If (\sec \theta=2) and (\theta) is in the first quadrant, what is (\cos \theta)?
Correct answer: B
( \sec \theta=\frac{1}{\cos \theta}), so ( \cos \theta=\frac{1}{2}). In exams remember reciprocal relations.
View question detailsMathematics
Trigonometric functions and their properties
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Medium · Level 72 · trigonometric-functions,reciprocal,secant
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- (2)
- (\frac{1}{2})
- (-\frac{1}{2})
- (0)
Medium · Level 72 · trigonometric-functions,reciprocal,cosecant
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- (\frac{5}{2})
- (-\frac{2}{5})
- (\frac{2}{5})
- (\frac{3}{5})
Medium · Level 72 · trigonometric-functions,reciprocal,product
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- (0)
- (1)
- (\sin^2 \theta)
- (\cosec^2 \theta)
Medium · Level 72 · trigonometric-functions,reciprocal,product
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- (\cos^2 \theta)
- (1)
- (0)
- (\sec^2 \theta)
Medium · Level 72 · trigonometric-functions,reciprocal,tangent-cotangent
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- (\tan^2 \theta)
- (\cot^2 \theta)
- (1)
- (0)
Medium · Level 72 · trigonometric-functions,odd-function,simplification
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- (2\sin \theta)
- (0)
- (\sin \theta)
- (1)
Medium · Level 72 · trigonometric-functions,even-function,simplification
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- (0)
- (2\cos \theta)
- (-2\cos \theta)
- (1)
Medium · Level 72 · trigonometric-functions,allied-angles,simplification
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- (\tan \theta)
- (2\tan \theta)
- (0)
- (1)
Medium · Level 72 · trigonometric-functions,double-angle,identity
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- (\cos 2\theta)
- (-\cos 2\theta)
- (1)
- (\sin 2\theta)
Medium · Level 72 · trigonometric-functions,max-value,first-quadrant
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- (\frac{\pi}{6})
- (\frac{\pi}{4})
- (\frac{\pi}{3})
- (\frac{\pi}{2})
Medium · Level 72 · trigonometric-functions,max-min,linear-combination
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- (1)
- \(\sqrt{2}\)
- (2)
- (0)
Medium · Level 72 · trigonometric-functions,max-min,amplitude
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- (-\sqrt{2})
- (0)
- (\sqrt{2})
- (1)
Medium · Level 72 · trigonometric-functions,max-value,linear-combination
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- (5)
- (7)
- (1)
- (\sqrt{7})
Medium · Level 72 · trigonometric-functions,min-value,linear-combination
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- (-17)
- (-13)
- (13)
- (17)
Medium · Level 72 · trigonometric-functions,quadrant,signs
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- (\sin \theta) positive and (\cos \theta) positive
- (\sin \theta) negative and (\cos \theta) negative
- (\sin \theta) positive and (\cos \theta) negative
- (\sin \theta) negative and (\cos \theta) positive
Medium · Level 72 · trigonometric-functions,periodicity,sine
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- (\frac{2\pi}{3})
- (\pi)
- (2\pi)
- (\frac{\pi}{3})
Medium · Level 72 · trigonometric-functions,general-solution,cosine
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- (\theta=2n\pi)
- (\theta=(2n+1)\pi)
- (\theta=n\pi+\frac{\pi}{2})
- (\theta=\frac{n\pi}{2})
Medium · Level 72 · trigonometric-functions,periodicity,cosine
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- (\frac{\pi}{2})
- (\pi)
- (2\pi)
- (\frac{\pi}{4})
Medium · Level 72 · trigonometric-functions,quadrant,reciprocal-signs
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- (\sec \theta) positive and (\cosec \theta) negative
- (\sec \theta) negative and (\cosec \theta) positive
- (\sec \theta) positive and (\cosec \theta) positive
- (\sec \theta) negative and (\cosec \theta) negative
Medium · Level 72 · trigonometric-functions,periodicity,tangent
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- (-\tan \theta)
- (\cot \theta)
- (\tan \theta)
- (-\cot \theta)
If (\cosec \theta=\frac{5}{2}) and (\theta) is in the first quadrant, what is (\sin \theta)?
Correct answer: C
( \cosec \theta=\frac{1}{\sin \theta}), so ( \sin \theta=\frac{2}{5}). In exams remember cosecant and sine are reciprocals.
View question detailsWhat is the value of (\sin \theta \cdot \cosec \theta), when both are defined?
Correct answer: B
( \cosec \theta=\frac{1}{\sin \theta}), so the product is (1). In exams do not ignore the defined condition.
View question detailsWhat is the value of (\cos \theta \cdot \sec \theta), when both are defined?
Correct answer: B
Since ( \sec \theta=\frac{1}{\cos \theta}), the product becomes (1). In exams identify reciprocal pairs quickly.
View question detailsWhat is the value of (\tan \theta \cdot \cot \theta), when both are defined?
Correct answer: C
( \cot \theta=\frac{1}{\tan \theta}), so the product is (1). In exams reciprocal formulas save time.
View question detailsIf (\sin \theta+\sin(-\theta)=p), what is the value of (p)?
Correct answer: B
( \sin(-\theta)=-\sin \theta), so both terms cancel. In exams odd function property solves such questions quickly.
View question detailsIf (\cos \theta-\cos(-\theta)=q), what is the value of (q)?
Correct answer: A
( \cos(-\theta)=\cos \theta), so the difference is (0). In exams keep the even function property in mind.
View question detailsIf (\tan \theta+\tan(\pi-\theta)=r), what is the value of (r)?
Correct answer: C
( \tan(\pi-\theta)=-\tan \theta), so the sum is (0). In exams allied angle sign gives the answer quickly.
View question detailsWhat is (\sin^2 \theta-\cos^2 \theta) equal to?
Correct answer: B
Because ( \cos 2\theta=\cos^2 \theta-\sin^2 \theta), the given form is (-\cos 2\theta). In exams read double angle identities in reverse too.
View question detailsIf (\sin \theta+\cos \theta=\sqrt{2}) and (0<\theta<\frac{\pi}{2}), what is the value of (\theta)?
Correct answer: B
In the first quadrant, the maximum of ( \sin \theta+\cos \theta) is ( \sqrt{2}) at ( \theta=\frac{\pi}{4}). In exams use symmetry.
View question detailsWhat is the maximum value of \(\sin \theta+\cos \theta\)?
Correct answer: B
\( \sin \theta+\cos \theta=\sqrt{2}\sin\left(\theta+\frac{\pi}{4}\right)\), so the maximum is \( \sqrt{2}\). In exams use maximum of (a\sin x+b\cos x) as \( \sqrt{a^2+b^2}\).
View question detailsWhat is the minimum value of (\sin \theta-\cos \theta)?
Correct answer: A
The amplitude of ( \sin \theta-\cos \theta) is ( \sqrt{1^2+(-1)^2}=\sqrt{2}). In exams the minimum is the negative amplitude.
View question detailsIf the maximum value of (3\sin \theta+4\cos \theta) is (M), what is (M)?
Correct answer: A
The maximum of (a\sin \theta+b\cos \theta) is ( \sqrt{a^2+b^2}), so (M=5). In exams add the squares of coefficients.
View question detailsIf the minimum value of (5\cos \theta-12\sin \theta) is (m), what is (m)?
Correct answer: B
The amplitude is ( \sqrt{5^2+(-12)^2}=13), so the minimum is (-13). In exams maximum is positive amplitude and minimum is negative amplitude.
View question detailsIf (\theta) is in the third quadrant, what are the signs of (\sin \theta) and (\cos \theta)?
Correct answer: B
In the third quadrant, both sine and cosine are negative. In exams use the ASTC rule to decide signs quickly.
View question detailsWhat is the fundamental period of (\sin 3\theta)?
Correct answer: A
The period of ( \sin k\theta) is ( \frac{2\pi}{k}), so here it is ( \frac{2\pi}{3}). In exams use the coefficient in the period formula.
View question detailsIf (\cos \theta=-1), what is the general solution for (\theta)?
Correct answer: B
( \cos \theta=-1) occurs at odd multiples of ( \pi). In exams distinguish the solutions of ( \cos \theta=1) and ( \cos \theta=-1).
View question detailsWhat is the fundamental period of (\cos(4\theta))?
Correct answer: A
The fundamental period of ( \cos k\theta) is ( \frac{2\pi}{k}), so here it is ( \frac{\pi}{2}). In exams put the coefficient in the denominator.
View question detailsIf (\theta) is in the fourth quadrant, what are the signs of (\sec \theta) and (\cosec \theta)?
Correct answer: A
In the fourth quadrant, (\cos \theta) is positive and (\sin \theta) is negative, so their reciprocals have the same signs. In exams decide the signs of basic functions first.
View question detailsWhat is (\tan(5\pi+\theta)) equal to?
Correct answer: C
The fundamental period of (\tan \theta) is (\pi), and (5\pi) is a multiple of the period, so the value does not change. In exams simplify tangent by removing multiples of (\pi).
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