15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )

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1 - TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the circumrdius of the tringle ABC, then ( r R ) equls b c b b c c [ AIEEE 00 ] ( ) If cos - - cos - y = α, then - y cos α y is equl to sin α sin α - sin α [ AIEEE 00 ] ( ) If in tringle ABC, the ltitudes from the vertices A, B, C on opposite sides re in H.P., then sin A, sin B, sin C re in G.P. A.P. Arithmetic-Geometric Progression H.P. [ AIEEE 00 ] ( ) Let α, β be such tht < α - β <. If sin α sin β = -, then the vlue of cos α - β is [ AIEEE 00 ] ( ) If u = sin cos θ b sin θ sin θ b sin θ, then difference between the mimum nd minimum vlues of u is given by ( b ) b ( b ) ( - b ) [ AIEEE 00 ] ( 7 ) The sides of tringle re sin α, cos α nd sin α cos α for some 0 < α < Then the gretest ngle of the tringle is [ AIEEE 00 ]

2 - TRIGONOMETRY Pge ( 8 ) A person stnding on the bnk of river observes tht the ngle of elevtion of the top of tree on the opposite bnk of river is 0 nd when he retires 0 m wy from the tree, the ngle of elevtion becomes 0. The bredth of the river is 0 m 0 m 0 m 0 m [ AIEEE 00 ] ( 9 ) If in tringle cos C c cos A = b, then the sides, b nd c re in A. P. in G. P. in H. P. stisfy b = c [ AIEEE 00 ] ( 0 ) The sum of the rdii of inscribed nd circumscribed circles, for n n sided regulr polygon of side, is cot n b cot n cot n cot n [ AIEEE 00 ] ( ) The upper th portion of verticl pole subtends n ngle tn - t point in the horizontl plne through its foot nd t distnce 0 m from the foot. The height of the verticl pole is 0 m 0 m 0 m 80 m [ AIEEE 00 ] ( ) The vlue of cos α cos ( α 0 ) cos ( α - 0 ) is 0 [ AIEEE 00 ] ( ) The trigonometric eqution sin - = sin - hs solution for l l < l l < l l < ll rel vlues of [ AIEEE 00 ] θ - φ ( ) If sin θ sin φ = nd cos θ cos φ = b, then the vlue of tn is - b - - b b b - b b b [ AIEEE 00 ] b

3 - TRIGONOMETRY Pge ( ) If tn - ( ) cot - ( ) =, then the vlue of is - [ AIEEE 00 ] ( ) The vlue of tn - tn - 7 tn - tn - n n is 0 [ AIEEE 00 ] ( 7 ) The ngles of elevtion of the top of tower ( A ) from the top ( B ) nd bottom ( D ) t building of height re 0 nd respectively. If the tower nd the building stnd t the sme level, then the height of the tower is - ( ) ( - ) [ AIEEE 00 ] ( 8 ) If cos ( α - β ) = nd cos ( α β ) = e, - α, β, then the number of ordered pirs ( α, β ) = 0 [ IIT 00 ] ( 9 ) Which of the following is correct for tringle ABC hving sides, b, c opposite to the ngles A, B, C respectively B - C sin B C ( b c ) sin = ( b - c ) cos A B C sin A B - C = cos sin = ( b c ) cos A = cos A [ IIT 00 ] ( 0 ) Three circles of unit rdii re inscribed in n equilterl tringle touching the sides of the tringle s shown in the figure. Then, the re of the tringle is [ IIT 00 ]

4 - TRIGONOMETRY Pge ( ) If θ nd φ re cute ngles such tht sin θ = nd cos θ =, then θ nd φ lies in,,,, [ IIT 00 ] ( ) For which vlue of, sin [ cot - ( ) ] = cos ( tn - )? 0 - [ IIT 00 ] ( ) If, b, c re the sides of tringle such tht : b : c = : :, then A : B : C is : : : : : : : : [ IIT 00 ] ( ) Vlue of equl to tn α, > 0, α 0, is lwys greter thn or tn α sec α [ IIT 00 ] ( ) If the ngles of tringle re in the rtio : :, then the rtio of the lrgest side to the perimeter is equl to : : : : [ IIT 00 ] ( ) The nturl domin of - sin ( ) for ll R, is -, -, -, -, [ IIT 00 ] ( 7 ) The length of longest intervl in which the function sin - sin is incresing is [ IIT 00 ] ( 8 ) Which of the following pieces of dt does NOT uniquely determine n cute-ngled tringle ABC ( R being the rdius of the circumcircle )? sin A, sin B, b, c, sin B, R, sin A, R [ IIT 00 ]

5 - TRIGONOMETRY Pge ( 9 ) The number of integrl vlues of k for which the eqution 7 cos sin = k hs solution is 8 0 [ IIT 00 ] ( 0 ) Let 0 < α < be fied ngle. If P = ( cos θ, sin θ ) nd Q = [ cos ( α - θ ), sin (α - θ ) ], then Q is obtined from P by clockwise rottion round origin through n ngle α nticlockwise rottion round origin through n ngle α reflection in the line through origin with slope tn α reflection in the line through origin with slope tn α [ IIT 00 ] ( ) Let PQ nd RS be tngents t the etremities of the dimeter PR of circle of rdius r. If PS nd RQ intersect t point X on the circumference of the circle, then r equls PQ RS PQ RS PQ RS PQ RS PQ RS [ IIT 00 ] ( ) A mn from the top of 00 metres high tower sees cr moving towrds the tower t n ngle of depression of 0. After some time, the ngle of depression becomes 0. The distnce in ( metres ) trveled by the cr during this time is [ IIT 00 ] ( ) If α β = nd β γ = α, then tn α equls ( tn β tn γ ) tn β tn γ tn β tn γ tn β tn γ [ IIT 00 ] ( ) If sin -... cos = for 0 < l l <, then equls - - [ IIT 00 ]

6 - TRIGONOMETRY Pge ( ) The mimum vlue of ( cos α ) ( cos α ).. ( cos α n ), under the restrictions 0 α, α,.. α n nd ( cos α ) ( cos α ).. ( cos α n ) = is n n n [ IIT 00 ] ( ) The number of distinct rel roots of sin cos cos cos sin cos cos cos sin = 0 in the intervl - is 0 [ IIT 00 ] ( 7 ) If f ( θ ) = sin θ ( sin θ sin θ ), then f ( θ ) 0 only when θ 0 0 for ll rel θ 0 for ll rel θ 0 only when θ 0 [ IIT 000 ] ( 8 ) In tringle ABC, c sin ( A - B C ) = b - c c - b b - c - c - - b [ IIT 000 ] ( 9 ) In tringle ABC, if C =, r = inrdius nd R = circum-rdius, then ( r R ) = b b c c b c [ IIT 000 ] ( 0 ) A pole stnds verticlly inside tringulr prk Δ ABC. If the ngle of elevtion of the top of the pole from ech corner of the prk is sme, then in Δ ABC, the foot of the pole is t the centroid circumcentre incentre orthocentre [ IIT 000 ]

7 - TRIGONOMETRY Pge 7 P ( ) In tringle PQR, R =. If tn eqution b c = 0 ( 0 ), then Q nd tn re the roots of the b = c b c = c = b b = c [ IIT 999 ] ( ) The number of rel solutions of tn - ( ) sin - = is zero one two infinite [ IIT 999 ] ( ) The number of vlues of where the function f ( ) = cos cos ( ) ttins its mimum is 0 infinite [ IIT 998 ] ( ) If, for positive integer n, θ f ( θ tn ( sec θ ) ( sec θ )... ( sec n n ) = θ ), then f = f = f = f 8 = [ IIT 999 ] ( ) If in tringle PQR, sin P, sin Q, sin R re in A. P., then the ltitudes re in A. P. the ltitudes re in H. P. the medins re in G. P. the medins re in A. P. [ IIT 998 ] ( ) The number of vlues of in the intervl [ 0, ] stisfying the eqution sin - 7 sin = 0 is 0 (b ) 0 [ IIT 998 ] ( 7 ) Which of the following number( s ) is / re rtionl? sin cos sin cos sin cos 7 [ IIT 998 ]

8 - TRIGONOMETRY Pge 8 n ( 8 ) Let n be n odd integer. If sin nθ = b r r sin r = 0 b respectively re θ, for every vlue of θ, then b 0 nd, 0, n -, n 0, n - n [ IIT 998 ] ( 9 ) The prmeter, on which the vlue of the determinnt cos ( p sin ( p - d ) cos p cos ( p d ) does not depend upon is - d ) sin p sin ( p d ) p d [ IIT 997 ] ( 0 ) The grph of the function cos cos ( ) - cos ( ) is stright line pssing through the point, - sin nd prllel to the X-is stright line pssing through ( 0, - sin ) with slope stright line pssing through ( 0, 0 ) prbol with verte (, - sin ) [ IIT 997 ] ( ) If A 0 A A A A A be regulr hegon inscribed in circle of unit rdius, then the product of the lengths of the line segments A 0 A, A 0 A nd A 0 A is [ IIT 998 ] ( ) sec θ = ( y y ) is true if nd only if y 0 = y, 0 = y 0, y 0 [ IIT 99 ] ( ) The minimum vlue of the epression sin α sin β sin γ, where α, β, γ re the rel numbers stisfying α β γ = is positive zero negtive ( D ) - [ IIT 99 ]

9 - TRIGONOMETRY Pge 9 ( ) In tringle ABC, B = nd C = sin BAD :, then equls sin CAD. If D divides BC internlly in the rtio [ IIT 99 ] ( ) Number of solutions of the eqution tn sec = cos, lying in the intervl [ 0, ], is 0 [ IIT 99 ] ( ) If = cos n φ, y = sin n φ, z = cos n φ sin n φ, for 0 < φ <, n = 0 n = 0 n = 0 then yz = z y yz = y z yz = y z yz = yz [ IIT 99 ] ( 7 ) If f ( ) = cos [ ] cos [ - ], where [ ] stnds for the gretest integer function, then f = - f ( ) = f ( - ) = 0 f = [ IIT 99 ] ( 8 ) The eqution ( cos p - ) ( cos p ) sin p = 0 in the vrible hs rel roots. Then p cn tke ny vlue in the intervl ( 0, ) ( -, 0 ), - ( 0, ) [ IIT 990 ] ( 9 ) In tringle ABC, ngle A is greter thn ngle B. If the mesures of ngles A nd B stisfy the eqution sin - sin - k = 0, 0 < k <, then the mesure of ngle C is [ IIT 990 ]

10 - TRIGONOMETRY Pge 0 ( 0 ) The number of rel solutions of the eqution sin ( e ) = is 0 infinitely mny [ IIT 990 ] ( ) The generl solution of sin - sin sin = cos - cos cos is n 8 ( - ) n n 8 n 8 n cos - [ IIT 989 ] ( ) The vlue of the epression cosec 0 - sec 0 is equl to sin 0 sin 0 o o sin 0 sin 0 o o [ IIT 988 ] ( ) The vlues of θ lying between θ = 0 nd θ = sin θ sin θ sin θ cos θ cos θ cos θ sin θ sin θ sin θ = 0 re nd stisfying the eqution 7 [ IIT 988 ] ( ) In tringle, the lengths of the two lrger sides re 0 nd 9 respectively. If the ngles re in A. P., then the lengths of the third side cn be - [ IIT 987 ] ( ) The smllest positive root of the eqution tn = lies in 0,,,, [ IIT 987 ] ( ) The number of ll triplets (,, ) such tht cos sin = 0 for ll is 0 infinite ( e ) none of these [ IIT 987 ]

11 - TRIGONOMETRY Pge ( 7 ) The principl vlue of sin sin is - ( e ) none of these [ IIT 98 ] ( 8 ) The epression sin - α sin ( α ) - sin α sin ( - α ) is equl to 0 sin α cos α ( e ) none of these [ IIT 98 ] ( 9 ) There eists tringle ABC stisfying the conditions b sin A =, A < b sin A >, A < ( e ) b sin A <, A > b sin A >, A > b sin A <, A <, b >, b = [ IIT 98 ] ( 70 ) 7 cos cos cos cos is equl to cos 8 8 [ IIT 98 ] ( 7 ) From the top of light-house 0 m high with its bse t the se-level, the ngle of depression of bot is. The distnce of the bot from the foot of the lighthouse is - 0 metres 0 metres - metres - None of these [ IIT 98 ] ( 7 ) The vlue of tn cos - tn - is None of these [ IIT 98 ]

12 - TRIGONOMETRY Pge ( 7 ) If f ( ) = cos ( ln ), then f ( ) f ( y ) - f f ( y ) hs the vlue y - - none of these [ IIT 98 ] ( 7 ) The generl solution of the trigonometric eqution sin cos = is given by = n, n = 0, ±, ±, = n, n = 0, ±, ±, n ( - ) n -, n = 0, ±, ±, none of these [ IIT 98 ] ( 7 ) If A = sin θ cos θ, then for ll rel vlues of θ A A A A [ IIT 980 ] ( 7 ) The eqution cos sin = -, 0 < hs no rel solution one rel solution more thn one rel solution [ IIT 980 ] ( 77 ) If tn θ = -, then sin θ is - but not but not - - or none of these [ IIT 979 ] ( 78 ) If α β γ =, then tn γ tn β tn α = tn α tn β tn γ tn α tn β tn β tn γ tn γ tn α = γ β α α β γ tn tn tn = - tn tn tn none of these [ IIT 979 ]

13 - TRIGONOMETRY Pge Answers b b c b d c c b b c b c d b d d c c d b d b c b c c b b c,b,c,d d c c b b c b c d b,c b c b b,c,c d e b,d c c d d c b b

SOLUTION OF TRIANGLES

SOLUTION OF TRIANGLES SOLUTION OF TIANGLES DPP by VK Sir B.TEH., IIT DELHI VK lsses, -9-40, Indr Vihr, Kot. Mob. No. 989060 . If cos A + cosb + cos = then the sides of the AB re in A.P. G.P H.P. none. If in tringle sin A :