac{\pi}{2} and sin⁡θ=−1\sin\theta = -1 at θ=3π2\theta = \frac{3\pi}{2}, with analogous extremes for cosine.

    Cosecant and secant are bounded away from zero but unbounded overall:

    ∣csc⁡(θ)∣≥1,∣sec⁡(θ)∣≥1|\csc(\theta)| \geq 1, \quad |\sec(\theta)| \geq 1


    Since ∣sin⁡θ∣≤1|\sin\theta| \leq 1, the reciprocal ∣csc⁡θ∣=1∣sin⁡θ∣≥1|\csc\theta| = \frac{1}{|\sin\theta|} \geq 1. As sin⁡θ\sin\theta approaches zero, csc⁡θ\csc\theta grows without bound. The range of both cosecant and secant is (−∞,−1]∪[1,∞)(-\infty, -1] \cup [1, \infty) — they can be arbitrarily large in magnitude but can never take a value between −1-1 and 11.

    Tangent and cotangent are entirely unbounded. Their ranges are (−∞,∞)(-\infty, \infty), and they take every real value on every interval between consecutive asymptotes.

    Boundedness has immediate consequences for equations and inequalities:

  • sin⁡(x)=1.5\sin(x) = 1.5 has no solution — the output cannot exceed 1.
  • csc⁡(x)=0.3\csc(x) = 0.3 has no solution — the output cannot be between −1-1 and 11.
  • tan⁡(x)=1000\tan(x) = 1000 has solutions — tangent can take any real value.
  • ∣sin⁡(x)∣≤1|\sin(x)| \leq 1 is always true — it constrains nothing.

    • Recognizing these range constraints before attempting to solve prevents wasted effort on impossible equations.

Zeros

The zeros of a trigonometric function — the values of θ\theta where the function equals zero — are determined by the unit circle coordinates and the algebraic definitions.

Sine equals zero when the yy-coordinate on the unit circle is zero — when the terminal side lies on the xx-axis:

sin⁡(θ)=0atθ=nπ(n∈Z)\sin(\theta) = 0 \quad \text{at} \quad \theta = n\pi \quad (n \in \mathbb{Z})


That is: …,−2π,−π,0,π,2π,…\ldots, -2\pi, -\pi, 0, \pi, 2\pi, \ldots

Cosine equals zero when the xx-coordinate is zero — when the terminal side lies on the yy-axis:

cos⁡(θ)=0atθ=π2+nπ(n∈Z)\cos(\theta) = 0 \quad \text{at} \quad \theta = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z})


That is: …,−3π2,−π2,π2,3π2,…\ldots, -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, \ldots

Tangent equals zero when sin⁡θ=0\sin\theta = 0 and cos⁡θ≠0\cos\theta \neq 0 — at the same points as sine:

tan⁡(θ)=0atθ=nπ\tan(\theta) = 0 \quad \text{at} \quad \theta = n\pi


Cotangent equals zero when cos⁡θ=0\cos\theta = 0 and sin⁡θ≠0\sin\theta \neq 0 — at the same points as cosine:

cot⁡(θ)=0atθ=π2+nπ\cot(\theta) = 0 \quad \text{at} \quad \theta = \frac{\pi}{2} + n\pi


Cosecant and secant have no zeros. Since csc⁡θ=1sin⁡θ\csc\theta = \frac{1}{\sin\theta}, a zero would require 1sin⁡θ=0\frac{1}{\sin\theta} = 0, which has no solution — no reciprocal of a real number equals zero. The same applies to secant. Their ranges, (−∞,−1]∪[1,∞)(-\infty, -1] \cup [1, \infty), confirm this: zero is not in the range.

The zeros of sine and cosine are evenly spaced at intervals of π\pi — a direct consequence of the 2π2\pi period and the two zero crossings per cycle. The zeros of tangent coincide with the zeros of sine, and the zeros of cotangent coincide with the zeros of cosine. On the graphs, these zeros are the xx-intercepts, and their regular spacing reflects the periodicity.

Continuity and Discontinuities

    Sine and cosine are continuous on the entire real line — their graphs are unbroken curves with no jumps, gaps, or asymptotes. This follows from the geometry: as an angle increases smoothly, the point on the unit circle moves smoothly, and the coordinates change continuously. There is no angle at which the xx- or yy-coordinate suddenly jumps.

    The remaining four functions all have vertical asymptotes — points where the function value grows without bound and the function is undefined.

    Tangent and secant are discontinuous at θ=π2+nπ\theta = \frac{\pi}{2} + n\pi (odd multiples of π2\frac{\pi}{2}). These are the zeros of cosine, and since both tangent (sin⁡θcos⁡θ\frac{\sin\theta}{\cos\theta}) and secant (1cos⁡θ\frac{1}{\cos\theta}) have cos⁡θ\cos\theta in the denominator, division by zero creates the discontinuity. As θ\theta approaches any of these points, the function value tends toward +∞+\infty or −∞-\infty, producing a vertical asymptote on the graph.

    Cotangent and cosecant are discontinuous at θ=nπ\theta = n\pi (integer multiples of π\pi). These are the zeros of sine, and both cotangent (cos⁡θsin⁡θ\frac{\cos\theta}{\sin\theta}) and cosecant (1sin⁡θ\frac{1}{\sin\theta}) have sin⁡θ\sin\theta in the denominator.

    These discontinuities are not removable — the function does not approach a finite limit from either side. They are infinite discontinuities, corresponding to vertical asymptotes. Between consecutive asymptotes, each function is continuous.

    The pattern is worth memorizing:

  • cos⁡θ=0\cos\theta = 0 → tangent and secant undefined
  • sin⁡θ=0\sin\theta = 0 → cotangent and cosecant undefined

    • This determines the domain of any expression involving these functions. For example, the expression tan⁡(x)+csc⁡(x)\tan(x) + \csc(x) is undefined wherever cos⁡(x)=0\cos(x) = 0 (from the tangent) or sin⁡(x)=0\sin(x) = 0 (from the cosecant) — that is, at every integer multiple of π2\frac{\pi}{2}.

Monotonicity on Principal Intervals

Monotonicity describes where a function is strictly increasing or strictly decreasing. For trigonometric functions, the intervals of monotonicity repeat with the period, so it suffices to identify them over one period — or, more precisely, over the intervals that will serve as restricted domains for the inverse functions.

Sine is strictly increasing on [−π2,π2]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right], where it rises from −1-1 to 11. It is strictly decreasing on [π2,3π2]\left[\frac{\pi}{2}, \frac{3\pi}{2}\right], where it falls from 11 back to −1-1. These two intervals together span one full period.

Cosine is strictly decreasing on [0,π][0, \pi], dropping from 11 to −1-1. It is strictly increasing on [π,2π][\pi, 2\pi], climbing from −1-1 back to 11.

Tangent is strictly increasing on each interval (−π2+nπ,π2+nπ)\left(-\frac{\pi}{2} + n\pi, \frac{\pi}{2} + n\pi\right) — that is, on every interval between consecutive asymptotes. Within each such interval, tangent runs continuously from −∞-\infty to +∞+\infty.

Cotangent is strictly decreasing on each interval (nπ,(n+1)π)(n\pi, (n+1)\pi) — between its consecutive asymptotes. It runs from +∞+\infty to −∞-\infty within each period.

Secant is decreasing on [0,π2)\left[0, \frac{\pi}{2}\right), increasing on (π2,π]\left(\frac{\pi}{2}, \pi\right], and so on — its behavior follows from the reciprocal of cosine.

Cosecant is decreasing on (0,π2]\left(0, \frac{\pi}{2}\right], increasing on [π2,π)\left[\frac{\pi}{2}, \pi\right), and so on — its behavior follows from the reciprocal of sine.

The principal intervals of monotonicity are precisely the domains used to construct the inverse trigonometric functions. Arcsine uses [−π2,π2]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] (where sine is increasing), arccosine uses [0,π][0, \pi] (where cosine is decreasing), and arctangent uses (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) (where tangent is increasing). The choice is not arbitrary — it is dictated by the requirement that the restricted function be one-to-one (which monotonicity guarantees) and that the restricted domain cover the entire range of the function.

Extrema

The maximum and minimum values of a trigonometric function — if they exist — occur at specific, predictable points determined by the unit circle geometry.

Sine achieves its maximum value of 11 at θ=π2+2nπ\theta = \frac{\pi}{2} + 2n\pi — the top of the unit circle, where the yy-coordinate is as large as it can be. It achieves its minimum of −1-1 at θ=3π2+2nπ\theta = \frac{3\pi}{2} + 2n\pi — the bottom of the circle. These are global (absolute) extrema, and they recur with period 2π2\pi.

Cosine achieves its maximum of 11 at θ=2nπ\theta = 2n\pi — the rightmost point of the unit circle. Its minimum of −1-1 occurs at θ=π+2nπ\theta = \pi + 2n\pi — the leftmost point.

Cosecant has local minima of 11 (where sin⁡θ=1\sin\theta = 1) and local maxima of −1-1 (where sin⁡θ=−1\sin\theta = -1). These labels may seem reversed, but they are correct: csc⁡θ=1\csc\theta = 1 is the smallest positive value cosecant achieves (on the upward-opening branches), and csc⁡θ=−1\csc\theta = -1 is the largest negative value (on the downward-opening branches). Between asymptotes, each branch has exactly one extremum.

Secant behaves analogously: local minima of 11 at cosine's maxima, local maxima of −1-1 at cosine's minima.

Tangent and cotangent have no extrema. They are unbounded in both directions, with no maximum or minimum value. Within each period, they increase (tangent) or decrease (cotangent) monotonically from −∞-\infty to +∞+\infty or vice versa, never leveling off or turning around.

For transformed functions like y=Asin⁡(Bx−C)+Dy = A\sin(Bx - C) + D, the extrema are:

maximum=D+∣A∣,minimum=D−∣A∣\text{maximum} = D + |A|, \quad \text{minimum} = D - |A|


These occur at the same relative positions within each period as for the standard function, shifted according to the phase shift and compressed or stretched according to the period.

Identity of the Six Functions: Period, Parity, Range

The three properties that fix the structural identity of each trigonometric function — period, parity, and range — collect into a single reference card. Reading down a column shows how each property partitions the six functions; reading across a row gives the basic profile of one function.
Function Period Parity Range
sin 2π odd [−1, 1]
cos 2π even [−1, 1]
tan π odd (−∞, ∞)
csc 2π odd (−∞, −1] ∪ [1, ∞)
sec 2π even (−∞, −1] ∪ [1, ∞)
cot π odd (−∞, ∞)

Critical-Point Landscape: Where Each Function Vanishes, Explodes, and Turns Around

The remaining four properties — zeros, asymptotes, principal monotonic intervals, and extrema — together describe where each function does something structurally interesting on the real line. The table below pairs with the identity card above to give the complete property-by-function profile of the family.
Function Zeros Asymptotes Principal monotonic interval Extrema
sin nπ none [−π/2, π/2] (increasing) max 1 at π/2 + 2nπ; min −1 at 3π/2 + 2nπ
cos π/2 + nπ none [0, π] (decreasing) max 1 at 2nπ; min −1 at π + 2nπ
tan nπ π/2 + nπ (−π/2, π/2) (increasing) none (unbounded)
csc none nπ [−π/2, 0) ∪ (0, π/2] (decreasing on each piece) local min 1 at π/2 + 2nπ; local max −1 at 3π/2 + 2nπ
sec none π/2 + nπ [0, π/2) ∪ (π/2, π] (increasing on each piece) local min 1 at 2nπ; local max −1 at π + 2nπ
cot π/2 + nπ nπ (0, π) (decreasing) none (unbounded)