If (\cos^{-1}x=0), what is the value of (x)?
Correct answer: A
If (\cos^{-1}x=0), then (x=\cos0=1). In an inverse trigonometric equation apply the corresponding trigonometric function on both sides.
View question detailsMathematics
Introduction to inverse trigonometric functions
प्रतिलोम त्रिकोणमितीय फलनों का परिचय
This Class 12 Mathematics topic introduces inverse trigonometric functions as the functions that reverse the action of sine, cosine, and tangent on suitably restricted domains. Students learn the meaning of sin⁻¹x, cos⁻¹x, and tan⁻¹x, along with their principal values, domains, and ranges. The topic also develops understanding of how these functions are represented, how their restrictions make them well-defined, and how to interpret basic relationships and expressions within the chapter on Inverse Trigonometric Functions.
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Easy · Level 33 · inverse-trigonometry,equation,arccos
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- (\1)
- (\0)
- (-1)
- (\frac{1}{2})
Easy · Level 33 · inverse-trigonometry,equation,arcsin
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- (\1)
- (\0) / (\0
- (-1)
- (\frac{1}{2})
Easy · Level 33 · inverse trigonometric functions, principal value, inverse sine, trigonometry, class 12 mathematics
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- \(\sin^{-1}(\sin(5\pi/6))=\pi/6\) because \(\pi/6\) is the related acute angle
- \(\sin^{-1}(\sin(5\pi/6))=\pi/6\) because the principal value of \(\sin^{-1}\) lies in \([-\pi/2,\pi/2]\)
- \(\sin^{-1}(\sin(5\pi/6))=5\pi/6\) because an inverse function always returns the original angle
- \(\sin^{-1}(\sin(5\pi/6))=-\pi/6\) because \(5\pi/6\) is in the second quadrant
Easy · Level 33 · inverse-trigonometry,odd-function,concept
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- (\sin^{-1}x)
- (\cos^{-1}x)
- (\sec^{-1}x)
- (\cos^{-1}(-x))
Easy · Level 33 · inverse-trigonometry,identity,arcsin
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- (-\sin^{-1}x)
- (\pi-\sin^{-1}x)
- (\frac{\pi}{2}-\sin^{-1}x)
- (\sin^{-1}x)
Easy · Level 33 · inverse-trigonometry,identity,arctan
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- (-\tan^{-1}x)
- (\pi-\tan^{-1}x)
- (\frac{\pi}{2}-\tan^{-1}x)
- (\tan^{-1}x)
Easy · Level 33 · inverse-trigonometry,composition,negative-angle
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- \(-\frac{\pi}{6}\)
- \(\frac{\pi}{6}\)
- \(\frac{5\pi}{6}\)
- \(-\frac{5\pi}{6}\)
Easy · Level 33 · inverse-trigonometry,definition,arcsin
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- \(\sin y=x\) and \(y\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\)
- \(\cos y=x\) and \(y\in[0,\pi]\)
- \(\tan y=x\) and \(y\in\mathbb{R}\)
- \(\sin x=y\) and \(x\in\mathbb{R}\)
Easy · Level 33 · inverse-trigonometry,definition,arctan
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- \(\tan y=x\) and \(y\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\)
- \(\cos y=x\) and \(y\in[0,\pi]\)
- \(\sin y=x\) and \(y\in[-1,1]\)
- \(\tan x=y\) and \(x\in\mathbb{R}\)
Easy · Level 33 · inverse-trigonometry,standard-values,arctan
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- \(\frac{\pi}{3}\)
- \(\frac{\pi}{6}\)
- \(\frac{\pi}{4}\)
- \(\frac{2\pi}{3}\)
Easy · Level 33 · inverse-trigonometry,standard-values,arctan
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- \(\frac{\pi}{6}\)
- \(\frac{\pi}{3}\)
- \(\frac{\pi}{4}\)
- \(\frac{\pi}{2}\)
Easy · Level 33 · inverse-trigonometry,composition,arccos
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- \(\frac{\pi}{4}\)
- \(-\frac{\pi}{4}\)
- \(\frac{3\pi}{4}\)
- \(\frac{5\pi}{4}\)
Easy · Level 33 · inverse-trigonometry,composition,arctan
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- \(\frac{\pi}{3}\)
- \(-\frac{\pi}{3}\)
- \(\frac{2\pi}{3}\)
- \(\frac{4\pi}{3}\)
Easy · Level 33 · inverse-trigonometry,standard-values,arccosec
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- (\frac{\pi}{6})
- (\frac{\pi}{3})
- (\frac{\pi}{4})
- (\frac{\pi}{2})
Easy · Level 33 · inverse-trigonometry,standard-values,arccot
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- (\frac{\pi}{6})
- (\frac{\pi}{3})
- (\frac{\pi}{4})
- (\frac{5\pi}{6})
Easy · Level 33 · inverse-trigonometry,domain-check,real-value
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- \(\cos^{-1}\left(\frac{7}{6}\right)\)
- \(\tan^{-1}7\)
- \(\sin^{-1}\left(-\frac{3}{4}\right)\)
- \(\cot^{-1}(-5)\)
Easy · Level 33 · inverse-trigonometry,range-check,arcsin
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- \(\frac{2\pi}{3}\)
- \(\frac{\pi}{6}\)
- \(-\frac{\pi}{4}\)
- \(\frac{\pi}{2}\)
Easy · Level 33 · inverse-trigonometry,range-check,arccos
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- (-\frac{\pi}{3})
- (\frac{\pi}{3})
- (\frac{\pi}{2})
- (\pi)
Easy · Level 33 · inverse-trigonometry,range-check,arctan
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- \(\frac{\pi}{2}\)
- \(\frac{\pi}{4}\)
- \(-\frac{\pi}{6}\)
- (\0)
Easy · Level 33 · inverse-trigonometry,application,right-triangle
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- \(\frac{4}{5}\)
- \(\frac{3}{5}\)
- \(-\frac{4}{5}\)
- \(\frac{5}{4}\)
If (\sin^{-1}x=\frac{\pi}{2}), what is the value of (x)?
Correct answer: A
Taking (\sin) on both sides gives (x=\sin\frac{\pi}{2}=1). This is the maximum principal value of (\sin^{-1}x).
View question detailsA student writes \(\sin^{-1}(\sin(5\pi/6))=5\pi/6\). What is the correct correction to this error?
Correct answer: B
Since \(\sin(5\pi/6)=1/2\), we need the angle in the principal range \([-\pi/2,\pi/2]\) whose sine is \(1/2\). It is \(\pi/6\), not \(5\pi/6\). Exam tip: always check the principal-value range first.
View question detailsWhich function is odd?
Correct answer: A
Since (\sin^{-1}(-x)=-\sin^{-1}x), (\sin^{-1}x) is odd. The function (\tan^{-1}x) is also odd but is not in the options.
View question detailsWhat is (\sin^{-1}(-x)) equal to when (x\in[-1,1])?
Correct answer: A
The function (\sin^{-1}x) is odd, so (\sin^{-1}(-x)=-\sin^{-1}x). This identity is useful in sign-change questions.
View question detailsWhat is (\tan^{-1}(-x)) equal to when (x\in\mathbb{R})?
Correct answer: A
The function (\tan^{-1}x) is odd, so (\tan^{-1}(-x)=-\tan^{-1}x). Keep the answer in the principal range.
View question detailsWhat is the value of \(\tan^{-1}\left(\tan\left(-\frac{\pi}{6}\right)\right)\)?
Correct answer: A
The angle \(-\frac{\pi}{6}\) lies in the principal range \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\), so the value is the same. For \(\tan^{-1}\), the interval is open.
View question detailsIf \(y=\sin^{-1}x\), what is the correct relation between (x) and (y)?
Correct answer: A
The statement \(y=\sin^{-1}x\) means \(\sin y=x\) and (y) lies in the principal range. Understanding inverse functions by definition is the safest method.
View question detailsIf \(y=\tan^{-1}x\), which relation is correct?
Correct answer: A
The expression \(y=\tan^{-1}x\) means \(\tan y=x\). Here (y) stays in the principal range \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\).
View question detailsWhat is the principal value of \(\tan^{-1}\sqrt{3}\)?
Correct answer: A
Because \(\tan\frac{\pi}{3}=\sqrt{3}\) and \(\frac{\pi}{3}\) lies in the principal range. For \(\tan^{-1}\), take the angle in \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\).
View question detailsWhat is the principal value of \(\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)\)?
Correct answer: A
Because \(\tan\frac{\pi}{6}=\frac{1}{\sqrt{3}}\). Inverse values are found quickly from basic trigonometric values.
View question detailsWhat is the value of \(\cos^{-1}\left(\cos\frac{\pi}{4}\right)\)?
Correct answer: A
Since \(\frac{\pi}{4}\in[0,\pi]\), \(\cos^{-1}\left(\cos\frac{\pi}{4}\right)=\frac{\pi}{4}\). If the angle lies in the principal range, it remains unchanged.
View question detailsWhat is the value of \(\tan^{-1}\left(\tan\frac{\pi}{3}\right)\)?
Correct answer: A
The angle \(\frac{\pi}{3}\) lies in the principal range \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\). Therefore the value of the expression is \(\frac{\pi}{3}\).
View question detailsWhat is the value of (\operatorname{cosec}^{-1}2)?
Correct answer: A
Because (\operatorname{cosec}\frac{\pi}{6}=2). In (\operatorname{cosec}^{-1}x), the magnitude of (x) cannot be less than (1).
View question detailsWhat is the principal value of (\cot^{-1}\sqrt{3})?
Correct answer: A
Because (\cot\frac{\pi}{6}=\sqrt{3}) and (\frac{\pi}{6}\in(0,\pi)). Remember ((0,\pi)) for the principal value of (\cot^{-1}).
View question detailsWhich expression does not give a real number?
Correct answer: A
The expression \(\cos^{-1}x\) is real only for \(x\in[-1,1]\), and \(\frac{7}{6}>1\). Hence it will not give a real value.
View question detailsWhich value is not possible from the range of \(\sin^{-1}x\)?
Correct answer: A
The range of \(\sin^{-1}x\) is \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\). The value \(\frac{2\pi}{3}\) lies outside this range.
View question detailsWhich value is not possible from the range of (\cos^{-1}x)?
Correct answer: A
The range of (\cos^{-1}x) is ([0,\pi]), so a negative value is not possible. In such questions check only the output range.
View question detailsWhich value is not possible from the range of \(\tan^{-1}x\)?
Correct answer: A
The range of \(\tan^{-1}x\) is \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\), which does not include \(\frac{\pi}{2}\). Pay attention to open intervals.
View question detailsWhat is the value of \(\sin\left(\cos^{-1}\frac{3}{5}\right)\)?
Correct answer: A
Let \(\theta=\cos^{-1}\frac{3}{5}\), then \(\cos\theta=\frac{3}{5}\) and \(\theta\in[0,\pi]\). Here \(\sin\theta=\frac{4}{5}\).
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