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\(y=\sin ^{-1}x\) if and only if \(x=\) _____, where \( -1\leq x\leq 1\) and \(-\dfrac{\pi }{2}\leq y\leq \dfrac{\pi }{2}\).

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True or False  \(\sin ^{-1} (\sin x) =x,\) where \(-1\leq x\) \(\leq 1\).

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True or False  The domain of \(y =\sin ^{-1}x\) is \(-\dfrac{\pi }{2}\leq x\leq \dfrac{\pi }{2}\).

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True or False  \(\sin (\sin ^{-1}0) =0\).

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True or False  \(y =\tan ^{-1}x\) means \(x=\tan y\), where \(-\infty <x<\infty \) and \(-\dfrac{\pi }{2}\leq y\leq \dfrac{\pi }{2}\).

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True or False  The domain of the inverse tangent function is the set of all real numbers.

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True or False  \(\sec ^{-1}0.5\) is not defined.

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True or False  Trigonometric equations can have multiple solutions.

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\(\sin ^{ - 1} \left( {\frac{{\sqrt 2 }}{2}} \right)\)

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\(\sin ^{-1}\left( -\dfrac{1}{2}\right)\)

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\(\sec^{-1}2\)

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\(\tan^{-1}\sqrt{3}\)

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\(\tan ^{-1}( -1) \)

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\(\sec ^{-1}\dfrac{2\sqrt{3}}{3}\)

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\(\tan ^{-1}\left( \tan \dfrac{\pi }{4}\right)\)

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\(\tan ^{-1} ( \sin 0)\)

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\(\sin \left( \sin ^{-1}\dfrac{3}{5}\right)\)

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\(\tan [ \sec ^{-1} ( -3) ]\)

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\(\sin ^{-1}\left( \sin \dfrac{4\pi }{5}\right)\)

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\(\sec ( \sec ^{-}{}^{1}3)\)

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Write \(\cos ( \sin ^{-1}u) \) as an algebraic expression containing \(u\), where \(\vert u \vert \leq 1\).

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Write \(\tan ( \sin ^{-1}u)\) as an algebraic expression containing \(u\), where \(\vert u \vert \leq 1\).

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Write \(\sec ( \tan ^{-1}u)\) as an algebraic expression containing \(u\).

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Show that \(y =\sin ^{-1}x\) is an odd function. That is, show \(\sin ^{-1} ( -x) =-\sin ^{-1}x\).

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Show that \(y =\tan ^{-1}x\) is an odd function. That is, show \(\tan ^{-1} ( -x) =-\tan ^{-1}x\).

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Given that \(x=\sin ^{-1}\dfrac{1}{2}\), find \(\cos x\), \(\tan x\), \(\cot x\), \(\sec x\), and \(\csc x\).

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\(\tan \theta =-\dfrac{\sqrt{3}}{3}\)

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\(\sec \dfrac{3\theta }{2}=-2\)

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\(2\sin \theta +3=2\)

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\(1-\cos \theta =\dfrac{1}{2}\)

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\(\sin ( 3\theta ) =-1\)

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\(\cos ( 2\theta ) =\dfrac{1}{2}\)

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\(4\cos ^{2}\theta =1\)

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\(2\sin ^{2}\theta -1=0\)

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\(2\sin ^{2}\theta -5\sin \theta +3=0\)

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\(2\cos ^{2}\theta -7\cos \theta -4=0\)

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\(1+\sin \theta =2\cos ^{2}\theta\)

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\(\sec ^{2}\theta +\tan \theta =0\)

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\(\sin \theta +\cos \theta = 0\)

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\(\tan \theta =\cot \theta\)

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\(\cos (2\theta ) +6\sin ^{2}\theta =4\)

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\(\cos ( 2\theta ) =\cos \theta\)

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\(\mathbf{\sin }( 2\theta ) +\sin ( 4\theta ) =0\)

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\(\cos ( 4\theta) -\cos ( 6\theta ) =0\)

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\(\tan \theta =5\)

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\(\cos \theta =0.6\)

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\(2+3\sin \theta =0\)

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\(4+\sec \theta =0\)

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1. On the same set of axes, graph \(f( x) = 3\sin ( 2x) +2\) and \(g( x) =\dfrac{7}{2}\) on the interval \( [ 0,\pi ]\).
2. Solve \(f( x) = g(x)\) on the interval \( [ 0,\pi ]\), and label the points of intersection on the graph drawn in (a).
3. Shade the region bounded by \(f( x) =3\sin (2x) + 2\) and \(g(x) = \dfrac{7}{2}\) between the points found in (b) on the graph drawn in (a).
4. Solve \(f(x) > g(x)\) on the interval \([ 0,\pi ]\).

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1. On the same set of axes, graph \(f( x) =-4 \cos x\) and \(g( x) =2\cos x+3\) on the interval \([ 0,2\pi ]\).
2. Solve \(f( x) =g( x)\) on the interval \([ 0,2\pi ]\), and label the points of intersection on the graph drawn in (a).
3. Shade the region bounded by \(f( x) = -4\cos x\) and \(g( x) =2\cos x+3\) between the points found in (b) on the graph drawn in (a).
4. Solve \(f(x) > g(x)\) on the interval \([ 0,2\pi ]\).