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Questions and Answers
What is the value of $ heta$ when $sin( heta) = -1$ in the interval $[- rac{ heta}{2}, rac{ heta}{2}]$?
What is the value of $ heta$ when $sin( heta) = -1$ in the interval $[- rac{ heta}{2}, rac{ heta}{2}]$?
Which interval is valid for finding values of $ heta$ such that $sin( heta) = rac{3}{2}$?
Which interval is valid for finding values of $ heta$ such that $sin( heta) = rac{3}{2}$?
If $ heta = sin^{-1}(-1)$, what is the resulting value of $ heta$?
If $ heta = sin^{-1}(-1)$, what is the resulting value of $ heta$?
What does the function $sin^{-1}( rac{3}{2})$ represent?
What does the function $sin^{-1}( rac{3}{2})$ represent?
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In which of the following intervals can you find an angle $ heta$ for $sin( heta) = 3$?
In which of the following intervals can you find an angle $ heta$ for $sin( heta) = 3$?
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What can be concluded about $sin^{-1}(sin(- rac{ heta}{2}))$?
What can be concluded about $sin^{-1}(sin(- rac{ heta}{2}))$?
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What is the sine value corresponding to $ heta = sin^{-1}( rac{3}{2})$?
What is the sine value corresponding to $ heta = sin^{-1}( rac{3}{2})$?
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Considering the angle $ heta$ where $sin( heta) = rac{3}{2}$, which statement is correct?
Considering the angle $ heta$ where $sin( heta) = rac{3}{2}$, which statement is correct?
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What is the value of $ an^{-1}(1)$ within the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$?
What is the value of $ an^{-1}(1)$ within the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$?
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What is the angle $ heta$ for which $ an(\theta) = -3$ in the range $(-\frac{\pi}{2}, \frac{\pi}{2})$?
What is the angle $ heta$ for which $ an(\theta) = -3$ in the range $(-\frac{\pi}{2}, \frac{\pi}{2})$?
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What is the value of the composite function sin(sin(0.5))?
What is the value of the composite function sin(sin(0.5))?
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Why is sin[sin(-2.5)] not defined?
Why is sin[sin(-2.5)] not defined?
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Which of the following is true about the function $f(\tan^{-1}(x))$?
Which of the following is true about the function $f(\tan^{-1}(x))$?
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What is the result of cos(cos^-1(-0.4))?
What is the result of cos(cos^-1(-0.4))?
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What is the range of the function $f(\sin^{-1}(x))$?
What is the range of the function $f(\sin^{-1}(x))$?
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In finding the exact value of $\tan^{-1}(-3)$, which interval is considered?
In finding the exact value of $\tan^{-1}(-3)$, which interval is considered?
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In what interval must x lie for the property ff^-1(x) = x to hold true for cos?
In what interval must x lie for the property ff^-1(x) = x to hold true for cos?
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What value does cos(cos^-1(15)) yield?
What value does cos(cos^-1(15)) yield?
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Which property must hold for an inverse tangent function $ an^{-1}(x)$?
Which property must hold for an inverse tangent function $ an^{-1}(x)$?
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Which of the following expressions involves an angle measurement that is not a valid input for inverse sine?
Which of the following expressions involves an angle measurement that is not a valid input for inverse sine?
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To find $\tan^{-1}(-\frac{3}{2})$, which reference angle should be used?
To find $\tan^{-1}(-\frac{3}{2})$, which reference angle should be used?
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What is the primary reason for choosing the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$ for the tangent function?
What is the primary reason for choosing the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$ for the tangent function?
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For the expression cos(cos^-1(15)), what would be a possible reason for it to be defined?
For the expression cos(cos^-1(15)), what would be a possible reason for it to be defined?
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What is the value of the composite function cos(π)?
What is the value of the composite function cos(π)?
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What is the domain of the inverse cosine function?
What is the domain of the inverse cosine function?
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What can be concluded about the composite function cos(π)?
What can be concluded about the composite function cos(π)?
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For which angle is the cosine function even?
For which angle is the cosine function even?
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What is the condition required for a function to have an inverse?
What is the condition required for a function to have an inverse?
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What is the range of the function f(x) = 2sin(x) - 1 given the interval for x?
What is the range of the function f(x) = 2sin(x) - 1 given the interval for x?
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Which property can be used to evaluate cos(-3π/4)?
Which property can be used to evaluate cos(-3π/4)?
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Which mathematical representation describes the relationship between a function and its inverse?
Which mathematical representation describes the relationship between a function and its inverse?
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In which of the following intervals is the function f(x) = tan(x) defined?
In which of the following intervals is the function f(x) = tan(x) defined?
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What is the inverse function of the equation $y = 2\sin x - 1$?
What is the inverse function of the equation $y = 2\sin x - 1$?
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What is the range of the function $f$ defined as $f(x) = 2\sin x - 1$?
What is the range of the function $f$ defined as $f(x) = 2\sin x - 1$?
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Which of the following describes the domain of the inverse function $f^{-1}(x)$?
Which of the following describes the domain of the inverse function $f^{-1}(x)$?
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How would you isolate the inverse sine in the equation $12\sin^{-1}x = 3\pi$?
How would you isolate the inverse sine in the equation $12\sin^{-1}x = 3\pi$?
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What does it mean for $y = \sin^{-1}x$ to imply $x = \sin y$?
What does it mean for $y = \sin^{-1}x$ to imply $x = \sin y$?
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What are the valid inputs for $\sin^{-1}(x)$ based on the domain limits outlined?
What are the valid inputs for $\sin^{-1}(x)$ based on the domain limits outlined?
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When solving for the value of $x$ in the equation $x = \sin^{-1}\left(\frac{\pi}{4}\right)$, what is the result?
When solving for the value of $x$ in the equation $x = \sin^{-1}\left(\frac{\pi}{4}\right)$, what is the result?
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What is the solution set for the equation $12\sin^{-1} x = 3\pi$?
What is the solution set for the equation $12\sin^{-1} x = 3\pi$?
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Study Notes
- Inverse Sine Function
- The inverse sine function is denoted as sin⁻¹(x) or arcsin(x).
- It's defined as the angle θ in the interval -π/2 ≤ θ ≤ π/2 whose sine equals x.
- For example, sin⁻¹(-1) = -π/2 because the sine of -π/2 is -1.
- It's important to note that the interval for θ is restricted to ensure that the inverse sine function is a function (meaning it has only one output for each input).
- Finding the Exact Values of Inverse Sine Function
- To find the exact value of sin⁻¹(x), we need to determine the angle θ in the interval -π/2 ≤ θ ≤ π/2, whose sine is equal to x.
- For example, sin⁻¹(√3/2) = π/3 because the sine of π/3 is √3/2 and π/3 lies within the specified interval.
- Finding the Exact Value of the Inverse Tangent Function
- To find the exact value of tan⁻¹(x), we need to determine the angle θ in the interval -π/2 < θ < π/2, whose tangent is equal to x.
- For example, tan⁻¹(-√3) = -π/3 because the tangent of -π/3 is -√3 and -π/3 lies within the specified interval.
-
Properties of Inverse Functions
- For the sine function, the following properties hold:
- sin(sin⁻¹(x)) = x for -1 ≤ x ≤ 1 (this is the cancellation property)
-
sin⁻¹(sin(x)) = x for -π/2 ≤ x ≤ π/2
- For the cosine function, the following properties hold:
- cos(cos⁻¹(x)) = x for -1 ≤ x ≤ 1
-
cos⁻¹(cos(x)) = x for 0 ≤ x ≤ π
- For the tangent function, the following properties hold:
- tan(tan⁻¹(x)) = x for all real numbers x
- tan⁻¹(tan(x)) = x for -π/2 < x < π/2
-
Finding the Inverse Function of a Trigonometric Function
- To find the inverse function of a trigonometric function, we need to follow these steps:
- Replace f(x) with y.
- Interchange x and y.
- Solve for y.
-
Replace y with f⁻¹(x).
- For example, to find the inverse function of f(x) = 2sinx - 1, we would follow these steps:
- y = 2sinx - 1.
- x = 2siny - 1.
- x + 1 = 2siny.
- siny = (x + 1)/2.
- y = sin⁻¹((x + 1)/2).
- f⁻¹(x) = sin⁻¹((x + 1)/2)
- Solving Inverse Trigonometric Equations
-
To solve equations involving inverse trigonometric functions, isolate the inverse trigonometric function and apply the definition of the inverse function.
- For example, to solve the equation 12sin⁻¹(x) = 3π, we would follow these steps:
- sin⁻¹(x) = π/4.
- x = sin(π/4).
- x = √2/2.
- Therefore, the solution set is {√2/2}.
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Description
This quiz covers the concepts and properties of inverse sine and tangent functions, including how to find their exact values. You'll learn about the intervals for the angles and how to apply these functions in various scenarios. Test your understanding of sin⁻¹(x) and tan⁻¹(x) with this challenging quiz.
