Extending Trigonometric Functions with Unit Circle - Pre-Calculus
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What is the radian measure of a half revolution around the unit circle?
What is the radian measure of a half revolution around the unit circle?
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$\pi$. Half of the circumference $2\pi$ equals $\pi$.
$\pi$. Half of the circumference $2\pi$ equals $\pi$.
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What is the radian measure of one full revolution around the unit circle?
What is the radian measure of one full revolution around the unit circle?
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$2\pi$. Circumference of unit circle is $2\pi r = 2\pi(1) = 2\pi$.
$2\pi$. Circumference of unit circle is $2\pi r = 2\pi(1) = 2\pi$.
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Which direction corresponds to negative angles on the unit circle?
Which direction corresponds to negative angles on the unit circle?
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Clockwise. Negative angles rotate in the opposite direction.
Clockwise. Negative angles rotate in the opposite direction.
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Identify the exact value of $\cos\left(-\frac{\pi}{3}\right)$ using unit-circle symmetry.
Identify the exact value of $\cos\left(-\frac{\pi}{3}\right)$ using unit-circle symmetry.
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$\frac{1}{2}$. By even property: $\cos(-\frac{\pi}{3})=\cos(\frac{\pi}{3})=\frac{1}{2}$.
$\frac{1}{2}$. By even property: $\cos(-\frac{\pi}{3})=\cos(\frac{\pi}{3})=\frac{1}{2}$.
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Identify the exact value of $\cos(\pi)$ using the unit circle.
Identify the exact value of $\cos(\pi)$ using the unit circle.
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$-1$. At $\pi$ radians, the point is $(-1,0)$, so $x=-1$.
$-1$. At $\pi$ radians, the point is $(-1,0)$, so $x=-1$.
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What is the symmetry identity for sine that follows from the unit circle?
What is the symmetry identity for sine that follows from the unit circle?
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$\sin(-t)=-\sin t$. Sine is odd: opposite $y$-coordinates for $\pm t$.
$\sin(-t)=-\sin t$. Sine is odd: opposite $y$-coordinates for $\pm t$.
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What is the symmetry identity for cosine that follows from the unit circle?
What is the symmetry identity for cosine that follows from the unit circle?
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$\cos(-t)=\cos t$. Cosine is even: same $x$-coordinate for $\pm t$.
$\cos(-t)=\cos t$. Cosine is even: same $x$-coordinate for $\pm t$.
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What is the period identity that extends sine to all real numbers $t$?
What is the period identity that extends sine to all real numbers $t$?
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$\sin(t+2\pi)=\sin t$. Full rotation returns to same point, so sine repeats.
$\sin(t+2\pi)=\sin t$. Full rotation returns to same point, so sine repeats.
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What is the period identity that extends cosine to all real numbers $t$?
What is the period identity that extends cosine to all real numbers $t$?
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$\cos(t+2\pi)=\cos t$. Full rotation returns to same point, so cosine repeats.
$\cos(t+2\pi)=\cos t$. Full rotation returns to same point, so cosine repeats.
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What is the radian measure of a quarter revolution around the unit circle?
What is the radian measure of a quarter revolution around the unit circle?
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$\frac{\pi}{2}$. One-fourth of the circumference $2\pi$ equals $\frac{\pi}{2}$.
$\frac{\pi}{2}$. One-fourth of the circumference $2\pi$ equals $\frac{\pi}{2}$.
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Which direction corresponds to positive angles on the unit circle?
Which direction corresponds to positive angles on the unit circle?
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Counterclockwise. Standard convention: positive angles rotate counterclockwise.
Counterclockwise. Standard convention: positive angles rotate counterclockwise.
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What does it mean to interpret $t$ as a radian measure on the unit circle?
What does it mean to interpret $t$ as a radian measure on the unit circle?
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$t$ is the signed arc length traveled on the unit circle. Radians equal arc length on unit circle (radius = 1).
$t$ is the signed arc length traveled on the unit circle. Radians equal arc length on unit circle (radius = 1).
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What is the unit-circle definition of $\tan t$ when $\cos t \ne 0$?
What is the unit-circle definition of $\tan t$ when $\cos t \ne 0$?
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$\tan t = \frac{\sin t}{\cos t}$. Ratio of $y$-coordinate to $x$-coordinate on the unit circle.
$\tan t = \frac{\sin t}{\cos t}$. Ratio of $y$-coordinate to $x$-coordinate on the unit circle.
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What are the coordinates on the unit circle at $t=\frac{\pi}{2}$?
What are the coordinates on the unit circle at $t=\frac{\pi}{2}$?
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$(0,1)$. Quarter turn counterclockwise from $(1,0)$ reaches top of circle.
$(0,1)$. Quarter turn counterclockwise from $(1,0)$ reaches top of circle.
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What point on the unit circle corresponds to an angle (radian measure) of $t$?
What point on the unit circle corresponds to an angle (radian measure) of $t$?
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$(\cos t, \sin t)$. Point reached by traveling $t$ radians counterclockwise from $(1,0)$.
$(\cos t, \sin t)$. Point reached by traveling $t$ radians counterclockwise from $(1,0)$.
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What is the definition of the unit circle in the coordinate plane?
What is the definition of the unit circle in the coordinate plane?
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Circle centered at $(0,0)$ with radius $1$. Standard form: center at origin with all points distance 1 away.
Circle centered at $(0,0)$ with radius $1$. Standard form: center at origin with all points distance 1 away.
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Identify the exact value of $\tan\left(\frac{\pi}{4}\right)$ using the unit circle.
Identify the exact value of $\tan\left(\frac{\pi}{4}\right)$ using the unit circle.
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$1$. At $\frac{\pi}{4}$, point is $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$, so $\tan=\frac{y}{x}=1$.
$1$. At $\frac{\pi}{4}$, point is $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$, so $\tan=\frac{y}{x}=1$.
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Identify the exact value of $\sin\left(\frac{3\pi}{2}\right)$ using the unit circle.
Identify the exact value of $\sin\left(\frac{3\pi}{2}\right)$ using the unit circle.
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$-1$. At $\frac{3\pi}{2}$, the point is $(0,-1)$, so $y=-1$.
$-1$. At $\frac{3\pi}{2}$, the point is $(0,-1)$, so $y=-1$.
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At which angles is $\tan\theta$ undefined on the unit circle?
At which angles is $\tan\theta$ undefined on the unit circle?
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$\theta=\frac{\pi}{2}+k\pi$, for integers $k$. Where $\cos\theta = 0$, making division undefined.
$\theta=\frac{\pi}{2}+k\pi$, for integers $k$. Where $\cos\theta = 0$, making division undefined.
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What is the unit circle in the coordinate plane?
What is the unit circle in the coordinate plane?
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The circle centered at $(0,0)$ with radius $1$. Defined by equation $x^2 + y^2 = 1$ in the coordinate plane.
The circle centered at $(0,0)$ with radius $1$. Defined by equation $x^2 + y^2 = 1$ in the coordinate plane.
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What point on the unit circle corresponds to angle $0$ radians?
What point on the unit circle corresponds to angle $0$ radians?
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$(1,0)$. Starting point where $\cos(0) = 1$ and $\sin(0) = 0$.
$(1,0)$. Starting point where $\cos(0) = 1$ and $\sin(0) = 0$.
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What is the direction of positive angle measure on the unit circle?
What is the direction of positive angle measure on the unit circle?
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Counterclockwise from the positive $x$-axis. Standard convention for measuring positive angles.
Counterclockwise from the positive $x$-axis. Standard convention for measuring positive angles.
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What is the relationship between arc length $s$ and angle $\theta$ on the unit circle?
What is the relationship between arc length $s$ and angle $\theta$ on the unit circle?
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$s=\theta$. On unit circle, arc length equals angle in radians.
$s=\theta$. On unit circle, arc length equals angle in radians.
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What ordered pair on the unit circle corresponds to angle $\theta$?
What ordered pair on the unit circle corresponds to angle $\theta$?
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$(\cos\theta,\sin\theta)$. Point on unit circle at angle $\theta$ from positive $x$-axis.
$(\cos\theta,\sin\theta)$. Point on unit circle at angle $\theta$ from positive $x$-axis.
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What is the definition of $\cos\theta$ using the unit circle?
What is the definition of $\cos\theta$ using the unit circle?
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$\cos\theta$ is the $x$-coordinate on the unit circle. Horizontal distance from origin to point on unit circle.
$\cos\theta$ is the $x$-coordinate on the unit circle. Horizontal distance from origin to point on unit circle.
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What is the definition of $\sin\theta$ using the unit circle?
What is the definition of $\sin\theta$ using the unit circle?
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$\sin\theta$ is the $y$-coordinate on the unit circle. Vertical distance from origin to point on unit circle.
$\sin\theta$ is the $y$-coordinate on the unit circle. Vertical distance from origin to point on unit circle.
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What is the definition of $\tan\theta$ using unit-circle coordinates?
What is the definition of $\tan\theta$ using unit-circle coordinates?
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$\tan\theta=\frac{\sin\theta}{\cos\theta}$ when $\cos\theta\ne 0$. Ratio of $y$-coordinate to $x$-coordinate on unit circle.
$\tan\theta=\frac{\sin\theta}{\cos\theta}$ when $\cos\theta\ne 0$. Ratio of $y$-coordinate to $x$-coordinate on unit circle.
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Which trig functions are defined for every real number $\theta$ via the unit circle?
Which trig functions are defined for every real number $\theta$ via the unit circle?
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$\sin\theta$ and $\cos\theta$. Unit circle coordinates always exist for any angle.
$\sin\theta$ and $\cos\theta$. Unit circle coordinates always exist for any angle.
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Identify the period statement for sine and cosine using the unit circle.
Identify the period statement for sine and cosine using the unit circle.
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$\sin(\theta+2\pi)=\sin\theta$ and $\cos(\theta+2\pi)=\cos\theta$. Functions repeat after one full rotation of $2\pi$.
$\sin(\theta+2\pi)=\sin\theta$ and $\cos(\theta+2\pi)=\cos\theta$. Functions repeat after one full rotation of $2\pi$.
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What is the reference angle for $\theta=\frac{7\pi}{6}$?
What is the reference angle for $\theta=\frac{7\pi}{6}$?
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$\frac{\pi}{6}$. Reference angle is acute angle to nearest $x$-axis.
$\frac{\pi}{6}$. Reference angle is acute angle to nearest $x$-axis.
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