Symmetry & Periodicity
of Trig Functions
Discover how the unit circle reveals that trigonometric functions are even, odd, or periodic: patterns that simplify every calculation you will ever do.
Precalculus
What Is Symmetry on the Unit Circle?
When we evaluate a trig function at an angle \(\theta\) and its opposite \(-\theta\), we are reflecting the terminal point across the x-axis. The coordinates change in a predictable way.
Even Functions: \(\cos(-\theta)=\cos\theta\)
A function \(f\) is even if \(f(-x)=f(x)\) for every \(x\) in its domain. Its graph is symmetric about the y-axis.
Cosine
\(\cos(-\theta)=\cos\theta\) because the x-coordinate stays the same when we reflect across the x-axis.
Secant
\(\sec(-\theta)=\sec\theta\) because \(\sec\theta=\frac{1}{\cos\theta}\), and cosine is even.
Odd Functions: \(\sin(-\theta)=-\sin\theta\)
A function \(f\) is odd if \(f(-x)=-f(x)\) for every \(x\) in its domain. Its graph has \(180^\circ\) rotational symmetry about the origin.
Sine
\(\sin(-\theta)=-\sin\theta\) because the y-coordinate negates when we reflect across the x-axis.
Tangent
\(\tan(-\theta)=-\tan\theta\) because \(\frac{-\sin\theta}{\cos\theta}=-\tan\theta\).
Cosecant & Cotangent
\(\csc(-\theta)=-\csc\theta\) and \(\cot(-\theta)=-\cot\theta\): reciprocals of odd functions are odd.
What Is Periodicity?
A function \(f\) is periodic with period \(T\) if \(f(x+T)=f(x)\) for all \(x\). The smallest such positive \(T\) is called the fundamental period.
Period \(2\pi\)
\(\sin\theta\), \(\cos\theta\), \(\csc\theta\), \(\sec\theta\): one full revolution around the unit circle.
Period \(\pi\)
\(\tan\theta\), \(\cot\theta\): the ratio repeats after a half revolution because both sine and cosine negate.
Core Symmetry & Periodicity Identities
1
Even: \(\cos(-\theta)=\cos\theta\) and \(\sec(-\theta)=\sec\theta\)
2
Odd: \(\sin(-\theta)=-\sin\theta\), \(\tan(-\theta)=-\tan\theta\), \(\csc(-\theta)=-\csc\theta\), \(\cot(-\theta)=-\cot\theta\)
3
Period \(2\pi\): \(\sin(\theta+2\pi)=\sin\theta\), \(\cos(\theta+2\pi)=\cos\theta\)
4
Period \(\pi\): \(\tan(\theta+\pi)=\tan\theta\), \(\cot(\theta+\pi)=\cot\theta\)
Seeing Even Symmetry in \(\cos\theta\)
Seeing Odd Symmetry in \(\sin\theta\)
Even vs Odd: Key Distinction
| Even Function | Odd Function | |
|---|---|---|
| Identity | \(f(-x)=f(x)\) | \(f(-x)=-f(x)\) |
| Symmetry | y-axis reflection | Origin rotation (\(180^\circ\)) |
| Trig examples | \(\cos, \sec\) | \(\sin, \tan, \csc, \cot\) |
| Graph test | Fold along y-axis | Rotate \(180^\circ\) about origin |
How to Apply These Properties
1
Identify the function. Is it \(\sin\), \(\cos\), \(\tan\), etc.?
2
Check for a negative angle. If the argument is \(-\theta\), apply the even/odd identity to remove the negative sign.
3
Check for extra full periods. Subtract \(2\pi\) (or \(\pi\) for tan/cot) until the angle is in the standard range \([0,\,2\pi)\).
4
Evaluate. Use the unit circle or known values to find the result.
Example 1: Negative Angle
Evaluate \(\sin\!\left(-\dfrac{\pi}{4}\right)\).
1
Sine is odd, so \(\sin(-\theta)=-\sin\theta\).
2
Apply: \(\sin\!\left(-\dfrac{\pi}{4}\right) = -\sin\!\left(\dfrac{\pi}{4}\right)\).
3
Evaluate: \(\sin\!\left(\dfrac{\pi}{4}\right) = \dfrac{\sqrt{2}}{2}\).
Example 2: Periodicity
Evaluate \(\cos\!\left(\dfrac{13\pi}{3}\right)\).
1
Cosine has period \(2\pi\). Subtract multiples: \(\dfrac{13\pi}{3} - 2\pi = \dfrac{13\pi}{3} - \dfrac{6\pi}{3} = \dfrac{7\pi}{3}\).
2
Still > \(2\pi\)? Yes. Subtract again: \(\dfrac{7\pi}{3} - 2\pi = \dfrac{7\pi}{3} - \dfrac{6\pi}{3} = \dfrac{\pi}{3}\).
3
Evaluate: \(\cos\!\left(\dfrac{\pi}{3}\right) = \dfrac{1}{2}\).
Example 3: Both Properties
Evaluate \(\tan\!\left(-\dfrac{7\pi}{4}\right)\).
1
Tangent is odd: \(\tan\!\left(-\dfrac{7\pi}{4}\right) = -\tan\!\left(\dfrac{7\pi}{4}\right)\).
2
Reduce: \(\dfrac{7\pi}{4}\) is in \([0,2\pi)\), so no period subtraction needed.
3
\(\dfrac{7\pi}{4}\) is in Q IV, reference angle \(\dfrac{\pi}{4}\). In Q IV, tangent is negative: \(\tan\!\left(\dfrac{7\pi}{4}\right)=-1\).
4
Combine: \(-(-1) = 1\).
\(\tan\theta\): Period \(\pi\) Visualized
Common Mistakes to Avoid
✕ Calling sine "even"
Only cosine and secant are even. Sine is odd: \(\sin(-\theta) = -\sin\theta\), not \(\sin\theta\).
✕ Wrong period for tangent
Tangent and cotangent repeat every \(\pi\), not \(2\pi\). Using \(2\pi\) will still be correct but misses the smallest period.
✕ Forgetting the negative sign
With odd functions, \(f(-x) = -f(x)\). Students often drop the leading negative after applying the identity.
Summary Table
| Function | Even / Odd | Period | Identity |
|---|---|---|---|
| \(\sin\theta\) | Odd | \(2\pi\) | \(\sin(-\theta)=-\sin\theta\) |
| \(\cos\theta\) | Even | \(2\pi\) | \(\cos(-\theta)=\cos\theta\) |
| \(\tan\theta\) | Odd | \(\pi\) | \(\tan(-\theta)=-\tan\theta\) |
| \(\csc\theta\) | Odd | \(2\pi\) | \(\csc(-\theta)=-\csc\theta\) |
| \(\sec\theta\) | Even | \(2\pi\) | \(\sec(-\theta)=\sec\theta\) |
| \(\cot\theta\) | Odd | \(\pi\) | \(\cot(-\theta)=-\cot\theta\) |
Question 1 of 5
Which of the following is an even trigonometric function?
Cosine is even because \(\cos(-\theta)=\cos\theta\). The x-coordinate on the unit circle does not change when we reflect across the x-axis. Sine, tangent, and cotangent are all odd.
Question 2 of 5
What is the value of \(\sin\!\left(-\dfrac{\pi}{6}\right)\)?
Sine is odd, so \(\sin\!\left(-\frac{\pi}{6}\right) = -\sin\!\left(\frac{\pi}{6}\right) = -\frac{1}{2}\).
Question 3 of 5
What is the fundamental period of \(\tan\theta\)?
Because \(\sin(\theta+\pi)=-\sin\theta\) and \(\cos(\theta+\pi)=-\cos\theta\), the ratio \(\tan(\theta+\pi) = \frac{-\sin\theta}{-\cos\theta} = \tan\theta\). So the fundamental period is \(\pi\).
Question 4 of 5
Simplify: \(\cos\!\left(-\dfrac{5\pi}{3}\right)\).
Cosine is even, so \(\cos\!\left(-\frac{5\pi}{3}\right) = \cos\!\left(\frac{5\pi}{3}\right)\). Since \(\frac{5\pi}{3}\) is in Q IV with reference angle \(\frac{\pi}{3}\), cosine is positive: \(\cos\!\left(\frac{\pi}{3}\right) = \frac{1}{2}\).
Question 5 of 5
If \(f(\theta) = \sin\theta + \cos\theta\), is \(f\) even, odd, or neither?
\(f(-\theta) = \sin(-\theta)+\cos(-\theta) = -\sin\theta + \cos\theta\). This equals neither \(f(\theta)\) nor \(-f(\theta)\), so \(f\) is neither even nor odd.