Solving Trigonometric Equations in Context - Geometry
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What is the principal range of $\arccos(x)$ in radians?
What is the principal range of $\arccos(x)$ in radians?
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$[0,\pi]$. Principal range where $\arccos$ returns unique values.
$[0,\pi]$. Principal range where $\arccos$ returns unique values.
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What is the principal range of $\arcsin(x)$ in radians?
What is the principal range of $\arcsin(x)$ in radians?
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$[-\frac{\pi}{2},\frac{\pi}{2}]$. Principal range where $\arcsin$ returns unique values.
$[-\frac{\pi}{2},\frac{\pi}{2}]$. Principal range where $\arcsin$ returns unique values.
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Solve $\sin(2x)=0$ for $x$ in $[0,\pi)$.
Solve $\sin(2x)=0$ for $x$ in $[0,\pi)$.
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$x=0,\frac{\pi}{2}$. Set $2x = 0, \pi$ and solve for $x$.
$x=0,\frac{\pi}{2}$. Set $2x = 0, \pi$ and solve for $x$.
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Solve for $x$ in $\tan(x)=-1$ on $[0,2\pi)$.
Solve for $x$ in $\tan(x)=-1$ on $[0,2\pi)$.
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$x=\frac{3\pi}{4},\frac{7\pi}{4}$. Negative tangent in Quadrants II and IV.
$x=\frac{3\pi}{4},\frac{7\pi}{4}$. Negative tangent in Quadrants II and IV.
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Identify the valid model output if an angle of elevation must satisfy $0\le\theta\le\frac{\pi}{2}$.
Identify the valid model output if an angle of elevation must satisfy $0\le\theta\le\frac{\pi}{2}$.
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Keep only solutions with $0\le\theta\le\frac{\pi}{2}$. Elevation angles must be between 0 and 90 degrees.
Keep only solutions with $0\le\theta\le\frac{\pi}{2}$. Elevation angles must be between 0 and 90 degrees.
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What identity connects $\cos(\arccos(x))$ for $x\in[-1,1]$?
What identity connects $\cos(\arccos(x))$ for $x\in[-1,1]$?
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$\cos(\arccos(x))=x$. Inverse cancels the function when input is in domain.
$\cos(\arccos(x))=x$. Inverse cancels the function when input is in domain.
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What identity connects $\tan(\arctan(x))$ for all real $x$?
What identity connects $\tan(\arctan(x))$ for all real $x$?
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$\tan(\arctan(x))=x$. Inverse cancels the function for all real inputs.
$\tan(\arctan(x))=x$. Inverse cancels the function for all real inputs.
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What is the range of $\arccos(x)$?
What is the range of $\arccos(x)$?
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$[0,\pi]$. Output angles from 0 to $\pi$ inclusive.
$[0,\pi]$. Output angles from 0 to $\pi$ inclusive.
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What identity connects $\tan(\arctan(x))$ for all real $x$?
What identity connects $\tan(\arctan(x))$ for all real $x$?
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$\tan(\arctan(x))=x$. Inverse cancels the function for all real inputs.
$\tan(\arctan(x))=x$. Inverse cancels the function for all real inputs.
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Identify the correct degree-to-radian conversion formula for an angle $\theta$.
Identify the correct degree-to-radian conversion formula for an angle $\theta$.
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$\theta_{rad}=\theta_{deg}\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ to get radians.
$\theta_{rad}=\theta_{deg}\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ to get radians.
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What is the domain of $\arcsin(x)$ and $\arccos(x)$?
What is the domain of $\arcsin(x)$ and $\arccos(x)$?
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$[-1,1]$. Both functions require input between -1 and 1 inclusive.
$[-1,1]$. Both functions require input between -1 and 1 inclusive.
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What is the principal range of $\arctan(x)$ in radians?
What is the principal range of $\arctan(x)$ in radians?
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$(-\frac{\pi}{2},\frac{\pi}{2})$. Principal range where $\arctan$ returns unique values.
$(-\frac{\pi}{2},\frac{\pi}{2})$. Principal range where $\arctan$ returns unique values.
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Find all solutions in $[0,2\pi)$ for $\cos(\theta)=0$.
Find all solutions in $[0,2\pi)$ for $\cos(\theta)=0$.
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$\theta=\frac{\pi}{2},\frac{3\pi}{2}$. Cosine equals zero at odd multiples of $\frac{\pi}{2}$.
$\theta=\frac{\pi}{2},\frac{3\pi}{2}$. Cosine equals zero at odd multiples of $\frac{\pi}{2}$.
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What is the midline of $y=A\cos(Bx+C)+D$?
What is the midline of $y=A\cos(Bx+C)+D$?
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$y=D$. Vertical shift $D$ moves the center line up or down.
$y=D$. Vertical shift $D$ moves the center line up or down.
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What condition must hold for $\arcsin(u)$ or $\arccos(u)$ to be real?
What condition must hold for $\arcsin(u)$ or $\arccos(u)$ to be real?
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$-1\le u\le 1$. Both inverse functions require inputs in $[-1,1]$.
$-1\le u\le 1$. Both inverse functions require inputs in $[-1,1]$.
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Identify the correct degree-to-radian conversion formula for an angle $\theta$.
Identify the correct degree-to-radian conversion formula for an angle $\theta$.
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$\theta_{rad}=\theta_{deg}\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ to get radians.
$\theta_{rad}=\theta_{deg}\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ to get radians.
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Identify the correct radian-to-degree conversion formula for an angle $\theta$.
Identify the correct radian-to-degree conversion formula for an angle $\theta$.
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$\theta_{deg}=\theta_{rad}\cdot\frac{180}{\pi}$. Multiply radians by $\frac{180}{\pi}$ to get degrees.
$\theta_{deg}=\theta_{rad}\cdot\frac{180}{\pi}$. Multiply radians by $\frac{180}{\pi}$ to get degrees.
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What is the inverse-trig step to solve $\sin(\theta)=a$ when $a\in[-1,1]$?
What is the inverse-trig step to solve $\sin(\theta)=a$ when $a\in[-1,1]$?
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Start with $\theta=\arcsin(a)$ as the principal angle. Find the reference angle first, then all solutions.
Start with $\theta=\arcsin(a)$ as the principal angle. Find the reference angle first, then all solutions.
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What is the inverse-trig step to solve $\cos(\theta)=a$ when $a\in[-1,1]$?
What is the inverse-trig step to solve $\cos(\theta)=a$ when $a\in[-1,1]$?
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Start with $\theta=\arccos(a)$ as the principal angle. Find the reference angle first, then all solutions.
Start with $\theta=\arccos(a)$ as the principal angle. Find the reference angle first, then all solutions.
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What is the inverse-trig step to solve $\tan(\theta)=a$ for real $a$?
What is the inverse-trig step to solve $\tan(\theta)=a$ for real $a$?
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Start with $\theta=\arctan(a)$ as the principal angle. Find the reference angle first, then all solutions.
Start with $\theta=\arctan(a)$ as the principal angle. Find the reference angle first, then all solutions.
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Find the principal solution for $\sin(\theta)=\frac{1}{2}$ in radians.
Find the principal solution for $\sin(\theta)=\frac{1}{2}$ in radians.
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$\theta=\frac{\pi}{6}$. Since $\arcsin(\frac{1}{2}) = \frac{\pi}{6}$ in Quadrant I.
$\theta=\frac{\pi}{6}$. Since $\arcsin(\frac{1}{2}) = \frac{\pi}{6}$ in Quadrant I.
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Find the second solution in $[0,2\pi)$ for $\sin(\theta)=\frac{1}{2}$.
Find the second solution in $[0,2\pi)$ for $\sin(\theta)=\frac{1}{2}$.
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$\theta=\frac{5\pi}{6}$. Since $\sin(\theta) = \frac{1}{2}$ in Quadrant II at $\pi - \frac{\pi}{6}$.
$\theta=\frac{5\pi}{6}$. Since $\sin(\theta) = \frac{1}{2}$ in Quadrant II at $\pi - \frac{\pi}{6}$.
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Solve for $x$ in $\cos(x)=0.2$ on $[0,2\pi)$ using inverse trig notation.
Solve for $x$ in $\cos(x)=0.2$ on $[0,2\pi)$ using inverse trig notation.
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$x=\arccos(0.2)$ or $x=2\pi-\arccos(0.2)$. General form using inverse trig for principal values.
$x=\arccos(0.2)$ or $x=2\pi-\arccos(0.2)$. General form using inverse trig for principal values.
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Find the principal solution for $\cos(\theta)=\frac{1}{2}$ in radians.
Find the principal solution for $\cos(\theta)=\frac{1}{2}$ in radians.
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$\theta=\frac{\pi}{3}$. Since $\arccos(\frac{1}{2}) = \frac{\pi}{3}$ in Quadrant I.
$\theta=\frac{\pi}{3}$. Since $\arccos(\frac{1}{2}) = \frac{\pi}{3}$ in Quadrant I.
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Solve for $x$ in $\sin(x)=0.5$ on $[0,2\pi)$ using inverse trig notation.
Solve for $x$ in $\sin(x)=0.5$ on $[0,2\pi)$ using inverse trig notation.
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$x=\arcsin(0.5)$ or $x=\pi-\arcsin(0.5)$. General form using inverse trig for principal values.
$x=\arcsin(0.5)$ or $x=\pi-\arcsin(0.5)$. General form using inverse trig for principal values.
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Solve for $x$ in $3\cos(x)+3=0$ on $[0,2\pi)$.
Solve for $x$ in $3\cos(x)+3=0$ on $[0,2\pi)$.
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$x=\pi$. Solve $\cos(x) = -1$ by isolating the cosine term.
$x=\pi$. Solve $\cos(x) = -1$ by isolating the cosine term.
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Solve for $x$ in $2\sin(x)-1=0$ on $[0,2\pi)$.
Solve for $x$ in $2\sin(x)-1=0$ on $[0,2\pi)$.
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$x=\frac{\pi}{6},\frac{5\pi}{6}$. Solve $\sin(x) = \frac{1}{2}$ by isolating the sine term.
$x=\frac{\pi}{6},\frac{5\pi}{6}$. Solve $\sin(x) = \frac{1}{2}$ by isolating the sine term.
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Solve for $x$ in $\tan(x)=1$ on $[0,\pi]$.
Solve for $x$ in $\tan(x)=1$ on $[0,\pi]$.
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$x=\frac{\pi}{4}$. Only one solution of $\tan(x) = 1$ lies in $[0,\pi]$.
$x=\frac{\pi}{4}$. Only one solution of $\tan(x) = 1$ lies in $[0,\pi]$.
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Identify the valid contextual time if a model gives $t=-2$ seconds and $t=3$ seconds.
Identify the valid contextual time if a model gives $t=-2$ seconds and $t=3$ seconds.
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$t=3$ seconds. Physical time cannot be negative in most contexts.
$t=3$ seconds. Physical time cannot be negative in most contexts.
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Solve for $t$ in $\cos(\pi t)=0$ on $[0,2]$.
Solve for $t$ in $\cos(\pi t)=0$ on $[0,2]$.
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$t=\frac{1}{2},\frac{3}{2}$. Set $\pi t = \frac{\pi}{2}, \frac{3\pi}{2}$ and solve for $t$.
$t=\frac{1}{2},\frac{3}{2}$. Set $\pi t = \frac{\pi}{2}, \frac{3\pi}{2}$ and solve for $t$.
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