Solving Trigonometric Equations in Context - Pre-Calculus
Card 1 of 30
Find all solutions on $[0,2\pi)$: $\cos(x)=-\frac{\sqrt{2}}{2}$.
Find all solutions on $[0,2\pi)$: $\cos(x)=-\frac{\sqrt{2}}{2}$.
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$x=\frac{3\pi}{4},\frac{5\pi}{4}$. Negative cosine occurs in quadrants II and III.
$x=\frac{3\pi}{4},\frac{5\pi}{4}$. Negative cosine occurs in quadrants II and III.
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What is the general solution to $\tan(x)=\tan(\theta)$?
What is the general solution to $\tan(x)=\tan(\theta)$?
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$x=\theta+\pi k$. Tangent repeats every $\pi$ radians.
$x=\theta+\pi k$. Tangent repeats every $\pi$ radians.
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What is the general solution to $\cos(x)=\cos(\theta)$?
What is the general solution to $\cos(x)=\cos(\theta)$?
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$x=\theta+2\pi k$ or $x=-\theta+2\pi k$. Cosine repeats every $2\pi$ with even function symmetry.
$x=\theta+2\pi k$ or $x=-\theta+2\pi k$. Cosine repeats every $2\pi$ with even function symmetry.
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What is the principal range of $\arcsin(x)$?
What is the principal range of $\arcsin(x)$?
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$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. Arcsin outputs angles from $-90°$ to $90°$ (quadrants I and IV).
$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. Arcsin outputs angles from $-90°$ to $90°$ (quadrants I and IV).
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What is the general solution to $\sin(x)=\sin(\theta)$?
What is the general solution to $\sin(x)=\sin(\theta)$?
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$x=\theta+2\pi k$ or $x=\pi-\theta+2\pi k$. Sine repeats every $2\pi$ with supplementary angle symmetry.
$x=\theta+2\pi k$ or $x=\pi-\theta+2\pi k$. Sine repeats every $2\pi$ with supplementary angle symmetry.
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What is the amplitude of $y=A\sin(Bx+C)+D$ or $y=A\cos(Bx+C)+D$?
What is the amplitude of $y=A\sin(Bx+C)+D$ or $y=A\cos(Bx+C)+D$?
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$|A|$. Amplitude is the absolute value of the coefficient $A$.
$|A|$. Amplitude is the absolute value of the coefficient $A$.
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What is the period of $y=A\sin(Bx+C)+D$ or $y=A\cos(Bx+C)+D$?
What is the period of $y=A\sin(Bx+C)+D$ or $y=A\cos(Bx+C)+D$?
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$\frac{2\pi}{|B|}$. Period formula divides $2\pi$ by the absolute value of $B$.
$\frac{2\pi}{|B|}$. Period formula divides $2\pi$ by the absolute value of $B$.
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What is the period of $y=A\tan(Bx+C)+D$?
What is the period of $y=A\tan(Bx+C)+D$?
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$\frac{\pi}{|B|}$. Tangent's period is half that of sine/cosine with same $B$.
$\frac{\pi}{|B|}$. Tangent's period is half that of sine/cosine with same $B$.
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Identify the inverse-trig step: Solve $\sin(x)=0.5$ for the principal value of $x$.
Identify the inverse-trig step: Solve $\sin(x)=0.5$ for the principal value of $x$.
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$x=\arcsin(0.5)=\frac{\pi}{6}$. Apply arcsin to isolate $x$ in the principal range.
$x=\arcsin(0.5)=\frac{\pi}{6}$. Apply arcsin to isolate $x$ in the principal range.
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Identify the inverse-trig step: Solve $\cos(x)=\frac{1}{2}$ for the principal value of $x$.
Identify the inverse-trig step: Solve $\cos(x)=\frac{1}{2}$ for the principal value of $x$.
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$x=\arccos\left(\frac{1}{2}\right)=\frac{\pi}{3}$. Apply arccos to find the principal angle.
$x=\arccos\left(\frac{1}{2}\right)=\frac{\pi}{3}$. Apply arccos to find the principal angle.
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Identify the inverse-trig step: Solve $\tan(x)=1$ for the principal value of $x$.
Identify the inverse-trig step: Solve $\tan(x)=1$ for the principal value of $x$.
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$x=\arctan(1)=\frac{\pi}{4}$. Apply arctan to find the principal angle.
$x=\arctan(1)=\frac{\pi}{4}$. Apply arctan to find the principal angle.
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Find all solutions on $[0,2\pi)$: $\sin(x)=\frac{1}{2}$.
Find all solutions on $[0,2\pi)$: $\sin(x)=\frac{1}{2}$.
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$x=\frac{\pi}{6},\frac{5\pi}{6}$. Reference angle $\frac{\pi}{6}$ occurs in quadrants I and II.
$x=\frac{\pi}{6},\frac{5\pi}{6}$. Reference angle $\frac{\pi}{6}$ occurs in quadrants I and II.
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Find all solutions on $[0,2\pi)$: $\tan(x)=-\sqrt{3}$.
Find all solutions on $[0,2\pi)$: $\tan(x)=-\sqrt{3}$.
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$x=\frac{2\pi}{3},\frac{5\pi}{3}$. Negative tangent occurs in quadrants II and IV.
$x=\frac{2\pi}{3},\frac{5\pi}{3}$. Negative tangent occurs in quadrants II and IV.
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Solve on $[0,2\pi)$: $2\sin(x)-1=0$.
Solve on $[0,2\pi)$: $2\sin(x)-1=0$.
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$x=\frac{\pi}{6},\frac{5\pi}{6}$. First isolate $\sin(x)=\frac{1}{2}$, then find all angles.
$x=\frac{\pi}{6},\frac{5\pi}{6}$. First isolate $\sin(x)=\frac{1}{2}$, then find all angles.
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Interpret feasibility: If a model gives $\sin(\theta)=1.2$, what conclusion is correct?
Interpret feasibility: If a model gives $\sin(\theta)=1.2$, what conclusion is correct?
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No real solution because $1.2\notin\left[-1,1\right]$. Sine values must be in $[-1,1]$ for real solutions.
No real solution because $1.2\notin\left[-1,1\right]$. Sine values must be in $[-1,1]$ for real solutions.
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What is the domain of $\arctan(x)$?
What is the domain of $\arctan(x)$?
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$x\in(-\infty,\infty)$. Tangent can take any real value as input.
$x\in(-\infty,\infty)$. Tangent can take any real value as input.
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What is the principal range of $\arctan(x)$?
What is the principal range of $\arctan(x)$?
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$\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Arctan outputs angles strictly between $-90°$ and $90°$.
$\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Arctan outputs angles strictly between $-90°$ and $90°$.
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What is the principal range of $\arccos(x)$?
What is the principal range of $\arccos(x)$?
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$\left[0,\pi\right]$. Arccos outputs angles from $0°$ to $180°$ (quadrants I and II).
$\left[0,\pi\right]$. Arccos outputs angles from $0°$ to $180°$ (quadrants I and II).
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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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$x\in\left[-1,1\right]$. Sine and cosine values range from $-1$ to $1$.
$x\in\left[-1,1\right]$. Sine and cosine values range from $-1$ to $1$.
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What is the vertical-range check for real solutions in $A\cos(\cdot)+D=y$?
What is the vertical-range check for real solutions in $A\cos(\cdot)+D=y$?
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$D-|A|\le y\le D+|A|$. Output range is midline $\pm$ amplitude.
$D-|A|\le y\le D+|A|$. Output range is midline $\pm$ amplitude.
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What domain restriction must be checked before using $\arcsin$ or $\arccos$ in an equation?
What domain restriction must be checked before using $\arcsin$ or $\arccos$ in an equation?
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$-1\le x\le 1$. Sine and cosine only output values between $-1$ and $1$.
$-1\le x\le 1$. Sine and cosine only output values between $-1$ and $1$.
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What general solution form solves $\sin(\theta)=\sin(\alpha)$?
What general solution form solves $\sin(\theta)=\sin(\alpha)$?
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$\theta=\alpha+2\pi k$ or $\theta=\pi-\alpha+2\pi k$. Sine has same value at supplementary angles, plus periodic repeats.
$\theta=\alpha+2\pi k$ or $\theta=\pi-\alpha+2\pi k$. Sine has same value at supplementary angles, plus periodic repeats.
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What general solution form solves $\tan(\theta)=\tan(\alpha)$?
What general solution form solves $\tan(\theta)=\tan(\alpha)$?
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$\theta=\alpha+\pi k$. Tangent repeats every $\pi$ radians (half the period of sine/cosine).
$\theta=\alpha+\pi k$. Tangent repeats every $\pi$ radians (half the period of sine/cosine).
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What contextual filtering is required after solving a trig model for time $t$?
What contextual filtering is required after solving a trig model for time $t$?
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Keep only $t$ values in the stated interval and with correct units. Discard solutions outside domain or with wrong units for the context.
Keep only $t$ values in the stated interval and with correct units. Discard solutions outside domain or with wrong units for the context.
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Identify the correct solutions in $[0,2\pi)$ if $\arcsin(0.8)=\alpha$ is used for $\sin(\theta)=0.8$.
Identify the correct solutions in $[0,2\pi)$ if $\arcsin(0.8)=\alpha$ is used for $\sin(\theta)=0.8$.
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$\theta=\alpha$ and $\theta=\pi-\alpha$. Arcsin gives acute angle; also need supplementary angle $\pi-\alpha$.
$\theta=\alpha$ and $\theta=\pi-\alpha$. Arcsin gives acute angle; also need supplementary angle $\pi-\alpha$.
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What are all solutions in $[0,2\pi)$ to $\tan(\theta)=\sqrt{3}$?
What are all solutions in $[0,2\pi)$ to $\tan(\theta)=\sqrt{3}$?
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$\theta=\frac{\pi}{3},\frac{4\pi}{3}$. Reference angle $\frac{\pi}{3}$; tangent positive in quadrants I and III.
$\theta=\frac{\pi}{3},\frac{4\pi}{3}$. Reference angle $\frac{\pi}{3}$; tangent positive in quadrants I and III.
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What are all solutions in $[0,2\pi)$ to $\cos(\theta)=-\frac{1}{2}$?
What are all solutions in $[0,2\pi)$ to $\cos(\theta)=-\frac{1}{2}$?
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$\theta=\frac{2\pi}{3},\frac{4\pi}{3}$. Reference angle $\frac{\pi}{3}$; cosine negative in quadrants II and III.
$\theta=\frac{2\pi}{3},\frac{4\pi}{3}$. Reference angle $\frac{\pi}{3}$; cosine negative in quadrants II and III.
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What are all solutions in $[0,2\pi)$ to $\sin(\theta)=\frac{1}{2}$?
What are all solutions in $[0,2\pi)$ to $\sin(\theta)=\frac{1}{2}$?
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$\theta=\frac{\pi}{6},\frac{5\pi}{6}$. Reference angle $\frac{\pi}{6}$; sine positive in quadrants I and II.
$\theta=\frac{\pi}{6},\frac{5\pi}{6}$. Reference angle $\frac{\pi}{6}$; sine positive in quadrants I and II.
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What is the vertical-range check for real solutions in $A\sin(\cdot)+D=y$?
What is the vertical-range check for real solutions in $A\sin(\cdot)+D=y$?
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$D-|A|\le y\le D+|A|$. Output range is midline $\pm$ amplitude.
$D-|A|\le y\le D+|A|$. Output range is midline $\pm$ amplitude.
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Identify the inverse-trig step to isolate $x$ in $A\cos(Bx+C)+D=y$.
Identify the inverse-trig step to isolate $x$ in $A\cos(Bx+C)+D=y$.
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$x=\frac{\arccos\left(\frac{y-D}{A}\right)-C}{B}$ (principal value). Subtract $D$, divide by $A$, apply arccos, then solve for $x$.
$x=\frac{\arccos\left(\frac{y-D}{A}\right)-C}{B}$ (principal value). Subtract $D$, divide by $A$, apply arccos, then solve for $x$.
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