Precalculus Flashcards: Symmetry And Periodicity Of Trigonometric Functions

What this deck covers

This deck focuses on Symmetry And Periodicity Of Trigonometric Functions, giving you a quick way to review the definitions, rules, and examples that matter most for Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What periodicity identity shows $\tan(\theta+\pi)$ in simplest form?

Answer: $\tan(\theta+\pi)=\tan\theta$. Tangent has period $\pi$, so adding $\pi$ returns the same value.

Flashcard 2: What symmetry identity relates $\tan(-\theta)$ to $\tan\theta$?

Answer: $\tan(-\theta)=-\tan\theta$. Tangent inherits odd symmetry from $\frac{\sin\theta}{\cos\theta}$.

Flashcard 3: What symmetry identity relates $\sin(-\theta)$ to $\sin\theta$?

Answer: $\sin(-\theta)=-\sin\theta$. Sine is odd, so negative angles give opposite values.

Flashcard 4: What is the unit-circle symmetry identity for sine with a $\pi$ shift?

Answer: $\sin(\theta+\pi)=-\sin\theta$. Adding $\pi$ rotates 180°, flipping the point through the origin.

Flashcard 5: What is the fundamental period of $\tan\theta$ on the unit circle?

Answer: Period $=\pi$. Tangent repeats every half rotation since $\tan(\theta+\pi)=\tan\theta$.

Flashcard 6: What identity defines tangent in terms of unit circle coordinates $(x,y)$?

Answer: $\tan\theta=\frac{y}{x}$. Tangent is the ratio of $y$ to $x$ coordinates on the unit circle.

Flashcard 7: What is $\tan(-\frac{\pi}{4})$ using odd symmetry?

Answer: $-1$. Since $\tan(-\theta)=-\tan\theta$ and $\tan(\frac{\pi}{4})=1$.

Flashcard 8: What is the cofunction identity for cosine using a $\frac{\pi}{2}$ shift?

Answer: $\cos\left(\frac{\pi}{2}-\theta\right)=\sin\theta$. The complement of an angle exchanges the roles of sine and cosine.

Flashcard 9: What symmetry identity relates $\cos(-\theta)$ to $\cos\theta$?

Answer: $\cos(-\theta)=\cos\theta$. Cosine is even, so negative angles give the same value.

Flashcard 10: What is the symmetry identity for cotangent using even-odd properties?

Answer: $\cot(-\theta)=-\cot\theta$. Cotangent is odd since $\cot\theta = \frac{\cos\theta}{\sin\theta}$ (even/odd = odd).

Flashcard 11: What unit-circle identity defines cosine and sine as coordinates at angle $\theta$?

Answer: Point is $(\cos\theta,\sin\theta)$ on the unit circle. The unit circle has radius 1, so any point at angle $\theta$ has these coordinates.

Flashcard 12: What is the unit-circle symmetry identity for tangent with a $\pi$ shift?

Answer: $\tan(\theta+\pi)=\tan\theta$. Since both sine and cosine flip signs with $\pi$, their ratio stays the same.

Flashcard 13: What is $\sin(-\frac{\pi}{6})$ using odd symmetry?

Answer: $-\frac{1}{2}$. Since $\sin(-\theta)=-\sin\theta$ and $\sin(\frac{\pi}{6})=\frac{1}{2}$.

Flashcard 14: What is the sine identity for reflecting across the $y$-axis: $\pi-\theta$?

Answer: $\sin(\pi-\theta)=\sin\theta$. Points at $\theta$ and $\pi-\theta$ have the same y-coordinate (sine value).

Flashcard 15: What periodicity identity shows $\sin(\theta+2\pi)$ in simplest form?

Answer: $\sin(\theta+2\pi)=\sin\theta$. Adding $2\pi$ completes one full rotation, returning to the same value.

Flashcard 16: What is the sine identity for $2\pi-\theta$ using unit-circle symmetry?

Answer: $\sin(2\pi-\theta)=-\sin\theta$. Going clockwise by $\theta$ from 0 gives the opposite y-coordinate.

Flashcard 17: What is the cosine identity for reflecting across the $y$-axis: $\pi-\theta$?

Answer: $\cos(\pi-\theta)=-\cos\theta$. Points at $\theta$ and $\pi-\theta$ have opposite x-coordinates.

Flashcard 18: What is $\sin(\frac{11\pi}{6})$ using periodicity and unit circle values?

Answer: $-\frac{1}{2}$. $\frac{11\pi}{6}=2\pi-\frac{\pi}{6}$, and $\sin(-\frac{\pi}{6})=-\frac{1}{2}$.

Flashcard 19: What is the even-odd symmetry identity for sine on the unit circle?

Answer: $\sin(-\theta)=-\sin\theta$. Sine is odd: reflecting across the y-axis negates the y-coordinate.

Flashcard 20: What is the even-odd symmetry identity for tangent?

Answer: $\tan(-\theta)=-\tan\theta$. Tangent is odd since $\tan\theta = \frac{\sin\theta}{\cos\theta}$ (odd/even = odd).

Flashcard 21: What is the even/odd classification of $\cos\theta$?

Answer: $\cos\theta$ is even. Even functions satisfy $f(-x)=f(x)$; cosine has this property.

Flashcard 22: What is the symmetry identity for secant using even-odd properties?

Answer: $\sec(-\theta)=\sec\theta$. Secant is even since $\sec\theta = \frac{1}{\cos\theta}$ and cosine is even.

Flashcard 23: What is the period of $\cos\theta$?

Answer: $2\pi$. Cosine completes one full cycle every $2\pi$ radians.

Flashcard 24: What is the unit-circle symmetry identity for cosine with a $\pi$ shift?

Answer: $\cos(\theta+\pi)=-\cos\theta$. A $\pi$ rotation negates both coordinates, changing the sign of cosine.

Flashcard 25: What is the even-odd symmetry identity for cosine on the unit circle?

Answer: $\cos(-\theta)=\cos\theta$. Cosine is even: reflecting across the y-axis doesn't change the x-coordinate.

Flashcard 26: What is the cosine identity for $2\pi-\theta$ using unit-circle symmetry?

Answer: $\cos(2\pi-\theta)=\cos\theta$. Going clockwise by $\theta$ from 0 gives the same x-coordinate.

Flashcard 27: What is the even/odd classification of $\tan\theta$?

Answer: $\tan\theta$ is odd. Since $\tan(-\theta)=-\tan\theta$, tangent is an odd function.

Flashcard 28: What periodicity identity shows $\cos(\theta+2\pi)$ in simplest form?

Answer: $\cos(\theta+2\pi)=\cos\theta$. Adding $2\pi$ completes one full rotation, returning to the same value.

Flashcard 29: What is the fundamental period of $\sin\theta$ on the unit circle?

Answer: Period $=2\pi$. One full rotation around the unit circle returns to the same point.

Flashcard 30: What is the symmetry identity for cosecant using even-odd properties?

Answer: $\csc(-\theta)=-\csc\theta$. Cosecant is odd since $\csc\theta = \frac{1}{\sin\theta}$ and sine is odd.

Flashcard 31: What is the even/odd classification of $\sin\theta$?

Answer: $\sin\theta$ is odd. Odd functions satisfy $f(-x)=-f(x)$; sine has this property.

Flashcard 32: What is the periodicity identity for sine for any integer $k$?

Answer: $\sin(\theta+2\pi k)=\sin\theta$. Adding any multiple of $2\pi$ completes full rotations, returning to the same point.

Flashcard 33: What is the fundamental period of $\cos\theta$ on the unit circle?

Answer: Period $=2\pi$. Cosine repeats every full rotation ($2\pi$ radians) around the circle.

Flashcard 34: What is the period of $\sin\theta$?

Answer: $2\pi$. Sine completes one full cycle every $2\pi$ radians.

Flashcard 35: What is the period of $\tan\theta$?

Answer: $\pi$. Tangent repeats every $\pi$ radians, half the period of sine/cosine.

Flashcard 36: What is the cofunction identity for sine using a $\frac{\pi}{2}$ shift?

Answer: $\sin\left(\frac{\pi}{2}-\theta\right)=\cos\theta$. Complementary angles swap sine and cosine values on the unit circle.