Where is (\cot x) undefined?
Correct answer: B
(\cot x=\frac{\cos x}{\sin x}), so it is undefined when (\sin x=0). Always check the denominator carefully.
View question detailsMathematics
Trigonometric functions and their properties
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Easy · Level 72 · domain,cotangent,undefined
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- (\cos x=0)
- (\sin x=0)
- (\tan x=1)
- (\sec x=0)
Easy · Level 72 · domain,secant,undefined
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- (\sin x=0)
- (\tan x=0)
- (\cos x=0)
- (\cot x=0)
Easy · Level 72 · domain,cosecant,undefined
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- (\cos x=0)
- (\sin x=0)
- (\tan x=0)
- (\sec x=1)
Easy · Level 72 · identity,first-quadrant,cosine
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- (\frac{3}{5})
- (\frac{5}{4})
- (\frac{4}{5})
- (-\frac{4}{5})
Easy · Level 72 · identity,first-quadrant,sine
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- (\frac{5}{13})
- (\frac{12}{13})
- (\frac{13}{5})
- -(\frac{5}{13}) / (-\frac{5}{13})
Easy · Level 72 · identity,tangent,secant
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- (\frac{24}{25})
- (\frac{25}{24})
- (\frac{7}{25})
- (\frac{25}{7})
Easy · Level 72 · identity,cotangent,cosecant
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- (\frac{15}{17})
- (\frac{17}{15})
- (\frac{8}{17})
- (\frac{17}{8})
Easy · Level 72 · allied-angles,sine,quadrants
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- (\sin x)
- (\cos x)
- (-\sin x)
- (-\cos x)
Easy · Level 72 · allied-angles,cosine,quadrants
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- (\cos x)
- (-\cos x)
- (\sin x)
- (-\sin x)
Easy · Level 72 · trigonometric-identities,tangent,secant
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- (\sec^2 x+\tan^2 x=1)
- (\sin^2 x-\cos^2 x=1)
- (\tan x=\frac{\cos x}{\sin x})
- (1+\tan^2 x=\sec^2 x)
Medium · Level 70 · trigonometric-identities,medium,sine-cosine
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- (0)
- (\frac{1}{2})
- (1)
- (-\frac{1}{2})
Medium · Level 70 · trigonometric functions,tangent ratio,trigonometric identities,class 11 mathematics,quotient of sine and cosine
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- 2
- \(\frac{1}{2}\)
- -2
- 1
Medium · Level 70 · trigonometric functions, tangent function, domain of tangent, undefined values, class 11 mathematics
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- \(x=n\pi,\; n\in\mathbb{Z}\)
- \(x=\frac{(2n+1)\pi}{2},\; n\in\mathbb{Z}\)
- \(x=\frac{n\pi}{2},\; n\in\mathbb{Z}\)
- \(x=2n\pi,\; n\in\mathbb{Z}\)
Medium · Level 70 · reciprocal-identities,cosecant,sine
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- (\frac{1}{2})
- (4)
- (-4)
- (\frac{1}{4})
Medium · Level 70 · trigonometric identities,pythagorean identity,algebraic simplification,class 11 mathematics
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- \(1\)
- \(\sin x\)
- \(\cos x\)
- \(\tan x\)
Medium · Level 70 · trigonometric-identities,simplification
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- (\sec x)
- (1)
- (\tan x)
- (\cot x)
Medium · Level 70 · period,sine,transformation
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- \(2\pi\)
- \(\pi\)
- \(\frac{2\pi}{3}\)
- \(\frac{\pi}{3}\)
Medium · Level 70 · period,tangent,transformation
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- \(\pi\)
- \(\frac{\pi}{2}\)
- \(\frac{\pi}{3}\)
- \(\frac{\pi}{4}\)
Medium · Level 70 · trigonometric functions, cosine range, range of function, amplitude, class 11 mathematics
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- \([-2,2]\)
- \([-1,1]\)
- \([0,2]\)
- \(( -\infty,\infty )\)
Medium · Level 70 · trigonometric functions, sine function, maximum value, range, class 11 mathematics
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- 2
- 3
- 4
- 1
Where is (\sec x) undefined?
Correct answer: C
(\sec x=\frac{1}{\cos x}), so it is undefined when (\cos x=0). In reciprocal functions, the denominator cannot be zero.
View question detailsWhere is (\cosec x) undefined?
Correct answer: B
(\cosec x=\frac{1}{\sin x}), so it is undefined when (\sin x=0). In a reciprocal, the denominator function must not be zero.
View question detailsIf (\sin x=\frac{3}{5}) and (x) is in the first quadrant, what is the value of (\cos x)?
Correct answer: C
Using (\sin^2 x+\cos^2 x=1), (\cos x=\frac{4}{5}). In the first quadrant, (\cos x) is positive.
View question detailsIf (\cos x=\frac{12}{13}) and (x) is in the first quadrant, what is the value of (\sin x)?
Correct answer: A
From (\sin^2 x=1-\cos^2 x), (\sin x=\frac{5}{13}). In the first quadrant, the positive value is taken.
View question detailsIf (\tan x=\frac{7}{24}) and (x) is in the first quadrant, what is the value of (\sec x)?
Correct answer: B
Using (\sec^2 x=1+\tan^2 x), (\sec x=\frac{25}{24}). In the first quadrant, (\sec x) is positive.
View question detailsIf (\cot x=\frac{8}{15}) and (x) is in the first quadrant, what is the value of (\cosec x)?
Correct answer: B
From (\cosec^2 x=1+\cot^2 x), (\cosec x=\frac{17}{15}). Take the positive value in the first quadrant.
View question detailsWhat is (\sin(\pi+x)) equal to?
Correct answer: C
(\pi+x) lies in the third quadrant and (\sin x) is negative there. Hence (\sin(\pi+x)=-\sin x).
View question detailsWhat is (\cos(\pi+x)) equal to?
Correct answer: B
(\pi+x) is in the third quadrant and (\cos x) is negative there. Therefore, (\cos(\pi+x)=-\cos x).
View question detailsWhich of the following identities is correct?
Correct answer: D
The correct identity is (1+\tan^2 x=\sec^2 x). In identity-based questions, check signs and fractions carefully.
View question detailsIf (\sin x+\cos x=\sqrt{2}), what is the value of (\sin x\cos x)?
Correct answer: B
Squaring both sides gives (1+2\sin x\cos x=2). Hence (\sin x\cos x=\frac{1}{2}); squaring is useful in such questions.
View question detailsIf \(\tan x=2\), what is the value of \(\frac{\sin x}{\cos x}\)?
Correct answer: A
By definition, \(\tan x=\frac{\sin x}{\cos x}\). Since \(\tan x=2\) is given, \(\frac{\sin x}{\cos x}=2\). The value \(\frac{1}{2}\) is the reciprocal and represents \(\cot x\), not \(\tan x\). Exam tip: Remember \(\tan x\) as \(\sin x/\cos x\).
View question detailsFor which of the following values is \(\tan x\) undefined?
Correct answer: B
\(\tan x=\frac{\sin x}{\cos x}\). It is undefined when \(\cos x=0\), i.e. at \(x=\frac{(2n+1)\pi}{2}\). At \(x=n\pi\), tan equals 0, so option A is not correct. Exam tip: link tan’s domain to non-zero cosine.
View question detailsIf (\cosec x=4), what is the value of (\sin x)?
Correct answer: D
(\cosec x=\frac{1}{\sin x}). Hence (\sin x=\frac{1}{4}); learn each reciprocal function as a pair.
View question detailsWhat is the simplified value of \(\frac{1-\cos^2 x}{\sin^2 x}\)?
Correct answer: A
Using the Pythagorean identity \(\sin^2 x+\cos^2 x=1\), we get \(1-\cos^2 x=\sin^2 x\). Therefore, \(\frac{1-\cos^2 x}{\sin^2 x}=\frac{\sin^2 x}{\sin^2 x}=1\), provided \(\sin x\ne 0\). Neither \(\sin x\), \(\cos x\), nor \(\tan x\) is the simplified value of this ratio. Exam tip: first replace \(1-\cos^2 x\) with \(\sin^2 x\) in such questions.
View question detailsWhat is the simplified value of (\frac{1-\sin^2 x}{\cos^2 x})?
Correct answer: B
Since (1-\sin^2 x=\cos^2 x), the value is (1). (\sin^2 x+\cos^2 x=1) is the most important identity.
View question detailsWhat is the period of the function (\sin 3x)?
Correct answer: C
The period of (\sin kx) is \(\frac{2\pi}{k}\). Here \(k=3\), so the period is \(\frac{2\pi}{3}\).
View question detailsWhat is the period of the function (\tan 4x)?
Correct answer: D
The period of (\tan kx) is \(\frac{\pi}{k}\). Substituting \(k=4\) gives period \(\frac{\pi}{4}\).
View question detailsWhat is the range of the function (2\cos x)?
Correct answer: A
Since \(\cos x\) lies between \(-1\) and \(1\), multiplying by 2 gives values of \(2\cos x\) from \(-2\) to \(2\). The endpoints are included because \(2\cos 0=2\) and \(2\cos \pi=-2\). Thus, \([-1,1]\) is the range of \(\cos x\), not of \(2\cos x\). Exam tip: the range of \(a\cos x\) is \([-|a|,|a|]\).
View question detailsWhat is the maximum value of the function (3\sin x+1)?
Correct answer: C
Since \(\sin x\) lies between \(-1\) and \(1\), its maximum value is \(1\). Therefore, the maximum of \(3\sin x+1\) is \(3\times1+1=4\). Option 3 is only the maximum of \(3\sin x\); adding the constant \(1\) shifts the maximum to 4. Exam tip: for \(a\sin x+b\), when \(a>0\), the maximum value is \(a+b\).
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