SSC Quantitative Aptitude Trigonometry Question Bank

done Trigonometry (I) Total Questions - 53

• question_answer1)
  If \[\sec A+\tan A=a,\] then the value of \[\cos A\] is [SSC CGL Tier II, 2017]
  A)
  \[\frac{{{a}^{2}}+1}{2a}\] done clear
  B)
  \[\frac{2a}{{{a}^{2}}+1}\] done clear
  C)
  \[\frac{{{a}^{2}}-1}{2a}\] done clear
  D)
  \[\frac{2a}{{{a}^{2}}-1}\] done clear
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• question_answer2)
  If \[\sin P+\text{cosec}\,P=2,\]then the value of \[{{\sin }^{7}}P+\cos e{{c}^{7}}P\] is   [SSC CGL Tier II, 2017]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer3)
  If \[\cos \theta =\frac{{{x}^{2}}-{{y}^{2}}}{{{x}^{2}}+{{y}^{2}}},\]then the value of \[\cot \theta \] is equal to \[[If\,0\le \,\,\theta \le \,90{}^\circ ]\][SSC CGL Tier II, 2017]
  A)
  \[\frac{2xy}{{{x}^{2}}-{{y}^{2}}}\] done clear
  B)
  \[\frac{2xy}{{{x}^{2}}+{{y}^{2}}}\] done clear
  C)
  \[\frac{{{x}^{2}}+{{y}^{2}}}{2xy}\] done clear
  D)
  \[\frac{{{x}^{2}}-{{y}^{2}}}{2xy}\] done clear
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• question_answer4)
  The distance between two pillars is 120 m. The height of one pillar is thrice the other. The angles   of elevation of their tops from the midpoint of the   line connecting their feet are complementary to each other. The height of the taller pillar is (Use\[\sqrt{3}=1.732)\][SSC CGL Tier II, 2017]
  A)
  34.64 m done clear
  B)
  51.96 m done clear
  C)
  69.28 m done clear
  D)
  103.92 m done clear
  View Solution play_arrow

• question_answer5)
  If tan\[A=n\tan B\]and sin \[A=m\sin B,\] \[\sin B,\] then the value of \[{{\cos }^{2}}\,A\]is  [SSC CGL Tier II, 2014]
  A)
  \[\frac{{{m}^{2}}-1}{{{n}^{2}}-1}\] done clear
  B)
  \[\frac{{{m}^{2}}+1}{{{n}^{2}}+1}\] done clear
  C)
  \[\frac{{{m}^{2}}-1}{{{n}^{2}}+1}\] done clear
  D)
  \[\frac{{{m}^{2}}+1}{{{n}^{2}}-1}\] done clear
  View Solution play_arrow

• question_answer6)
  If\[\sin A+{{\sin }^{2}}A=1,\]then the value \[{{\cos }^{2}}A+{{\cos }^{4}}A\]is [SSC CGL Tier II, 2015]
  A)
  2 done clear
  B)
  1 done clear
  C)
   \[1\frac{2}{3}\] done clear
  D)
  \[1\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer7)
  If \[\tan \theta -\cot \theta =0\]and \[\theta \] is positive acute angle, then the value of \[\frac{\tan \,(\theta +15{}^\circ )}{\tan \,(\theta -15{}^\circ )}\] is [SSC CGL Tier II, 2015]
  A)
  \[\sqrt{3}\] done clear
  B)
  \[\frac{1}{\sqrt{3}}\] done clear
  C)
  \[3\] done clear
  D)
  \[\frac{1}{3}\] done clear
  View Solution play_arrow

• question_answer8)
  The value of\[\cot 41{}^\circ \cdot cot42{}^\circ \cdot cot43{}^\circ \cdot cot44{}^\circ .\]\[cot45{}^\circ \cdot cot46{}^\circ \cdot cot47{}^\circ \cdot cot48{}^\circ \cdot cot49{}^\circ .\]is [SSC CGL Tier II, 2015]
  A)
  \[\frac{\sqrt{3}}{2}\] done clear
  B)
  \[0\] done clear
  C)
  \[\frac{1}{\sqrt{2}}\] done clear
  D)
  \[1\] done clear
  View Solution play_arrow

• question_answer9)
  If \[\sin \theta +cos\theta -\sqrt{2}\,\cos \theta ,\]then the value of \[(cos\theta -sin\theta )\] is
  A)
  \[\sqrt{3}\,\cos \theta \] done clear
  B)
  \[\sqrt{2}\,\sin \theta \] done clear
  C)
  \[\sqrt{2}\,\cos \theta \] done clear
  D)
  \[\sqrt{3}\,\sin \theta \] done clear
  View Solution play_arrow

• question_answer10)
  \[\frac{\sin A}{1+\cos \,A}+\frac{\sin A}{1-\cos A}\]is \[(0{}^\circ <A<90{}^\circ )\]
  A)
  \[2\,\text{cosec}\,A\] done clear
  B)
  \[2\sec A\] done clear
  C)
  \[2\,\sin A\] done clear
  D)
  \[2\cos A\] done clear
  View Solution play_arrow

• question_answer11)
  If \[r\sin \theta =1,\]\[r\cos \theta =\sqrt{3,}\]then the value of \[(\sqrt{3}\,tan\theta +1)\]is
  A)
  \[\sqrt{3}\] done clear
  B)
  \[\frac{1}{\sqrt{3}}\] done clear
  C)
  1 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer12)
  A tower standing on a horizontal plane subtends   certain angle at a point 160 m apart from the foot of the tower. On advancing 100 m towards it, the tower is found to subtend an angle twice as before. The height of the tower is
  A)
  80 m done clear
  B)
  100 m done clear
  C)
  160 m done clear
  D)
  200 m done clear
  View Solution play_arrow

• question_answer13)
  The angle of elevation of a tower from a distance 50 m from its foot is \[30{}^\circ \]. The height of the tower is
  A)
  \[50\sqrt{3\,m}\] done clear
  B)
  \[50\sqrt{3\,m}\] done clear
  C)
  \[75\sqrt{3}\,m\] done clear
  D)
  \[\frac{75}{\sqrt{3}}\,m\] done clear
  View Solution play_arrow

• question_answer14)
  . The value of 9, which satisfies the equality \[{{\tan }^{2}}\theta +3=3\sec \theta ,\]\[0{}^\circ \le \theta <90{}^\circ ,\]is
  A)
  \[15{}^\circ \] or\[\text{ }0{}^\circ \] done clear
  B)
  \[30{}^\circ \]or \[0{}^\circ \] done clear
  C)
  \[45{}^\circ ~\]or \[0{}^\circ \] done clear
  D)
  \[60{}^\circ \]or\[0{}^\circ \] done clear
  View Solution play_arrow

• question_answer15)
  Value of \[\left( \begin{align}   & {{\sin }^{2}}7\frac{1{}^\circ }{2}+{{\sin }^{2}}82\frac{1{}^\circ }{2}+ \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\tan }^{2}}2{}^\circ \cdot ta{{n}^{2}}88 \\ \end{align} \right)\]is
  A)
  1 done clear
  B)
  2 done clear
  C)
  0 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer16)
  If \[(\sin \theta -\cos \theta )=0,\]then,\[({{\sin }^{4}}\theta -{{\cos }^{4}}\theta )\]is equal to
  A)
  \[1\] done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  \[\frac{1}{4}\] done clear
  D)
  \[\frac{3}{4}\] done clear
  View Solution play_arrow

• question_answer17)
  If \[\tan (A+B)=\sqrt{3}\]and \[\tan (A+B)=\frac{1}{\sqrt{3}},\] where A B and (A + B) is acute, then A is equal to
  A)
  \[15{}^\circ \] done clear
  B)
  \[30{}^\circ \] done clear
  C)
  \[45{}^\circ \] done clear
  D)
  \[60{}^\circ \] done clear
  View Solution play_arrow

• question_answer18)
  \[(si{{n}^{4}}\theta -co{{s}^{4}}\theta +1)\,cose{{c}^{2}}\theta \]is equal to
  A)
  4 done clear
  B)
  3 done clear
  C)
  2 done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer19)
  If \[x\,\sin 45{}^\circ =y\,\text{cosec}\,30{}^\circ ,\]then \[\frac{{{x}^{4}}}{{{y}^{4}}}\] is equal to
  A)
  \[{{4}^{3}}\] done clear
  B)
  \[{{6}^{3}}\] done clear
  C)
  \[{{2}^{3}}\] done clear
  D)
  \[{{8}^{3}}\] done clear
  View Solution play_arrow

• question_answer20)
  The maximum value of \[{{\sin }^{8}}\theta +{{\cos }^{14}}\theta ,\] for all real values of \[\theta \] is
  A)
  \[1\] done clear
  B)
  \[\sqrt{2}\] done clear
  C)
  \[\frac{1}{\sqrt{2}}\] done clear
  D)
  \[0\] done clear
  View Solution play_arrow

• question_answer21)
  If\[=\frac{{{\cos }^{2}}\theta \,(cose{{c}^{2}}\theta +1)}{\text{cose}{{\text{c}}^{\text{2}}}-1}\]\[0{}^\circ <\theta <90{}^\circ \] then tan \[\theta \] equal to
  A)
  \[\sqrt{\frac{l-2}{1-m}}\] done clear
  B)
  \[\sqrt{\frac{2-l}{1-m}}\] done clear
  C)
  \[\sqrt{\frac{l-2}{m-1}}\] done clear
  D)
  \[\sqrt{\frac{l-1}{2-m}}\] done clear
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• question_answer22)
              If \[\sin \,(10{}^\circ 6'32')=a,\] then the value of \[\cos \,(79{}^\circ 53'28'')+tan\,(10{}^\circ 6'32'')\]is
  A)
  \[\frac{a\,(1+\sqrt{1-{{a}^{2}}})}{\sqrt{1-{{a}^{2}}}}\] done clear
  B)
  \[\frac{1+\sqrt{1-{{a}^{2}}}}{\sqrt{1-{{a}^{2}}}}\] done clear
  C)
  \[\frac{\sqrt{1-{{a}^{2}}}+a}{\sqrt{1-{{a}^{2}}}}\] done clear
  D)
  \[\frac{a\sqrt{1-{{a}^{2}}}+1}{\sqrt{1-{{a}^{2}}}}\] done clear
  View Solution play_arrow

• question_answer23)
  The value of\[(tan1{}^\circ tan2{}^\circ tan3{}^\circ ...tan89{}^\circ )\]is
  A)
  undefined done clear
  B)
  0 done clear
  C)
  1 done clear
  D)
  89 done clear
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• question_answer24)
  If \[\sin \,\theta +\text{cosec}\,\theta =2,\]then the value of \[{{\sin }^{7}}\theta +\text{cose}{{\text{c}}^{7}}\theta \]is
  A)
  1 done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  2 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer25)
  \[\tan \frac{\pi }{8}\tan \frac{\pi }{12}\tan \frac{3\pi }{8}\tan \frac{5\pi }{12}-{{\sin }^{2}}\frac{\pi }{6}\]is equal to
  A)
  \[\frac{1}{2}\] done clear
  B)
  \[\frac{2-\sqrt{3}}{2}\] done clear
  C)
  \[\frac{1}{4}\] done clear
  D)
  \[\frac{3}{4}\]

  [d] \[\tan \frac{\pi }{8}\cdot \tan \frac{\pi }{12}\cdot tan\frac{3\pi }{8}.\] \[\tan \frac{5\pi }{12}-{{\sin }^{2}}\frac{\pi }{6}\] \[=\tan \frac{\pi }{8}\cdot \tan \frac{\pi }{12}\cdot \cot \left( \frac{\pi }{2}-\frac{3\pi }{8} \right).\cot \left( \frac{\pi }{2}-\frac{5\pi }{12} \right)-\frac{1}{4}\] \[=\tan \frac{\pi }{8}\cdot tan\frac{\pi }{12}\cot \frac{\pi }{8}\cot \frac{\pi }{12}-\frac{1}{4}\] \[=1-\frac{1}{4}=\frac{3}{4}\]

  done clear
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• question_answer26)
  If\[x\,{{\sin }^{3}}\alpha +y\,{{\cos }^{3}}\alpha =\cos \,\alpha \ne 0\]and\[x\sin \alpha -y\,\cos \,\alpha =0,\]then the value of \[{{x}^{2}}+{{y}^{2}}\]is
  A)
  1 done clear
  B)
  2 done clear
  C)
  4 done clear
  D)
  9 done clear
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• question_answer27)
  If \[5\tan \,\theta =4,\]then the value of \[\left( \frac{5\sin \theta -3\cos \theta }{5\sin \theta +3\cos \theta } \right)\]is
  A)
  \[\frac{1}{7}\] done clear
  B)
  \[\frac{2}{7}\] done clear
  C)
  \[\frac{5}{7}\] done clear
  D)
  \[\frac{2}{5}\] done clear
  View Solution play_arrow

• question_answer28)
  Number of integral values of x for which \[\sin \theta =\frac{4x-3}{9},\] where \[0{}^\circ <9<90{}^\circ ,\]is
  A)
  5 done clear
  B)
  4 done clear
  C)
  3 done clear
  D)
  2 done clear
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• question_answer29)
  Two angles of a triangle are \[\frac{1}{2}\] radian and \[\frac{1}{3}\]radian. The measure of the third angle in degree\[\left( take\,\pi =\frac{22}{7} \right)\]is
  A)
  \[132\frac{2{}^\circ }{11}\] done clear
  B)
  \[132\frac{3{}^\circ }{11}\] done clear
  C)
  \[132{}^\circ \] done clear
  D)
  \[132\frac{1{}^\circ }{11}\] done clear
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• question_answer30)
  If \[x=\sin \theta +\cos \theta \]and\[y=\text{sec}\theta +\text{cosec}\theta ,\]find y in terms of x
  A)
  \[\frac{x}{{{x}^{2}}+1}\] done clear
  B)
  \[\frac{x}{{{x}^{2}}-1}\] done clear
  C)
  \[\frac{2x}{{{x}^{2}}-1}\] done clear
  D)
  \[\frac{2x}{{{x}^{2}}+1}\] done clear
  View Solution play_arrow

• question_answer31)
  At the foot of mountain, the elevation of its summit is \[45{}^\circ .\] After ascending 2 km towards the mountain upon an incline of \[30{}^\circ ,\]the elevation changes to\[60{}^\circ \]. The height of the mountain is
  A)
  \[(\sqrt{3}-1)\,km\] done clear
  B)
  \[(\sqrt{3}+1)\,km\] done clear
  C)
  \[(\sqrt{3}-2)\,km\] done clear
  D)
  \[(\sqrt{3}+2)\,km\] done clear
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• question_answer32)
  The length of a shadow of a vertical tower is \[\frac{1}{\sqrt{3}}\]times its height. The angle of elevation of the Sun is
  A)
  \[30{}^\circ \] done clear
  B)
  \[46{}^\circ \] done clear
  C)
  \[60{}^\circ \] done clear
  D)
  \[90{}^\circ \] done clear
  View Solution play_arrow

• question_answer33)
  If \[{{\sin }^{2}}\alpha +{{\sin }^{2}}\beta =2,\]then the value of \[\cos \frac{\alpha +\beta }{2}\]is
  A)
  1 done clear
  B)
  \[-1\] done clear
  C)
  0 done clear
  D)
  0.5 done clear
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• question_answer34)
  If \[\theta \] be acute angle and \[\cos \theta =\frac{15}{17},\]then the value of \[(90{}^\circ -\theta )\]is
  A)
  \[\frac{2\sqrt{8}}{15}\] done clear
  B)
  \[\frac{8}{15}\] done clear
  C)
  \[\frac{\sqrt{2}}{17}\] done clear
  D)
  \[\frac{8\sqrt{2}}{17}\] done clear
  View Solution play_arrow

• question_answer35)
  The value of \[\cot \,\frac{\pi }{20}\,\cot \frac{3\pi }{20}\,\cot \frac{5\pi }{20}\,\cot \frac{7\pi }{20}\,\cot \frac{9\pi }{20}\]is
  A)
  \[-1\] done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  0 done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer36)
  If \[\sin \theta +\cos \theta =\frac{17}{13},\]\[0{}^\circ <\theta <90{}^\circ ,\]then the value of \[{{\sin }^{2}}\theta -\cos \theta \] is
  A)
  \[\frac{5}{17}\] done clear
  B)
  \[\frac{3}{19}\] done clear
  C)
  \[\frac{7}{10}\] done clear
  D)
  \[\frac{7}{13}\] done clear
  View Solution play_arrow

• question_answer37)
  If\[x{{\sin }^{3}}\theta +y{{\cos }^{^{3}}}\theta =\sin \theta \cos \theta \ne 0\]and \[x\sin \theta -y\cos \theta =0,\]then value of is
  A)
  \[\sin \theta -\cos \theta \] done clear
  B)
  \[\sin \theta +\cos \theta \] done clear
  C)
  \[0\] done clear
  D)
  \[1\] done clear
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• question_answer38)
  If \[0\le \alpha \le \frac{\pi }{2}\]and \[2\sin \alpha +15{{\cos }^{2}}\alpha =7,\] then the value of \[\cot \alpha \] is
  A)
  \[\frac{5}{4}\] done clear
  B)
  \[\frac{3}{4}\] done clear
  C)
  \[\frac{1}{4}\] done clear
  D)
  \[\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer39)
  If  \[\tan \theta \cdot \tan 2\theta =1,\]then the value of \[{{\sin }^{2}}2\theta +{{\tan }^{2}}2\theta \] is equal to
  A)
  \[\frac{3}{4}\] done clear
  B)
  \[\frac{10}{3}\] done clear
  C)
  \[3\frac{3}{4}\] done clear
  D)
  \[3\] done clear
  View Solution play_arrow

• question_answer40)
  If \[\sin \theta +{{\sin }^{2}}\theta -1,\]then the value of \[{{\cos }^{12}}\theta +3{{\cos }^{10}}\theta +3{{\cos }^{8}}\theta +{{\cos }^{6}}\theta -1\]is
  A)
  \[0\] done clear
  B)
  \[1\] done clear
  C)
  \[-1~\] done clear
  D)
  \[2\] done clear
  View Solution play_arrow

• question_answer41)
  If \[x=\cos ec\,\theta -\sin \theta \] and \[y=\sec \theta -\cos \theta ,\] then the value of \[{{z}^{2}}{{y}^{2}}({{x}^{2}}+{{y}^{2}}+3)\] is
  A)
  0 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  3 done clear
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• question_answer42)
  If \[\tan (x+y)tan(x-y)=1,\]then the value of \[\tan \left( \frac{2x}{3} \right)\]is
  A)
  \[\frac{1}{\sqrt{3}}\] done clear
  B)
  \[\frac{2}{\sqrt{3}}\] done clear
  C)
  \[\sqrt{3}\] done clear
  D)
  \[1\] done clear
  View Solution play_arrow

• question_answer43)
  If \[0\le \theta \le \frac{\pi }{2},\]\[2y\cos \theta =x\]and \[2\,x\sec \theta -y\,\text{cosec}\,\theta =3,\] then the value of \[{{x}^{2}}+4{{y}^{2}}\] is
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer44)
  The least value of \[(4\text{ se}{{\text{c}}^{2}}\theta +9\,\text{cose}{{\text{c}}^{2}}\theta )\]is
  A)
  1 done clear
  B)
  19 done clear
  C)
  25 done clear
  D)
  7 done clear
  View Solution play_arrow

• question_answer45)
  From the top of a cliff 90 m high, the angles of depression of the top and bottom of a tower are observed to be \[30{}^\circ \]and \[60{}^\circ \]respectively. The height of the y tower is
  A)
  60 m done clear
  B)
  75 m done clear
  C)
  30 m done clear
  D)
  45 m done clear
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• question_answer46)
  When the angle of elevation of the sun increases from \[30{}^\circ \] to \[60{}^\circ ,\]the shadow of a post is diminished by 5 m. Then, the height of the post is
  A)
  \[\frac{5\sqrt{3}}{2}\,m\] done clear
  B)
  \[\frac{2\sqrt{3}}{5}\,m\] done clear
  C)
  \[\frac{2}{5\sqrt{3}}\,m\] done clear
  D)
  \[\frac{4}{5\sqrt{3}}\,m\] done clear
  View Solution play_arrow

• question_answer47)
  One flies a kite with a thread 150 m long. If the thread of the kite makes an angle of \[60{}^\circ \] with the horizontal line, then the height of the kite from the ground (assuming the thread to be in a straight line) is
  A)
  \[50\,m\] done clear
  B)
  \[75\sqrt{3}\,m\] done clear
  C)
  \[25\sqrt{3}\,m\] done clear
  D)
  \[80\,m\] done clear
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• question_answer48)
  A man from the top of a 100 m high tower sees a car moving towards the tower at an angle of depression of\[30{}^\circ \]. After some time, the angle of depression becomes\[60{}^\circ \]. The distance (in metre) travelled by the car during this time is
  A)
  \[100\sqrt{3}\] done clear
  B)
  \[\frac{200\sqrt{3}}{3}\] done clear
  C)
  \[\frac{100\sqrt{3}}{3}\] done clear
  D)
  \[200\sqrt{3}\] done clear
  View Solution play_arrow

• question_answer49)
  The angles of elevation of the top of a tower from two points A and B lying on the horizontal through the foot of the tower are \[15{}^\circ \] and \[30{}^\circ \] respectively. If A and B are on the same side of the tower and AB = 48 m, then the height of the tower is
  A)
  \[24\sqrt{3}\text{ }m\] done clear
  B)
  \[24\,m\] done clear
  C)
  \[24\sqrt{2}\,m\] done clear
  D)
  \[96\,m\] done clear
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• question_answer50)
  The angles of elevation of the top of a building and the top of the chimney on the roof of the building from a point on the ground are x and\[45{}^\circ \], respectively. The height of building is h m. Then, the height of the chimney is (in metre)
  A)
  \[h\,\,\cot x+h\] done clear
  B)
  \[h\,\cot x-h\] done clear
  C)
  \[h\tan x-h\] done clear
  D)
  \[h\tan x+h\] done clear
  View Solution play_arrow

• question_answer51)
  The angles of elevation of the top of a building from the top and bottom of a tree are x and y, respectively. If the height of the tree is h m, then the height of the building is (in metre)
  A)
  \[\frac{h\cot x}{\cot x+\cot y}\] done clear
  B)
  \[\frac{h\cot \,y}{\cot x+\cot y}\] done clear
  C)
  \[\frac{h\cot x}{\cot x-\cot y}\] done clear
  D)
  \[\frac{h\cot y}{\cot y-\cot y}\] done clear
  View Solution play_arrow

• question_answer52)
  If the angle of elevation of the sun changes from \[30{}^\circ \] to \[45{}^\circ ,\]the length of the shadow of a pillar decreases by 20 m. The height of the pillar is
  A)
  \[20\,(\sqrt{3}-1)\,m\] done clear
  B)
  \[20\,(\sqrt{3}+1)\,m\] done clear
  C)
  \[10\,(\sqrt{3}-1)\,m\] done clear
  D)
  \[10\,(\sqrt{3}+1)\,m\] done clear
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• question_answer53)
  The shadow of a tower is 15 m when the sun's elevation is \[30{}^\circ .\] What is the length of the shadow when the sun's elevation is\[60{}^\circ ?\]
  A)
  3 m done clear
  B)
  4 m done clear
  C)
  5 m done clear
  D)
  6 m done clear
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