Which is the range of (\cos \theta)?
Correct answer: A
The range of (\cos \theta) is ([-1,1]). (\sin \theta) and (\cos \theta) have the same range.
View question detailsMathematics
Trigonometric functions and their properties
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Easy · Level 70 · trigonometric-functions,range,cosine
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- ([-1,1]) / ([-1,1]
- ([0,\infty))
- ([1,\infty))
- ((-\infty,-1])
Easy · Level 70 · trigonometric functions,range of tangent,tangent function,class 11 mathematics,real numbers
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- \([-1,1]\)
- \([0,\infty)\)
- \(( -\infty,\infty )\)
- \((0,1)\)
Easy · Level 70 · trigonometric-functions,range,secant
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- ([-1,1])
- ((-\infty,-1]\cup[1,\infty))
- ([0,1])
- ((-\infty,\infty))
Easy · Level 70 · trigonometric functions,range of cosecant,reciprocal trigonometric functions,class 11 mathematics,range and domain
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- \([-1,1]\)
- \(( -\infty,0)\cup(0,\infty)\)
- \(( -\infty,-1)\cup(1,\infty)\)
- \(( -\infty,-1]\cup[1,\infty)\)
Easy · Level 70 · trigonometric-functions,range,cotangent
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- ((-\infty,\infty))
- ([-1,1])
- ([1,\infty))
- ([0,1])
Easy · Level 70 · trigonometric-functions,domain,tangent
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- when (\sin \theta=0)
- when (\cos \theta=0)
- when (\tan \theta=0)
- when (\sec \theta=1)
Easy · Level 70 · trigonometric-functions,domain,secant
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- when (\cos \theta=0)
- when (\sin \theta=0)
- when (\cot \theta=0)
- when (\cos \theta=1)
Easy · Level 70 · trigonometric-functions,domain,cosecant
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- when (\cos \theta=0)
- when (\tan \theta=1)
- when (\sin \theta=0)
- when (\sec \theta=1)
Easy · Level 70 · trigonometric-functions,domain,cotangent
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- when (\cos \theta=0)
- when (\sec \theta=0)
- when (\tan \theta=1)
- when (\sin \theta=0)
Easy · Level 70 · trigonometric-functions,angle-transformation,sine
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- -\(\sin \theta\)
- \(\sin \theta\)
- \(\cos \theta\)
- -\(\cos \theta\)
Easy · Level 70 · trigonometric-functions,angle-transformation,cosine
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- \(\cos \theta\)
- -\(\cos \theta\)
- \(\sin \theta\)
- -\(\sin \theta\)
Easy · Level 70 · trigonometric-functions,period,angle-transformation
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- -(\tan \theta)
- (\cot \theta)
- (\tan \theta)
- -(\cot \theta)
Easy · Level 70 · trigonometric-functions,period,sine
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- -(\sin \theta)
- (\cos \theta)
- -(\cos \theta)
- (\sin \theta)
Easy · Level 70 · trigonometric functions,cosine,periodicity,trigonometric identities,class 11 mathematics
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- \(\cos \theta\)
- \(-\cos \theta\)
- \(\sin \theta\)
- \(-\sin \theta\)
Easy · Level 70 · trigonometric-functions,angle-transformation,sine
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- -\(\sin \theta\)
- \(\sin \theta\)
- \(\cos \theta\)
- -\(\cos \theta\)
Easy · Level 70 · trigonometric-functions,angle-transformation,cosine
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- \(\cos \theta\)
- \(\sin \theta\)
- -\(\cos \theta\)
- -\(\sin \theta\)
Easy · Level 70 · trigonometric-functions,angle-transformation,tangent
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- \(\tan \theta\)
- \(\cot \theta\)
- -\(\cot \theta\)
- -\(\tan \theta\)
Easy · Level 70 · trigonometric-functions,cofunction,sine-cosine
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- \(\cos \theta\)
- \(\sin \theta\)
- -\(\cos \theta\)
- \(\tan \theta\)
Easy · Level 70 · trigonometric-functions,cofunction,cosine-sine
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- \(\cos \theta\)
- \(\sin \theta\)
- -\(\sin \theta\)
- \(\cot \theta\)
Easy · Level 70 · trigonometric-functions,cofunction,tangent-cotangent
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- \(\tan \theta\)
- -\(\tan \theta\)
- \(\cot \theta\)
- -\(\cot \theta\)
What is the range of \(\tan \theta\)?
Correct answer: C
\(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\). For values of \(\theta\) where \(\cos\theta\ne0\), \(\tan\theta\) can take every real value. Hence its range is \(( -\infty,\infty )\). The interval \([-1,1]\) is the range of \(\sin\theta\) and \(\cos\theta\), not \(\tan\theta\). Exam tip: the range of \(\tan\theta\) is all real numbers, although it is undefined at \(\theta=\frac{\pi}{2}+n\pi\).
View question detailsWhich is the range of (\sec \theta)?
Correct answer: B
Since (\sec \theta=\frac{1}{\cos \theta}), its values do not lie inside ([-1,1]). Its range is ((-\infty,-1]\cup[1,\infty)).
View question detailsWhat is the range of \(\cosec \theta\)?
Correct answer: D
\(\cosec\theta=\frac{1}{\sin\theta}\). Since \(-1\leq\sin\theta\leq1\) and \(\sin\theta\neq0\), the value of \(\cosec\theta\) is either less than or equal to \(-1\), or greater than or equal to 1. At \(\sin\theta=-1\) and \(1\), we get \(\cosec\theta=-1\) and \(1\), so both endpoints are included. Option C incorrectly excludes these endpoints. Exam tip: for reciprocal trigonometric functions, check values for which the denominator becomes zero.
View question detailsWhat is the range of (\cot \theta)?
Correct answer: A
(\cot \theta) can take all real values. (\tan \theta) and (\cot \theta) have the same range.
View question detailsWhen is (\tan \theta) undefined?
Correct answer: B
(\tan \theta=\frac{\sin \theta}{\cos \theta}), so it is undefined when (\cos \theta=0). The denominator must not be zero.
View question detailsWhen is (\sec \theta) undefined?
Correct answer: A
(\sec \theta=\frac{1}{\cos \theta}), so it is undefined when (\cos \theta=0). Check the denominator in reciprocal functions.
View question detailsWhen is (\cosec \theta) undefined?
Correct answer: C
(\cosec \theta=\frac{1}{\sin \theta}), so it is undefined when (\sin \theta=0). A function is not defined when the denominator is zero.
View question detailsWhen is (\cot \theta) undefined?
Correct answer: D
(\cot \theta=\frac{\cos \theta}{\sin \theta}), so it is undefined when (\sin \theta=0). In quotient functions, identify the denominator.
View question detailsWhat is \(\sin(\pi+\theta)\) equal to?
Correct answer: A
\(\sin(\pi+\theta)=-\sin \theta\). In the third quadrant, \(\sin \theta\) is negative.
View question detailsWhat is \(\cos(\pi+\theta)\) equal to?
Correct answer: B
\(\cos(\pi+\theta)=-\cos \theta\). In the third quadrant, \(\cos \theta\) is negative.
View question detailsWhat is (\tan(\pi+\theta)) equal to?
Correct answer: C
The period of (\tan \theta) is (\pi), so (\tan(\pi+\theta)=\tan \theta). Knowing periods makes transformations easier.
View question detailsWhat is (\sin(2\pi+\theta)) equal to?
Correct answer: D
The period of (\sin \theta) is (2\pi), so the value remains (\sin \theta). Adding a full cycle does not change (\sin \theta).
View question detailsWhat is \(\cos(2\pi+\theta)\) equal to?
Correct answer: A
The cosine function has period \(2\pi\). Therefore, \(\cos(\theta+2\pi)=\cos\theta\), so \(\cos(2\pi+\theta)=\cos\theta\). The expression \(-\cos\theta\) results when \(\pi\) is added to the angle: \(\cos(\theta+\pi)=-\cos\theta\). Exam tip: Adding or subtracting any integral multiple of \(2\pi\) leaves the values of sine and cosine unchanged.
View question detailsWhat is \(\sin(\pi-\theta)\) equal to?
Correct answer: B
\(\sin(\pi-\theta)=\sin \theta\). In the second quadrant, \(\sin \theta\) is positive.
View question detailsWhat is \(\cos(\pi-\theta)\) equal to?
Correct answer: C
\(\cos(\pi-\theta)=-\cos \theta\). In the second quadrant, \(\cos \theta\) is negative.
View question detailsWhat is \(\tan(\pi-\theta)\) equal to?
Correct answer: D
\(\tan(\pi-\theta)=-\tan \theta\). In the second quadrant, \(\tan \theta\) is negative.
View question detailsWhat is \(\sin\left(\frac{\pi}{2}-\theta\right)\) equal to?
Correct answer: A
\(\sin\left(\frac{\pi}{2}-\theta\right)=\cos \theta\) is a cofunction relation. In transformations with \(\frac{\pi}{2}\), the function changes.
View question detailsWhat is \(\cos\left(\frac{\pi}{2}-\theta\right)\) equal to?
Correct answer: B
\(\cos\left(\frac{\pi}{2}-\theta\right)=\sin \theta\). In cofunction relations, \(\sin \theta\) and \(\cos \theta\) interchange.
View question detailsWhat is \(\tan\left(\frac{\pi}{2}-\theta\right)\) equal to?
Correct answer: C
\(\tan\left(\frac{\pi}{2}-\theta\right)=\cot \theta\). The cofunction of \(\tan \theta\) is \(\cot \theta\).
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