Constructing Inverse Trigonometric Functions - Pre-Calculus
Card 1 of 30
What is the simplified value of $\arctan\left(\tan\left(\frac{3\pi}{4}\right)\right)$?
What is the simplified value of $\arctan\left(\tan\left(\frac{3\pi}{4}\right)\right)$?
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$-\frac{\pi}{4}$. $\tan(\frac{3\pi}{4})=-1$, and $\arctan(-1)=-\frac{\pi}{4}$ in range.
$-\frac{\pi}{4}$. $\tan(\frac{3\pi}{4})=-1$, and $\arctan(-1)=-\frac{\pi}{4}$ in range.
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What is the range of $\arccos(x)$?
What is the range of $\arccos(x)$?
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$\left[0,\pi\right]$. This matches the restricted domain of cosine used to define its inverse.
$\left[0,\pi\right]$. This matches the restricted domain of cosine used to define its inverse.
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What is the range of $\arctan(x)$?
What is the range of $\arctan(x)$?
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$\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. This matches the restricted domain of tangent used to define its inverse.
$\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. This matches the restricted domain of tangent used to define its inverse.
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What property must a trigonometric function have on a restricted domain to have an inverse?
What property must a trigonometric function have on a restricted domain to have an inverse?
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It must be one-to-one (strictly increasing or strictly decreasing). This ensures each output maps to exactly one input, making the function invertible.
It must be one-to-one (strictly increasing or strictly decreasing). This ensures each output maps to exactly one input, making the function invertible.
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What does it mean for a function to be one-to-one in terms of the horizontal line test?
What does it mean for a function to be one-to-one in terms of the horizontal line test?
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Every horizontal line intersects the graph at most once. This test verifies that each y-value corresponds to at most one x-value.
Every horizontal line intersects the graph at most once. This test verifies that each y-value corresponds to at most one x-value.
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What restricted domain is used to define $\arcsin(x)$ as the inverse of $\sin(x)$?
What restricted domain is used to define $\arcsin(x)$ as the inverse of $\sin(x)$?
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$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. On this interval, sine increases from -1 to 1, passing the horizontal line test.
$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. On this interval, sine increases from -1 to 1, passing the horizontal line test.
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What restricted domain is used to define $\arccos(x)$ as the inverse of $\cos(x)$?
What restricted domain is used to define $\arccos(x)$ as the inverse of $\cos(x)$?
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$\left[0,\pi\right]$. On this interval, cosine decreases from 1 to -1, making it one-to-one.
$\left[0,\pi\right]$. On this interval, cosine decreases from 1 to -1, making it one-to-one.
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What restricted domain is used to define $\arctan(x)$ as the inverse of $\tan(x)$?
What restricted domain is used to define $\arctan(x)$ as the inverse of $\tan(x)$?
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$\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Tangent is strictly increasing on this interval without vertical asymptotes.
$\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Tangent is strictly increasing on this interval without vertical asymptotes.
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What is the range of $\arcsin(x)$?
What is the range of $\arcsin(x)$?
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$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. This matches the restricted domain of sine used to define its inverse.
$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. This matches the restricted domain of sine used to define its inverse.
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What is the domain of $\arcsin(x)$?
What is the domain of $\arcsin(x)$?
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$\left[-1,1\right]$. Since sine's range is $[-1,1]$, this becomes arcsine's domain.
$\left[-1,1\right]$. Since sine's range is $[-1,1]$, this becomes arcsine's domain.
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Evaluate $\arccos\left(-\frac{1}{2}\right)$ using the principal range of $\arccos$.
Evaluate $\arccos\left(-\frac{1}{2}\right)$ using the principal range of $\arccos$.
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$\frac{2\pi}{3}$. Since $\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}$ and $\frac{2\pi}{3} \in [0, \pi]$.
$\frac{2\pi}{3}$. Since $\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}$ and $\frac{2\pi}{3} \in [0, \pi]$.
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Evaluate $\arcsin\left(\frac{1}{2}\right)$ using the principal range of $\arcsin$.
Evaluate $\arcsin\left(\frac{1}{2}\right)$ using the principal range of $\arcsin$.
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$\frac{\pi}{6}$. Since $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$ and $\frac{\pi}{6} \in [-\frac{\pi}{2}, \frac{\pi}{2}]$.
$\frac{\pi}{6}$. Since $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$ and $\frac{\pi}{6} \in [-\frac{\pi}{2}, \frac{\pi}{2}]$.
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What identity expresses the inverse relationship on the domain of $\arctan$?
What identity expresses the inverse relationship on the domain of $\arctan$?
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$\tan(\arctan(x))=x$ for $x\in\left(-\infty,\infty\right)$. Composing a function with its inverse yields the identity function.
$\tan(\arctan(x))=x$ for $x\in\left(-\infty,\infty\right)$. Composing a function with its inverse yields the identity function.
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What identity expresses the inverse relationship on the domain of $\arccos$?
What identity expresses the inverse relationship on the domain of $\arccos$?
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$\cos(\arccos(x))=x$ for $x\in\left[-1,1\right]$. Composing a function with its inverse yields the identity function.
$\cos(\arccos(x))=x$ for $x\in\left[-1,1\right]$. Composing a function with its inverse yields the identity function.
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What identity expresses the inverse relationship on the domain of $\arcsin$?
What identity expresses the inverse relationship on the domain of $\arcsin$?
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$\sin(\arcsin(x))=x$ for $x\in\left[-1,1\right]$. Composing a function with its inverse yields the identity function.
$\sin(\arcsin(x))=x$ for $x\in\left[-1,1\right]$. Composing a function with its inverse yields the identity function.
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What is the relationship between the graphs of $y=f(x)$ and $y=f^{-1}(x)$?
What is the relationship between the graphs of $y=f(x)$ and $y=f^{-1}(x)$?
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They are reflections across the line $y=x$. Inverse functions are symmetric about the line $y=x$.
They are reflections across the line $y=x$. Inverse functions are symmetric about the line $y=x$.
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Identify the restricted interval where $\tan(x)$ is strictly increasing and invertible.
Identify the restricted interval where $\tan(x)$ is strictly increasing and invertible.
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$\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Tangent increases continuously without vertical asymptotes here.
$\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Tangent increases continuously without vertical asymptotes here.
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Identify the restricted interval where $\cos(x)$ is strictly decreasing and invertible.
Identify the restricted interval where $\cos(x)$ is strictly decreasing and invertible.
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$\left[0,\pi\right]$. Cosine decreases monotonically from 1 to -1 on this interval.
$\left[0,\pi\right]$. Cosine decreases monotonically from 1 to -1 on this interval.
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Identify the restricted interval where $\sin(x)$ is strictly increasing and invertible.
Identify the restricted interval where $\sin(x)$ is strictly increasing and invertible.
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$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. Sine increases monotonically from -1 to 1 on this interval.
$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. Sine increases monotonically from -1 to 1 on this interval.
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What is the domain of $\arccos(x)$?
What is the domain of $\arccos(x)$?
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$\left[-1,1\right]$. Since cosine's range is $[-1,1]$, this becomes arccosine's domain.
$\left[-1,1\right]$. Since cosine's range is $[-1,1]$, this becomes arccosine's domain.
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What property must a trigonometric function have on a restricted domain to have an inverse there?
What property must a trigonometric function have on a restricted domain to have an inverse there?
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It must be one-to-one (strictly increasing or strictly decreasing). This ensures each input maps to exactly one output, allowing inverse construction.
It must be one-to-one (strictly increasing or strictly decreasing). This ensures each input maps to exactly one output, allowing inverse construction.
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Which restricted interval makes $\sin(x)$ one-to-one: $\left[0,2\pi\right]$ or $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$?
Which restricted interval makes $\sin(x)$ one-to-one: $\left[0,2\pi\right]$ or $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$?
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$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. Only the second interval has sine strictly monotonic throughout.
$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. Only the second interval has sine strictly monotonic throughout.
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Identify whether $\tan(x)$ is increasing or decreasing on $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$.
Identify whether $\tan(x)$ is increasing or decreasing on $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$.
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Increasing on $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Derivative $\sec^2(x)>0$ on this interval confirms monotonic increase.
Increasing on $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Derivative $\sec^2(x)>0$ on this interval confirms monotonic increase.
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Identify whether $\cos(x)$ is increasing or decreasing on $\left[0,\pi\right]$.
Identify whether $\cos(x)$ is increasing or decreasing on $\left[0,\pi\right]$.
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Decreasing on $\left[0,\pi\right]$. Derivative $-\sin(x)<0$ on this interval confirms monotonic decrease.
Decreasing on $\left[0,\pi\right]$. Derivative $-\sin(x)<0$ on this interval confirms monotonic decrease.
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Identify whether $\sin(x)$ is increasing or decreasing on $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$.
Identify whether $\sin(x)$ is increasing or decreasing on $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$.
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Increasing on $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. Derivative $\cos(x)>0$ on this interval confirms monotonic increase.
Increasing on $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. Derivative $\cos(x)>0$ on this interval confirms monotonic increase.
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What is the simplified value of $\arccos\left(\cos\left(\frac{5\pi}{3}\right)\right)$?
What is the simplified value of $\arccos\left(\cos\left(\frac{5\pi}{3}\right)\right)$?
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$\frac{\pi}{3}$. $\cos(\frac{5\pi}{3})=\frac{1}{2}$, and $\arccos(\frac{1}{2})=\frac{\pi}{3}$ in range.
$\frac{\pi}{3}$. $\cos(\frac{5\pi}{3})=\frac{1}{2}$, and $\arccos(\frac{1}{2})=\frac{\pi}{3}$ in range.
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What is the simplified value of $\arcsin\left(\sin\left(\frac{3\pi}{4}\right)\right)$?
What is the simplified value of $\arcsin\left(\sin\left(\frac{3\pi}{4}\right)\right)$?
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$\frac{\pi}{4}$. $\sin(\frac{3\pi}{4})=\frac{\sqrt{2}}{2}$, and $\arcsin(\frac{\sqrt{2}}{2})=\frac{\pi}{4}$ in range.
$\frac{\pi}{4}$. $\sin(\frac{3\pi}{4})=\frac{\sqrt{2}}{2}$, and $\arcsin(\frac{\sqrt{2}}{2})=\frac{\pi}{4}$ in range.
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Identify the principal-value identity for $\tan(\arctan(x))$.
Identify the principal-value identity for $\tan(\arctan(x))$.
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$\tan(\arctan(x))=x$ for $x\in\left(-\infty,\infty\right)$. Applying tangent to its inverse returns the original value.
$\tan(\arctan(x))=x$ for $x\in\left(-\infty,\infty\right)$. Applying tangent to its inverse returns the original value.
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Identify the principal-value identity for $\cos(\arccos(x))$.
Identify the principal-value identity for $\cos(\arccos(x))$.
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$\cos(\arccos(x))=x$ for $x\in\left[-1,1\right]$. Applying cosine to its inverse returns the original value.
$\cos(\arccos(x))=x$ for $x\in\left[-1,1\right]$. Applying cosine to its inverse returns the original value.
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Identify the principal-value identity for $\sin(\arcsin(x))$.
Identify the principal-value identity for $\sin(\arcsin(x))$.
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$\sin(\arcsin(x))=x$ for $x\in\left[-1,1\right]$. Applying sine to its inverse returns the original value.
$\sin(\arcsin(x))=x$ for $x\in\left[-1,1\right]$. Applying sine to its inverse returns the original value.
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