Trigonometry Questions and answers

Topics

General Knowledge
General Science
General English
Aptitude
General Computer Science
General Intellingence and Reasoning
Current Affairs
Exams
Elementary Mathematics
English Literature

Elementary Mathematics

41). If \( \Large tan \ 2 \theta . tan \ 4 \theta =1 \), then the value of \( \Large 3 \theta \) is

A). \( \Large \sqrt{3} \)

B). 0

C). 1

D). \( \Large \frac{1}{\sqrt{3}} \)

View Answer

Correct Answer: 1

\( \Large tan \ 2 \theta =\frac{1}{tan4 \theta } \)

\( \Large tan2 \theta =cot4 \theta \)

\( \Large => \ tan2 \theta =tan(90 ^{\circ} -4 \theta ) \)

\( \Large => \ 2 \theta =90 ^{\circ} -4 \theta \)

\( \Large => \ 6 \theta =90 ^{\circ} \)

\( \Large => \ \theta =15 ^{\circ} \)

\( \Large tan3 \theta =tan(3 \times 15 ^{\circ} )=tan45 ^{\circ} =1 \)

42). If \( \Large sin \theta +cosec \theta =2 \), then the value of \( \Large sin^{5} \theta +cosec^{5} \theta \), when \( \Large 0 ^{\circ} \le \theta \le 90 ^{\circ} \), is

A). 0

B). 1

C). 10

D). 2

View Answer

Correct Answer: 2

\( \Large sin \theta +cosec \theta =2 \)

\( \Large => \ sin \theta +\frac{1}{sin \theta }=2 \)

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

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

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

\( \Large => \ sin \theta =1 \)

\( \Large => \ cosec \theta =1 \)

Value of \( \Large sin^{5} \theta +cosec^{5} \theta \)

\( \Large = \ (1)^{5}+(1)^{5} \)

= 1+1 = 2

43). If A=\( \Large sin^{2} \theta +cos^{4} \theta \) ; for any value of \( \Large \theta \), then the value of A is

A). \( \Large 1 \le A \le 2 \)

B). \( \Large \frac{3}{4} \le A \le 1 \)

C). \( \Large \frac{13}{16} \le A \le 1 \)

D). \( \Large \frac{3}{4} \le A \le \frac{13}{16} \)

View Answer

Correct Answer: \( \Large \frac{3}{4} \le A \le 1 \)

When \( \Large \theta =0 ^{\circ} \)

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

When \( \Large \theta =45 ^{\circ} \)

\( \Large sin^{2} \theta +cos^{4} \theta =\frac{1}{2}+\frac{1}{4}=\frac{3}{4} \)

When \( \Large \theta =30 ^{\circ} \)

\( \Large sin^{2} \theta +cos^{4} \theta =\frac{1}{4}+\frac{9}{16}=\frac{13}{16} \)

Average value of A

\( \Large => \ \frac{3}{4} \le A \le 1 \)

44). The minimum value of \( \Large 2 \ sin^{2} \theta +3 \ cos^{2} \theta \) is

A). 0

B). 3

C). 2

D). 1

View Answer

Correct Answer: 3

\( \Large 2 \ sin^{2} \theta +3 \ cos^{2} \theta \)

\( \Large 2sin^{2} \theta +2cos^{2} \theta+cos^{2} \theta \)

\( \Large 2(sin^{2} \theta +cos^{2} \theta )+cos^{2} \theta \)

\( \Large 2+cos^{2} \theta \)

Minimum value of \( \Large cos \theta =-1 \)

Required minimum value

=\( \Large 2+(-1)^{2}=2+1=3 \)

45). The value of cot 10\( ^{\circ} \) . cot 20\( ^{\circ} \) . cot 60 \( ^{\circ} \) . cot 70\( ^{\circ} \) . cot 80 \( ^{\circ} \) is

A). 1

B). -1

C). \( \Large \sqrt{3} \)

D). \( \Large \frac{1}{\sqrt{3}} \)

View Answer

Correct Answer: \( \Large \frac{1}{\sqrt{3}} \)

cot 10\( ^{\circ} \) .cot 80 \( ^{\circ} \) .cot 20\( ^{\circ} \) .cot 70 \( ^{\circ} \) .cot 60\( ^{\circ} \)

= cot 10\( ^{\circ} \) .tan10\( ^{\circ} \) .cot 20\( ^{\circ} \) .tan 20\( ^{\circ} \) .cot 60\( ^{\circ} \)

= \( \Large 1.1.\frac{1}{\sqrt{3}} =\frac{1}{\sqrt{3}} \)

46). If \( \Large cos \theta \) cosec 23\( ^{\circ} \) = 1, then the value of \( \Large \theta \) is

A). 23 \( ^{\circ} \)

B). 37\( ^{\circ} \)

C). 63\( ^{\circ} \)

D). 67\( ^{\circ} \)

View Answer

Correct Answer: 67\( ^{\circ} \)

\( \Large cos \theta .cosec23 ^{\circ} =1 \)

\( \Large => \ cosec23 ^{\circ} =\frac{1}{cos \theta }=sec \theta \)

\( \Large => \ cosec23 ^{\circ} =cosec(90 ^{\circ} - \theta ) \)

\( \Large => \ 23 ^{\circ} =90 ^{\circ} - \theta \)

\( \Large => \ \theta =90 ^{\circ} -23 ^{\circ} =67 ^{\circ} \)

47). If \( \Large \theta \) is a positive acute angle and \( \Large tan \ 2 \theta \ tan \ 3 \theta =1 \) , then the value of \( \Large \left(2 \ cos^{2}\frac{5 \theta }{2}-1\right) \) is

A). \( \Large -\frac{1}{2} \)

B). 1

C). 0

D). \( \Large \frac{1}{2} \)

View Answer

Correct Answer: 0

\( \Large tan \ 2 \theta .tan \ 3 \theta =1 \)

\( \Large => \ tan3 \theta =\frac{1}{tan2 \theta }=cot2 \theta \)

\( \Large => \ tan3 \theta =tan(90 ^{\circ} -2 \theta ) \)

\( \Large 3 \theta =90 ^{\circ} -2 \theta \)

\( \Large => \ 5 \theta =90 ^{\circ} \ => \theta =\frac{90 ^{\circ} }{5}=18 ^{\circ} \)

\( \Large 2cos^{2} \frac{5 \theta }{2}-1=2cos^{2}\frac{5 \times 18}{2}-1 \)

= \( \Large 2cos^{2}45 ^{\circ} -1 \)

= \( \Large 2 \times \left(\frac{1}{\sqrt{5}}\right)^{2}-1 \)

\( \Large => \ 2 \times \frac{1}{2}-1=0 \)

48). If \( \Large 0 ^{\circ} < \theta <90 ^{\circ} \), then the value of \( \Large sin \theta +cos \theta \) is

A). equal to 1

B). greater than 1

C). less than 1

D). equal to 2

View Answer

Correct Answer: greater than 1

Let X =\( \Large sin \theta +cos \theta \)

On squaring both side, we get

\( \Large X^{2}=sin^{2} \theta +cos^{2} \theta +2sin \theta .cos \theta \)

= \( \Large 1+2sin \theta .cos \theta \)

\( \Large 0< \theta <90 ^{\circ} \)

Value of both \( \Large sin \theta \) and \( \Large cos \theta \) will be less than 1.

\( \Large 2sin \theta .cos \theta <1 \)

\( \Large X^{2}<2 \ => \ X<\sqrt{2} \)

\( \Large X<1.414 \)

49). If \( \Large sin \theta +cosec \theta=2 \), then value of \( \Large sin^{100} \theta +cosec^{100} \theta \) is equal to

A). 1

B). 2

C). 3

D). 100

View Answer

Correct Answer: 2

\( \Large sin \theta +cosec \theta =2 \)

\( \Large => \ sin \theta +\frac{1}{sin \theta }=2 \)

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

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

\( \Large => \ sin \theta -1=0 \)

\( \Large => \ sin \theta =1 \)

\( \Large => \ cosec \theta =1 \)

\( \Large sin^{100} \theta +cosec^{100} \theta \)

=\( \Large (1)^{100}+(1)^{100} \)

=1+1=2

50). If \( \Large \frac{sin \theta +cos \theta }{sin \theta -cos \theta }=3 \),then the value of \( \Large sin \theta -cos^{4} \theta \) is

A). \( \Large \frac{1}{5} \)

B). \( \Large \frac{2}{5} \)

C). \( \Large \frac{3}{5} \)

D). \( \Large \frac{4}{5} \)

View Answer

Correct Answer: \( \Large \frac{3}{5} \)

\( \Large \frac{sin \theta +cos \theta }{sin \theta -cos \theta }=3 \)

\( \Large => \ sin \theta +cos \theta =3sin \theta -3cos \theta \)

\( \Large => \ 4cos \theta =2sin \theta \)

\( \Large => \ tan \theta =2 \)

Value of \( \Large sin ^{4} \theta -cos^{4} \theta \)

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

= \( \Large sin^{2} \theta -cos^{2} \theta \)

= \( \Large cos^{2} \theta (tan^{2} \theta -1) \)

= \( \Large \frac{tan^{2} \theta -1}{sec^{2} \theta } \)

= \( \Large \frac{tan^{2} \theta -1}{1+tan^{2} \theta }=\frac{4-1}{1+4}=\frac{3}{5} \)

2	3	4	5	6	7	8

Go to :

Total Pages : 39