Periodicity Identities

Periodicity Identities

Trig identities showing the periodic behavior of the six trig functions.

Periodicity Identities, radians		Periodicity Identities, degrees
sin (x + 2π) = sin x	csc (x + 2π) = csc x	sin (x + 360°) = sin x	csc (x + 360°) = csc x
cos (x + 2π) = cos x	sec (x + 2π) = sec x	cos (x + 360°) = cos x	sec (x + 360°) = sec x
tan (x + π) = tan x	cot (x + π) = cot x	tan (x + 180°) = tan x	cot (x + 180°) = cot x

Key Formula

\begin{aligned} \sin(x + 2\pi) &= \sin x & \csc(x + 2\pi) &= \csc x \\ \cos(x + 2\pi) &= \cos x & \sec(x + 2\pi) &= \sec x \\ \tan(x + \pi) &= \tan x & \cot(x + \pi) &= \cot x \end{aligned}

Where:

$x$ = Any angle, measured in radians (or degrees if 2π is replaced by 360° and π by 180°)
$2\pi$ = The period of sine, cosine, cosecant, and secant (one full revolution, or 360°)
$\pi$ = The period of tangent and cotangent (half a revolution, or 180°)

Worked Example

Problem: Use a periodicity identity to find the exact value of sin(13π/6).

Step 1: Recognize that 13π/6 is greater than 2π. Subtract 2π (one full period) from the angle to bring it into a familiar range.

\frac{13\pi}{6} - 2\pi = \frac{13\pi}{6} - \frac{12\pi}{6} = \frac{\pi}{6}

Step 2: Apply the periodicity identity for sine: sin(x + 2π) = sin x.

\sin\!\left(\frac{13\pi}{6}\right) = \sin\!\left(\frac{\pi}{6} + 2\pi\right) = \sin\!\left(\frac{\pi}{6}\right)

Step 3: Evaluate sin(π/6) using the known exact value.

\sin\!\left(\frac{\pi}{6}\right) = \frac{1}{2}

Answer: sin(13π/6) = 1/2

Another Example

This example uses the tangent periodicity identity with a period of π (not 2π) and removes multiple periods at once, showing that tan(x + nπ) = tan x for any integer n.

Problem: Use a periodicity identity to simplify tan(7π/4 + 3π) and find its exact value.

Step 1: Notice that 3π is a multiple of the tangent period π. We can write 3π = 3 · π, so we remove three full periods.

\tan\!\left(\frac{7\pi}{4} + 3\pi\right) = \tan\!\left(\frac{7\pi}{4}\right)

Step 2: This works because tan(x + nπ) = tan x for any integer n. Here n = 3.

\tan(x + n\pi) = \tan x, \quad n \in \mathbb{Z}

Step 3: Evaluate tan(7π/4). Since 7π/4 is in the fourth quadrant with a reference angle of π/4, tangent is negative there.

\tan\!\left(\frac{7\pi}{4}\right) = -\tan\!\left(\frac{\pi}{4}\right) = -1

Answer: tan(7π/4 + 3π) = −1

Frequently Asked Questions

Why do tangent and cotangent have a period of π instead of 2π?

Tangent is defined as sin x / cos x. After adding π radians, both sine and cosine change sign (both become negative), so their ratio stays the same. This means the tangent function completes one full cycle in just π radians, which is half the period of sine or cosine. The same reasoning applies to cotangent since cot x = cos x / sin x.

Can you subtract a period instead of adding one?

Yes. Periodicity works in both directions. For example, sin(x − 2π) = sin x and tan(x − π) = tan x. More generally, sin(x + 2πn) = sin x for any integer n, whether positive or negative. Subtracting periods is useful when the angle is negative and you want to shift it into a positive range.

What is the difference between periodicity identities and cofunction identities?

Periodicity identities show that a trig function repeats when you add its period (e.g., sin(x + 2π) = sin x). Cofunction identities relate complementary functions to each other, such as sin(π/2 − x) = cos x. Periodicity keeps the same function with a shifted angle, while cofunction identities switch between paired functions like sine and cosine.

Periodicity Identities vs. Cofunction Identities

	Periodicity Identities	Cofunction Identities
What they describe	A trig function repeats after one full period	A trig function equals its complement's cofunction
Key formula (example)	sin(x + 2π) = sin x	sin(π/2 − x) = cos x
Function changes?	No — same function on both sides	Yes — switches to the paired function (sin ↔ cos, tan ↔ cot, sec ↔ csc)
When to use	Simplify angles that exceed one revolution or are highly negative	Convert between complementary trig functions

Why It Matters

Periodicity identities appear constantly when you solve trig equations, because every trig equation has infinitely many solutions spaced one period apart. They are also essential in calculus and physics when simplifying integrals or analyzing waves, signals, and circular motion. On standardized tests and in precalculus courses, you will use these identities to reduce large or negative angles to values you can evaluate from memory.

Common Mistakes

Mistake: Using a period of 2π for tangent or cotangent.

Correction: Tangent and cotangent repeat every π radians (180°), not every 2π. Writing tan(x + 2π) = tan x is technically true but misses the smaller period. The fundamental period is π, so always use tan(x + π) = tan x when simplifying.

Mistake: Thinking the function value changes sign after adding one period.

Correction: Adding a full period never changes the output. For instance, sin(x + 2π) = sin x, not −sin x. Confusing this with the identity sin(x + π) = −sin x (which adds only half of sine's period) is a common source of sign errors.

Related Terms

Trig Identities — Broader family of all trigonometric identities
Periodic Function — Defines the concept of period that these identities use
Trig Functions — The six functions whose periodicity is described
Radian — Unit of angle measure used in these identities
Degree — Alternative angle unit (360° = 2π radians)
Cofunction Identities — Related identities pairing complementary trig functions
Unit Circle — Visual basis for understanding trig periodicity
Reference Angle — Used alongside periodicity to evaluate trig values