Understanding Radian Measure of Angles - Pre-Calculus
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What is the radius of the unit circle used to define radian measure?
What is the radius of the unit circle used to define radian measure?
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$r=1$. Unit circle has radius 1 by definition.
$r=1$. Unit circle has radius 1 by definition.
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What is the sign of the arc length $s$ on the unit circle for a clockwise angle $\theta$?
What is the sign of the arc length $s$ on the unit circle for a clockwise angle $\theta$?
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$s$ is negative because $\theta<0$ implies $s=\theta<0$. Clockwise angles are negative, so arc length is negative.
$s$ is negative because $\theta<0$ implies $s=\theta<0$. Clockwise angles are negative, so arc length is negative.
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What is the definition of radian measure on the unit circle in terms of arc length?
What is the definition of radian measure on the unit circle in terms of arc length?
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$\theta=s$ on the unit circle, where $s$ is the subtended arc length. Arc length equals angle in radians when radius is 1.
$\theta=s$ on the unit circle, where $s$ is the subtended arc length. Arc length equals angle in radians when radius is 1.
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What is the general formula relating radian measure $\theta$, arc length $s$, and radius $r$?
What is the general formula relating radian measure $\theta$, arc length $s$, and radius $r$?
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$s=r\theta$. Arc length equals radius times angle in radians.
$s=r\theta$. Arc length equals radius times angle in radians.
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What is the formula for radian measure $\theta$ in terms of arc length $s$ and radius $r$?
What is the formula for radian measure $\theta$ in terms of arc length $s$ and radius $r$?
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$\theta=\frac{s}{r}$. Rearranged from $s=r\theta$ by dividing both sides by $r$.
$\theta=\frac{s}{r}$. Rearranged from $s=r\theta$ by dividing both sides by $r$.
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What is the radius of the unit circle used in defining radian measure?
What is the radius of the unit circle used in defining radian measure?
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$r=1$. Unit circle has radius 1 by definition.
$r=1$. Unit circle has radius 1 by definition.
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What is the radian measure of a full revolution around a circle?
What is the radian measure of a full revolution around a circle?
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$2\pi$. Full circle's circumference is $2\pi r$, so angle is $2\pi$ when $r=1$.
$2\pi$. Full circle's circumference is $2\pi r$, so angle is $2\pi$ when $r=1$.
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What is the radian measure of a straight angle (a half revolution)?
What is the radian measure of a straight angle (a half revolution)?
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$\pi$. Half of a full revolution ($2\pi$) equals $\pi$.
$\pi$. Half of a full revolution ($2\pi$) equals $\pi$.
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What is the radian measure of a right angle (a quarter revolution)?
What is the radian measure of a right angle (a quarter revolution)?
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$\frac{\pi}{2}$. Quarter of a full revolution ($2\pi$) equals $\frac{\pi}{2}$.
$\frac{\pi}{2}$. Quarter of a full revolution ($2\pi$) equals $\frac{\pi}{2}$.
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What is the radian measure of $60^\circ$?
What is the radian measure of $60^\circ$?
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$\frac{\pi}{3}$. Convert using $60^\circ \cdot \frac{\pi}{180} = \frac{\pi}{3}$.
$\frac{\pi}{3}$. Convert using $60^\circ \cdot \frac{\pi}{180} = \frac{\pi}{3}$.
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What is the radian measure of $45^\circ$?
What is the radian measure of $45^\circ$?
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$\frac{\pi}{4}$. Convert using $45^\circ \cdot \frac{\pi}{180} = \frac{\pi}{4}$.
$\frac{\pi}{4}$. Convert using $45^\circ \cdot \frac{\pi}{180} = \frac{\pi}{4}$.
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What is the radian measure of $30^\circ$?
What is the radian measure of $30^\circ$?
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$\frac{\pi}{6}$. Convert using $30^\circ \cdot \frac{\pi}{180} = \frac{\pi}{6}$.
$\frac{\pi}{6}$. Convert using $30^\circ \cdot \frac{\pi}{180} = \frac{\pi}{6}$.
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What is the degree measure of $\pi$ radians?
What is the degree measure of $\pi$ radians?
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$180^\circ$. Convert using $\pi \cdot \frac{180}{\pi} = 180^\circ$.
$180^\circ$. Convert using $\pi \cdot \frac{180}{\pi} = 180^\circ$.
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What is the degree measure of $\frac{\pi}{2}$ radians?
What is the degree measure of $\frac{\pi}{2}$ radians?
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$90^\circ$. Convert using $\frac{\pi}{2} \cdot \frac{180}{\pi} = 90^\circ$.
$90^\circ$. Convert using $\frac{\pi}{2} \cdot \frac{180}{\pi} = 90^\circ$.
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State the conversion formula from degrees $D$ to radians.
State the conversion formula from degrees $D$ to radians.
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$\text{radians}=D\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ to get radians.
$\text{radians}=D\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ to get radians.
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State the conversion formula from radians $\theta$ to degrees.
State the conversion formula from radians $\theta$ to degrees.
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$\text{degrees}=\theta\cdot\frac{180}{\pi}$. Multiply radians by $\frac{180}{\pi}$ to get degrees.
$\text{degrees}=\theta\cdot\frac{180}{\pi}$. Multiply radians by $\frac{180}{\pi}$ to get degrees.
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Find the arc length $s$ on a circle of radius $4$ subtended by $\theta=\frac{\pi}{3}$.
Find the arc length $s$ on a circle of radius $4$ subtended by $\theta=\frac{\pi}{3}$.
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$s=\frac{4\pi}{3}$. Use $s=r\theta=4\cdot\frac{\pi}{3}=\frac{4\pi}{3}$.
$s=\frac{4\pi}{3}$. Use $s=r\theta=4\cdot\frac{\pi}{3}=\frac{4\pi}{3}$.
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Find the arc length $s$ on the unit circle subtended by $\theta=\frac{5\pi}{6}$.
Find the arc length $s$ on the unit circle subtended by $\theta=\frac{5\pi}{6}$.
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$s=\frac{5\pi}{6}$. On unit circle, $s=\theta$, so $s=\frac{5\pi}{6}$.
$s=\frac{5\pi}{6}$. On unit circle, $s=\theta$, so $s=\frac{5\pi}{6}$.
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Find the radian measure $\theta$ if arc length is $s=10$ on a circle of radius $r=5$.
Find the radian measure $\theta$ if arc length is $s=10$ on a circle of radius $r=5$.
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$\theta=2$. Use $\theta=\frac{s}{r}=\frac{10}{5}=2$.
$\theta=2$. Use $\theta=\frac{s}{r}=\frac{10}{5}=2$.
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Identify the arc length on the unit circle subtended by an angle of $1$ radian.
Identify the arc length on the unit circle subtended by an angle of $1$ radian.
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$s=1$. On unit circle, arc length equals angle in radians.
$s=1$. On unit circle, arc length equals angle in radians.
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On the unit circle, what arc length corresponds to $\theta=2\pi$ radians?
On the unit circle, what arc length corresponds to $\theta=2\pi$ radians?
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$s=2\pi$. On unit circle, $s=\theta=2\pi$ (full circumference).
$s=2\pi$. On unit circle, $s=\theta=2\pi$ (full circumference).
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What is $\frac{3\pi}{2}$ expressed in degrees?
What is $\frac{3\pi}{2}$ expressed in degrees?
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$270^\circ$. $\frac{3\pi}{2} \times \frac{180}{\pi} = 270$ degrees.
$270^\circ$. $\frac{3\pi}{2} \times \frac{180}{\pi} = 270$ degrees.
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What is $45^\circ$ expressed in radians?
What is $45^\circ$ expressed in radians?
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$\frac{\pi}{4}$. $45^\circ \times \frac{\pi}{180} = \frac{\pi}{4}$ radians.
$\frac{\pi}{4}$. $45^\circ \times \frac{\pi}{180} = \frac{\pi}{4}$ radians.
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What is the radian measure of a right angle (a quarter rotation) on the unit circle?
What is the radian measure of a right angle (a quarter rotation) on the unit circle?
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$\frac{\pi}{2}$. Quarter of full rotation ($2\pi$) gives $\frac{\pi}{2}$ radians.
$\frac{\pi}{2}$. Quarter of full rotation ($2\pi$) gives $\frac{\pi}{2}$ radians.
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What is the degree measure equivalent to $1$ radian (to the nearest tenth of a degree)?
What is the degree measure equivalent to $1$ radian (to the nearest tenth of a degree)?
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$\approx 57.3^\circ$. $1$ radian = $\frac{180}{\pi} \approx 57.3$ degrees.
$\approx 57.3^\circ$. $1$ radian = $\frac{180}{\pi} \approx 57.3$ degrees.
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What is the conversion formula from degrees to radians for an angle $x^\circ$?
What is the conversion formula from degrees to radians for an angle $x^\circ$?
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$x^\circ\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ to convert to radians.
$x^\circ\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ to convert to radians.
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What is the angle $\theta$ (in radians) if $s=10$ and $r=5$ using $s=r\theta$?
What is the angle $\theta$ (in radians) if $s=10$ and $r=5$ using $s=r\theta$?
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$\theta=2$. $\theta = \frac{s}{r} = \frac{10}{5} = 2$ radians.
$\theta=2$. $\theta = \frac{s}{r} = \frac{10}{5} = 2$ radians.
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What is the radian measure of a full rotation around the unit circle?
What is the radian measure of a full rotation around the unit circle?
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$2\pi$. Full circle circumference is $2\pi r = 2\pi(1) = 2\pi$.
$2\pi$. Full circle circumference is $2\pi r = 2\pi(1) = 2\pi$.
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What is the radian measure of a central angle that subtends an arc length of $1$ on the unit circle?
What is the radian measure of a central angle that subtends an arc length of $1$ on the unit circle?
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$1\text{ rad}$. When $r=1$, angle in radians equals arc length.
$1\text{ rad}$. When $r=1$, angle in radians equals arc length.
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What is the radian measure of a straight angle (a half rotation) on the unit circle?
What is the radian measure of a straight angle (a half rotation) on the unit circle?
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$\pi$. Half of full rotation ($2\pi$) gives $\pi$ radians.
$\pi$. Half of full rotation ($2\pi$) gives $\pi$ radians.
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