Symmetry and Periodicity of Trigonometric Functions - Pre-Calculus
Card 1 of 30
What unit-circle identity defines cosine and sine as coordinates at angle $\theta$?
What unit-circle identity defines cosine and sine as coordinates at angle $\theta$?
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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.
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.
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What is the even-odd symmetry identity for sine on the unit circle?
What is the even-odd symmetry identity for sine on the unit circle?
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$\sin(-\theta)=-\sin\theta$. Sine is odd: reflecting across the y-axis negates the y-coordinate.
$\sin(-\theta)=-\sin\theta$. Sine is odd: reflecting across the y-axis negates the y-coordinate.
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What is the even-odd symmetry identity for tangent?
What is the even-odd symmetry identity for tangent?
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$\tan(-\theta)=-\tan\theta$. Tangent is odd since $\tan\theta = \frac{\sin\theta}{\cos\theta}$ (odd/even = odd).
$\tan(-\theta)=-\tan\theta$. Tangent is odd since $\tan\theta = \frac{\sin\theta}{\cos\theta}$ (odd/even = odd).
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What is the symmetry identity for secant using even-odd properties?
What is the symmetry identity for secant using even-odd properties?
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$\sec(-\theta)=\sec\theta$. Secant is even since $\sec\theta = \frac{1}{\cos\theta}$ and cosine is even.
$\sec(-\theta)=\sec\theta$. Secant is even since $\sec\theta = \frac{1}{\cos\theta}$ and cosine is even.
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What is the symmetry identity for cosecant using even-odd properties?
What is the symmetry identity for cosecant using even-odd properties?
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$\csc(-\theta)=-\csc\theta$. Cosecant is odd since $\csc\theta = \frac{1}{\sin\theta}$ and sine is odd.
$\csc(-\theta)=-\csc\theta$. Cosecant is odd since $\csc\theta = \frac{1}{\sin\theta}$ and sine is odd.
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What is the symmetry identity for cotangent using even-odd properties?
What is the symmetry identity for cotangent using even-odd properties?
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$\cot(-\theta)=-\cot\theta$. Cotangent is odd since $\cot\theta = \frac{\cos\theta}{\sin\theta}$ (even/odd = odd).
$\cot(-\theta)=-\cot\theta$. Cotangent is odd since $\cot\theta = \frac{\cos\theta}{\sin\theta}$ (even/odd = odd).
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What is the unit-circle symmetry identity for sine with a $\pi$ shift?
What is the unit-circle symmetry identity for sine with a $\pi$ shift?
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$\sin(\theta+\pi)=-\sin\theta$. Adding $\pi$ rotates 180°, flipping the point through the origin.
$\sin(\theta+\pi)=-\sin\theta$. Adding $\pi$ rotates 180°, flipping the point through the origin.
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What is the even/odd classification of $\cos\theta$?
What is the even/odd classification of $\cos\theta$?
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$\cos\theta$ is even. Even functions satisfy $f(-x)=f(x)$; cosine has this property.
$\cos\theta$ is even. Even functions satisfy $f(-x)=f(x)$; cosine has this property.
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What is the even/odd classification of $\sin\theta$?
What is the even/odd classification of $\sin\theta$?
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$\sin\theta$ is odd. Odd functions satisfy $f(-x)=-f(x)$; sine has this property.
$\sin\theta$ is odd. Odd functions satisfy $f(-x)=-f(x)$; sine has this property.
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What is the even/odd classification of $\tan\theta$?
What is the even/odd classification of $\tan\theta$?
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$\tan\theta$ is odd. Since $\tan(-\theta)=-\tan\theta$, tangent is an odd function.
$\tan\theta$ is odd. Since $\tan(-\theta)=-\tan\theta$, tangent is an odd function.
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What identity defines tangent in terms of unit circle coordinates $(x,y)$?
What identity defines tangent in terms of unit circle coordinates $(x,y)$?
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$\tan\theta=\frac{y}{x}$. Tangent is the ratio of $y$ to $x$ coordinates on the unit circle.
$\tan\theta=\frac{y}{x}$. Tangent is the ratio of $y$ to $x$ coordinates on the unit circle.
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What is $\sin(\frac{11\pi}{6})$ using periodicity and unit circle values?
What is $\sin(\frac{11\pi}{6})$ using periodicity and unit circle values?
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$-\frac{1}{2}$. $\frac{11\pi}{6}=2\pi-\frac{\pi}{6}$, and $\sin(-\frac{\pi}{6})=-\frac{1}{2}$.
$-\frac{1}{2}$. $\frac{11\pi}{6}=2\pi-\frac{\pi}{6}$, and $\sin(-\frac{\pi}{6})=-\frac{1}{2}$.
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What is $\tan(-\frac{\pi}{4})$ using odd symmetry?
What is $\tan(-\frac{\pi}{4})$ using odd symmetry?
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$-1$. Since $\tan(-\theta)=-\tan\theta$ and $\tan(\frac{\pi}{4})=1$.
$-1$. Since $\tan(-\theta)=-\tan\theta$ and $\tan(\frac{\pi}{4})=1$.
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What is $\sin(-\frac{\pi}{6})$ using odd symmetry?
What is $\sin(-\frac{\pi}{6})$ using odd symmetry?
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$-\frac{1}{2}$. Since $\sin(-\theta)=-\sin\theta$ and $\sin(\frac{\pi}{6})=\frac{1}{2}$.
$-\frac{1}{2}$. Since $\sin(-\theta)=-\sin\theta$ and $\sin(\frac{\pi}{6})=\frac{1}{2}$.
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What periodicity identity shows $\tan(\theta+\pi)$ in simplest form?
What periodicity identity shows $\tan(\theta+\pi)$ in simplest form?
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$\tan(\theta+\pi)=\tan\theta$. Tangent has period $\pi$, so adding $\pi$ returns the same value.
$\tan(\theta+\pi)=\tan\theta$. Tangent has period $\pi$, so adding $\pi$ returns the same value.
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What periodicity identity shows $\cos(\theta+2\pi)$ in simplest form?
What periodicity identity shows $\cos(\theta+2\pi)$ in simplest form?
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$\cos(\theta+2\pi)=\cos\theta$. Adding $2\pi$ completes one full rotation, returning to the same value.
$\cos(\theta+2\pi)=\cos\theta$. Adding $2\pi$ completes one full rotation, returning to the same value.
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What periodicity identity shows $\sin(\theta+2\pi)$ in simplest form?
What periodicity identity shows $\sin(\theta+2\pi)$ in simplest form?
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$\sin(\theta+2\pi)=\sin\theta$. Adding $2\pi$ completes one full rotation, returning to the same value.
$\sin(\theta+2\pi)=\sin\theta$. Adding $2\pi$ completes one full rotation, returning to the same value.
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What is the period of $\tan\theta$?
What is the period of $\tan\theta$?
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$\pi$. Tangent repeats every $\pi$ radians, half the period of sine/cosine.
$\pi$. Tangent repeats every $\pi$ radians, half the period of sine/cosine.
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What is the period of $\cos\theta$?
What is the period of $\cos\theta$?
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$2\pi$. Cosine completes one full cycle every $2\pi$ radians.
$2\pi$. Cosine completes one full cycle every $2\pi$ radians.
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What is the period of $\sin\theta$?
What is the period of $\sin\theta$?
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$2\pi$. Sine completes one full cycle every $2\pi$ radians.
$2\pi$. Sine completes one full cycle every $2\pi$ radians.
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What symmetry identity relates $\tan(-\theta)$ to $\tan\theta$?
What symmetry identity relates $\tan(-\theta)$ to $\tan\theta$?
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$\tan(-\theta)=-\tan\theta$. Tangent inherits odd symmetry from $\frac{\sin\theta}{\cos\theta}$.
$\tan(-\theta)=-\tan\theta$. Tangent inherits odd symmetry from $\frac{\sin\theta}{\cos\theta}$.
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What symmetry identity relates $\sin(-\theta)$ to $\sin\theta$?
What symmetry identity relates $\sin(-\theta)$ to $\sin\theta$?
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$\sin(-\theta)=-\sin\theta$. Sine is odd, so negative angles give opposite values.
$\sin(-\theta)=-\sin\theta$. Sine is odd, so negative angles give opposite values.
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What symmetry identity relates $\cos(-\theta)$ to $\cos\theta$?
What symmetry identity relates $\cos(-\theta)$ to $\cos\theta$?
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$\cos(-\theta)=\cos\theta$. Cosine is even, so negative angles give the same value.
$\cos(-\theta)=\cos\theta$. Cosine is even, so negative angles give the same value.
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What is the even-odd symmetry identity for cosine on the unit circle?
What is the even-odd symmetry identity for cosine on the unit circle?
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$\cos(-\theta)=\cos\theta$. Cosine is even: reflecting across the y-axis doesn't change the x-coordinate.
$\cos(-\theta)=\cos\theta$. Cosine is even: reflecting across the y-axis doesn't change the x-coordinate.
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What is the fundamental period of $\sin\theta$ on the unit circle?
What is the fundamental period of $\sin\theta$ on the unit circle?
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Period $=2\pi$. One full rotation around the unit circle returns to the same point.
Period $=2\pi$. One full rotation around the unit circle returns to the same point.
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What is the fundamental period of $\cos\theta$ on the unit circle?
What is the fundamental period of $\cos\theta$ on the unit circle?
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Period $=2\pi$. Cosine repeats every full rotation ($2\pi$ radians) around the circle.
Period $=2\pi$. Cosine repeats every full rotation ($2\pi$ radians) around the circle.
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What is the fundamental period of $\tan\theta$ on the unit circle?
What is the fundamental period of $\tan\theta$ on the unit circle?
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Period $=\pi$. Tangent repeats every half rotation since $\tan(\theta+\pi)=\tan\theta$.
Period $=\pi$. Tangent repeats every half rotation since $\tan(\theta+\pi)=\tan\theta$.
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What is the periodicity identity for sine for any integer $k$?
What is the periodicity identity for sine for any integer $k$?
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$\sin(\theta+2\pi k)=\sin\theta$. Adding any multiple of $2\pi$ completes full rotations, returning to the same point.
$\sin(\theta+2\pi k)=\sin\theta$. Adding any multiple of $2\pi$ completes full rotations, returning to the same point.
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What is the unit-circle symmetry identity for cosine with a $\pi$ shift?
What is the unit-circle symmetry identity for cosine with a $\pi$ shift?
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$\cos(\theta+\pi)=-\cos\theta$. A $\pi$ rotation negates both coordinates, changing the sign of cosine.
$\cos(\theta+\pi)=-\cos\theta$. A $\pi$ rotation negates both coordinates, changing the sign of cosine.
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What is the unit-circle symmetry identity for tangent with a $\pi$ shift?
What is the unit-circle symmetry identity for tangent with a $\pi$ shift?
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$\tan(\theta+\pi)=\tan\theta$. Since both sine and cosine flip signs with $\pi$, their ratio stays the same.
$\tan(\theta+\pi)=\tan\theta$. Since both sine and cosine flip signs with $\pi$, their ratio stays the same.
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