Page 1 :
JEE(Adv.)-Mathematics, , Trigonometry, , TRIGONOMETRY, &, 1., , INTRODUCTION TO TRIGONOMETRY :, The word 'trigonometry' is derived from the Greek words 'trigon' and 'metron' and it means, 'measuring the sides of a triangle'. The subject was originally developed to solve geometric problems involving, triangles. It was studied by sea captains for navigation, surveyor to map out the new lands, by engineers and, others. Currently, trigonometry is used in many areas such as the science of seismology, designing electric, circuits, describing the state of an atom, predicting the heights of tides in the ocean, analysing a musical, tone and in many other areas., (a), Measurement of angles : Commonly two systems of measurement of angles are used., (i), Sexagesimal or English System : Here 1 right angle = 90° (degrees), 1° = 60' (minutes), 1' = 60" (seconds), (ii), Circular system : Here an angle is measured in radians. One radian corresponds to the, angle subtended by an arc of length 'r ' at the centre of the circle of radius r. It is a constant, quantity and does not depend upon the radius of the circle., p, radian = 90°, 2, , (b), , Relation between the these systems :, , (c), , If q is the angle subtended at the centre of a circle of radius 'r',, by an arc of length 'l' then, , l, =q., r, , l, •, , q, r, , Note that here l, r are in the same units and q is always in radians., , S OLVED E XAMPLE, Example 1 :, Solution :, , If the arcs of same length in two circles subtend angles of 60° and 75° at their centres. Find the ratio, of their radii., Let r1 and r2 be the radii of the given circles and let their arcs of same length 's' subtend angles of 60°, and 75° at their centres., , æ, è, , Now, 60° = ç 60 ´, , p ö æpö, p ö æ 5p ö, æ, ÷ = ç ÷ and 75° = ç 75 ´, ÷=ç ÷, 180 ø è 3 ø, 180 ø è 12 ø, è, , \, , p s, 5p s, = and, =, 3 r1, 12 r2, , Þ, , p, 5p, p, 5p, r1 = s and, r2 = s Þ r1 =, r2 Þ 4r1 = 5r2 Þ r1 : r2 = 5 : 4 Ans., 3, 12, 3, 12, , Problems for Self Practise - 01 :, (1), , The radius of a circle is 30 cm. Find the length of an arc of this circle if the length of the chord of the, arc is 30 cm., Answers :, (1), 10 p cm, , E, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , 1

Page 3 :
JEE(Adv.)-Mathematics, , Trigonometry, , &, 4., , NEW DEFINITION OF T-RATIOS :, By using rectangular coordinates the definitions of trigonometric functions, can be extended to angles of any size in the following way (see diagram). A, point P is taken with coordinates (x, y). The radius vector OP has length r, and the angle q is taken as the directed angle measured anticlockwise from, the x-axis. The three main trigonometric functions are then defined in terms, of r and the coordinates x and y., sinq = y/r,, cosq = x/r, tanq = y/x,, (The other function are reciprocals of these), This can give negative values of the trigonometric functions., , y, , P(x, y), r, , q, , •O, , x, , &, 5., , SIGNS OF TRIGONOMETRIC FUNCTIONS IN DIFFERENT QUADRANTS :, II quadrant, , 180°,p, , 90°, p/2, I quadrant, , only sine, & cosec +ve, , All +ve, , only tan & cot, +ve, , only cos, & sec +ve, , III quadrant, , IV quadrant, , 0°, 360°, 2p, , 270°, 3p/2, , &, 6., , TRIGONOMETRIC FUNCTIONS OF ALLIED ANGLES :, If q is any angle, then - q,, (a), l, , l, , l, , p, 3p, ± q, 2p ± q etc. are called allied angles., ± q, p ± q,, 2, 2, , sin (2np + q) = sin q, cos (2np + q) = cos q, where n Î I, Trigonometric Ratios of ( – q ) :, sin (– q) = – sin q, cos (– q) = cos q, tan (– q) = – tan q, cot (– q) = – cot q,, sec (– q) = sec q, cosec (– q) = – cosec q., Trigonometric Ratios of (p – q) :, sin (p – q) = sin q, cos (p – q) = – cos q, tan (p – q) = – tan q, cot (p – q) = – cot q,, sec (p – q) = – sec q, cosec (p – q) = cosec q., ö, æp, Trigonometric Ratios of ç – q ÷ :, ø, è2, ö, ö, ö, æp, æp, æp, ö, æp, sin ç – q ÷ = cos q , cos ç – q ÷ = sin q, tan ç – q ÷ = cot q, cot ç – q ÷ = tan q,, 2, 2, 2, 2, ø, ø, ø, è, è, è, ø, è, , ö, ö, æp, æp, sec ç – q ÷ = cosec q, cosec ç – q ÷ = sec q,, ø, ø, è2, è2, , E, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , 3

Page 5 :
JEE(Adv.)-Mathematics, , Trigonometry, , N.D. ® Not Defined, (a), sin np = 0 ; cos np =(–1)n; tan np = 0 where n Î I, (b), , sin(2n+1), , p, p, = (–1)n; cos(2n+1) = 0 where n Î I, 2, 2, , S OLVED E XAMPLE, Example 4 : cos (540° – q) – sin (630° – q) is equal to, (A) 0, (B) 2 cos q, (C) 2 sin q, Solution :, cos (540º – q) – sin (630º – q) = – cos q + cos q = 0 Ans. (A), , (D) sin q – cos q, , Problems for Self Practise - 03 :, (1), , If cosq = –, , 1, 3p, , then find the value of 4tan2q – 3cosec2q., and p < q <, 2, 2, , (2), Find value of cos570° sin510° + sin(–330°) cos(–390°), Answers :, (1), 8, (2), 0, , &, 8., , 8.1, , TRIGONOMETRIC RATIOS OF THE SUM & DIFFERENCE OF TWO ANGLES :, (a), (c), , sin (A + B) = sin A cos B + cos A sin B. (b), cos (A + B) = cos A cos B – sin A sin B (d), , (e), , tan (A + B) =, , tan A + tan B, 1 - tan A tan B, , (f), , tan (A – B) =, , tan A - tan B, 1 + tan A tan B, , (g), , cot (A + B) =, , cot B cot A - 1, cot B + cot A, , (h), , cot (A – B) =, , cot B cot A + 1, cot B - cot A, , Some more results :, (a), (b), , 8.2, , sin (A – B) = sin A cos B – cos A sin B., cos (A – B) = cos A cos B + sin A sin B, , sin2 A – sin2 B = sin (A + B). sin(A – B) = cos2 B – cos2 A., cos2 A – sin2 B = cos (A+B). cos (A – B)., , Trigonometric ratios of sum of more than two angles :, (a), , (b), , sin (A+B+C) = sinAcosBcosC + sinBcosAcosC + sinCcosAcosB – sinAsinBsinC, = SsinA cosB cosC – Psin A, = cosA cosB cosC [tanA + tanB + tanC – tanA tanB tanC], cos (A+B+C) = cosA cosB cosC – sinA sinB cosC – sinA cosB sinC – cosA sinB sinC, = Pcos A – Ssin A sin B cos C, = cos A cos B cos C [1 – tan A tan B – tan B tan C – tan C tan A ], , tan A + tan B + tan C - tan A tan Btan C S1 - S3, =, 1 - tan A tan B - tan Btan C - tan C tan A 1 - S2, , (c), , tan (A + B+ C) =, , (d), , tan (q1 + q2 + q3 + ....... + qn) =, , S1 - S 3 + S 5 - ......, 1 - S 2 + S 4 - ......., , where Si denotes sum of product of tangent of angles taken i at a time, , E, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , 5

Page 8 :
JEE(Adv.)-Mathematics, , Trigonometry, , Problems for Self Practise - 05 :, (1), , Prove that :, sin A + sin 2A, A, = cot, cos A - cos 2A, 2, , (a), (2), , (b), , sin A + 2 sin 3 A + sin 5 A, sin 3 A, =, sin 3A + 2 sin 5 A + sin 7A, sin 5 A, , Prove that :, (a), cos A sin (B – C) + cos B sin (C – A) + cos C sin (A – B) = 0, (b), (sin3A + sinA)sinA + (cos3A – cosA)cosA = 0, , &, 11., , TRIGONOMETRIC RATIOS OF MULTIPLE ANGLES :, , 11.1, , Trigonometric ratios of an angle 2q in terms of the angle q :, , 11.2, , 11.3, , 2 tan q, 1 + tan 2 q, , (a), , sin 2q = 2 sin q cos q =, , (b), , cos 2q = cos2 q – sin2 q = 2 cos2 q – 1 = 1 – 2 sin2 q =, , (c), (d), , 1 + cos 2q = 2 cos2 q, 1 – cos2q = 2 sin2 q, , (e), , tan q =, , (f), , tan 2q =, , 1 - tan 2 q, 1 + tan 2 q, , 1 - cos 2q, sin 2q, =, sin 2q, 1 + cos 2q, , 2 tan q, 1 - tan 2 q, , Trigonometric ratios of an angle 3q in terms of the angle q :, (a), (b), , sin3q = 3sinq – 4sin3q., cos3q = 4cos3q – 3cosq., , (c), , tan 3q =, , 3tan q - tan 3 q, 1 - 3 tan 2 q, , Trigonometric ratios of sub multiple angles :, Since the trigonometric relations are true for all values of angle q, they will be true if instead of q be, substitute, , (a), , 8, , q, 2, , q, 2 tan, q, q, 2, sin q = 2 sin cos =, 2, 2, 2 q, 1 + tan, 2, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , E

Page 9 :
JEE(Adv.)-Mathematics, , Trigonometry, , (b), , q, q, q, q, q, 2, cosq = cos2 – sin2 = 2 cos2 – 1 = 1 – 2 sin2 =, q, 2, 2, 2, 2, 1 + tan 2, 2, , (c), , 1 + cosq = 2 cos2, , (d), , 1 – cosq = 2 sin2, , (e), , tan, , (f), , q, 2, tan q =, q, 1 - tan 2, 2, , (g), , sin, , (h), , q, 1 + cos q, cos = ±, 2, 2, , (i), , tan, , 1 - tan 2, , q, 2, , q, 2, , q 1 - cos q, sin q, =, =, 2, sin q, 1 + cos q, , 2 tan, , 11.4, , 11.5, , E, , q, 1 - cos q, =±, 2, 2, , q, 1 - cos q, =±, 2, 1 + cos q, , Important results :, 1, sin 3q, 4, , (a), , sin q sin (60° – q) sin (60° + q) =, , (b), , cos q. cos (60° – q) cos (60° + q) =, , (c), (d), , tan q tan (60° – q) tan (60° + q) = tan 3q, (i), cotA – tanA = 2cot2A, (ii), , 1, cos3q, 4, cotA + tanA = 2cosec2A, , Trigonometric ratios of some standard angles :, p, 5 -1, 2p, =, = cos 72° = cos, 10, 4, 5, , (a), , sin18° = sin, , (b), , cos36° = cos, , p, 5 +1, 3p, =, = sin 54° = sin, 5, 4, 10, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , 9

Page 10 :
JEE(Adv.)-Mathematics, , Trigonometry, , (c), , sin 72° = sin, , 2p, =, 5, , 10 + 2 5, p, = cos18° = cos, 4, 10, , (d), , sin 36° = sin, , p, =, 5, , (e), , sin15° = sin, , p, 3 -1, 5p, =, = cos 75° = cos, 12 2 2, 12, , (f), , cos15° = cos, , p, 3 +1, 5p, =, = sin 75° = sin, 12, 12, 2 2, , (g), , tan15° = tan, , p, = 2- 3 =, 12, , 3 -1, 5p, = cot 75° = cot, 12, 3 +1, , (h), , tan 75° = tan, , 5p, = 2+ 3 =, 12, , 3 +1, p, = cot15° = cot, 12, 3 -1, , (i), , tan ( 22.5° ) = tan, , p, =, 8, , (j), , tan ( 67.5° ) = tan, , 3p, =, 8, , 10 - 2 5, 3p, = cos 54° = cos, 4, 10, , 2 - 1 = cot ( 67.5° ) = cot, , 3p, 8, , 2 + 1 = cot ( 22.5° ) = cot, , p, 8, , S OLVED E XAMPLE, Example 12 : Prove that :, Solution :, , 2 cos 2A + 1, = tan(60° + A) tan(60° - A) ., 2 cos 2A - 1, , R.H.S. = tan(60° + A) tan(60° – A), , æ tan 60° + tan A ö æ tan 60° - tan A ö æ 3 + tan A ö æ 3 - tan A ö, ÷ç, ÷, ÷ç, ÷ =ç, =ç, è 1 - tan 60° tan A ø è 1 + tan 60° tan A ø çè 1 - 3 tan A ÷ø çè 1 + 3 tan A ÷ø, sin 2 A, 2, 2, 2, 2, 2, 2, 3 - tan 2 A, cos2 A = 3 cos A - sin A == 2 cos A + cos A - 2 sin A + sin A, =, =, 2, 2, 2, 2, 2, 2, 2, 2, 1 - 3 tan A, sin A cos A - 3sin A, 2 cos A - 2 sin A - sin A - cos A, 1-3 2, cos A, 3-, , 2(cos2 A - sin 2 A) + cos 2 A + sin 2 A 2 cos 2A + 1, =, =, = L.H.S., 2(cos2 A - sin 2 A) - (sin 2 A + cos2 A) 2 cos 2A - 1, , 10, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , E

Page 15 :
JEE(Adv.)-Mathematics, (v), , Trigonometry, , y = secx, , (vi), , y = cosecx, Y, , Y, , (-2p,1), , X', , –5 p/2,0 –3p/2,0 –p/2,0, , o, , (–p,–1), , p/2,0, , Y=1, , Y=1, , (2p,1), , (0,1), , 3p/2,0 5p /2,0, , (p,–1), , X', , X, , –p,0, , o, , p,0, , X, Y=–1, , Y=–1, , Y', , Y', , &, 15., , DOMAINS, RANGES AND PERIODICITY OF TRIGONOMETRIC FUNCTIONS :, T-Ratio, sin x, cos x, tan x, cot x, sec x, cosec x, , Domain, , Range, , Period, , R, R, R–{(2n+1)p/2 ; nÎI}, R–{np : n Î I}, R– {(2n+1) p/2 : n Î I}, R– {np : n Î I}, , [–1,1], [–1,1], R, R, (–¥,–1] È[1,¥), (–¥,–1] È[1,¥), , 2p, 2p, p, p, 2p, 2p, , &, 16., , MAXIMUM & MINIMUM VALUES OF TRIGONOMETRIC EXPRESSIONS :, (a), , E = a sin q + b cos q, , Þ, , E=, , Let, , Þ, , 2, , a +b, , 2, , b, 2, , a +b, , 2, , ìï, üï, a, b, sin q +, cos qý, í, ïî a 2 + b 2, ïþ, a2 + b2, , = sin a &, , a, a2 + b 2, , E = a 2 + b 2 sin (q + a), where tan a =, , = cos a, b, a, , Hence for any real value of q,, - a 2 + b2 £ E £, , a2 + b2, , Hence acosq + bsinq will always lie in the interval, values are, (b), (c), , E, , [- a 2 + b 2 , a 2 + b2 ] i.e. the maximum and minimum, , a 2 + b2 , - a 2 + b 2 respectively.., , Minimum value of a2 tan2 q + b2 cot2 q = 2ab where a, b > 0, In case a quadratic in sin q & cos q is given then the maximum or minimum values can be obtained, by making perfect square., Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , 15

Page 19 :
JEE(Adv.)-Mathematics, , Trigonometry, , Problems for Self Practise - 10 :, (1), Find general solutions of the following equations :, (a), , (e), , Answers :, , sin q =, , 1, 2, , (b), , 3 sec 2q = 2 (f), , æ 3q ö, cos ç ÷ = 0 (c), è 2 ø, , (a), , (c), , q=, , (e), , q = np ±, , cos22q = 1, , æqö, cosec ç ÷ = -1, è2ø, , q = np + (-1) n, , (1), , æ 3q ö, tan ç ÷ = 0 (d), è 4 ø, , 4np, , nÎI, 3, p, , nÎI, 12, , p, , n ÎI, 6, (d), , (b), , q=, , p, q = (2n + 1) , n Î I, 3, , np, , nÎI, 2, , (f) q = 2np + ( -1) n +1 p , n Î I, , Note : Important points to be remembered while solving trigonometric equations :, (a), Many trigonometrical equations can be solved by different methods. The form of solution obtained in, different methods may be different. From these different forms of solutions, the students should not, think that the answer obtained by one method are wrong and those obtained by another method are, correct. The solutions obtained by different methods may be shown to be equivalent by some, supplementary transformations., To test the equivalence of two solutions obtained from two methods, the simplest way is to put values, of n = .......–2, –1, 0, 1, 2, 3....... etc. and then to find the angles in [0, 2p]. If all the angles in both, solutions are same, the solutions are equivalent., (b), , For equations of the type sin q = k or cos q = k, one must check that | k | < 1., , (c), , Avoid squaring the equations, if possible, because it may lead to extraneous solutions. Reject extra, solutions if they do not satisfy the given equation., , (d), , Do not cancel the common variable factor from the two sides of the equations which are in a product, because we may loose some solutions., , (e), , Some necessary restrictions :, If the equation involves tanx, secx, take cosx ¹ 0. If cot x or cosec x appear, take sinx ¹ 0., If log appear in the equation, i.e. log [f(q)] appear in the equation, use f(q) > 0 and base of, log > 0, ¹ 1., Also note that, , (f), , E, , [ f ( q)] is always positive, for example, , sin 2 q = |sin q|, not ± sin q ., , Verification : Student are advice to check whether all the roots obtained by them satisfy the equation, and lie in the domain of the variable of the given equation., Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , 19

Page 29 :
JEE(Adv.)-Mathematics, , Trigonometry, , &, 19., , HEIGHTS AND DISTANCES, , 19.1, , introduction :, One of the important application of trigonometry is in finding the height and distance of the point which are not, directly measurable. This is done with the help of trigonometric ratios., , 19.2, , Angles of Elevation and Depression :, Let OP be a horizontal line in the vertical plane in which an object R is given and let OR be joined., , R, , O, , angle of elevation, Fig. (a), , O, , P, angle of depression, , P, , R, Fig. (b), , In Fig. (a), where the object R is above the horizontal line OP, the angle POR is called the angle of elevation, of the object R as seen from the point O. In Fig. (b) where the object R is below the horizontal line OP, the, angle POR is called the angle of depression of the object R as seen from the point O., Remark :, Unless stated to the contrary, it is assumed that the height of the observer is neglected, and that the angles, of elevation are measured from the ground., , S OLVED E XAMPLE, Example 42:, Solution :, , Find the angle of elevation of the sum when the length of shadow of a vertical pole is equal to its, height., Let height of the pole AB = h and, length of the shadow of the Pole (AC) = h, , In DABC tan q =, Þ, Þ, Þ, , E, , AB h, = =1, AC h, , tan q = 1, tan q = tan 45°, q = 45°, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , 29

Page 30 :
JEE(Adv.)-Mathematics, Example 43 :, , Trigonometry, , The shadow of the tower standing on a level ground is found to be 60 metres longer when the sun's, altitude is 30° than when it is 45°. The height of the tower is(2) 30( 3 –1)m, , (1) 60 m, , (3) 60 3 m, , (4) 30( 3 +1) m., , Solution :, , AC = h cot 30° = 3 h, AB = h cot 45° = h, , Example 44 :, , Solution :, , \, , BC = AC – AB = h ( 3 –1), , Þ, , 60 = h ( 3 –1), , \, , h=, , 60, 3 -1, , =, , 60( 3 + 1), = 30 ( 3 +1), 3 -1, , The angle of elevation of the tower observed from each of the three point A,B,C on the ground,, forming a triangle is the same angle a . If R is the circum - radius of the triangle ABC, then the height, of the tower is(1) R sin a, (2) R cos a, (3) R cot a, (4) R tan a, The tower makes equal angles at the vertices of the triangle, therefore foot of the tower is at the, circumcentre., From D OCP, OP is perpendicular to OC., P, , A, a, , O, B, , a, C, , ÐOCP = a, so tan a =, , OP, Þ OP = OA tan a, OA, , OP = R tan a, , 30, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , E

Page 32 :
JEE(Adv.)-Mathematics, , Trigonometry, , 9q, q, 5q, cos 3q – sin 2q cos = cos 5 q sin, ., 2, 2, 2, , B-5., , Show that : sin, , B-6., , If A + B = 450, prove that (1 + tanA)(1 + tan B) = 2 and hence deduce that tan 22, , B-7., , Eliminate q from the relations a sec q – x tan q = y and b sec q + y tan q = x, , 10, =, 2, , 2 –1, , Section (C) : Multiple & sub-multiple angle formula, C-1., , Prove that :, (i) (cosec q – sin q) (sec q – cos q) (tan q + cot q) = 1, (ii), , C-2., , 2 sin q tan q (1 - tan q) + 2 sin q sec 2 q, (1 + tan q), , 2, , 2 sin q, (1 + tan q), , (iii), , cos A cos ecA - sin A sec A, = cosec A – sec A, cos A + sin A, , (iv), , 1, 1, 1, 1, –, =, –, sec a - tan a, cos a, cos a sec a + tan a, , Prove that, (i), , sin 2 A - sin2 B, = tan (A + B), sin A cos A - sin B cos B, , (ii) cot (A + 15º) – tan (A – 15º) =, , C-3., , Prove that, , C-4., , Prove that, , 32, , =, , 4 cos 2A, 1 + 2 sin 2A, , sin 3A cos3A, =2, sin A, cos A, , (i), , ü, ì, 2 æ a -p ö, ÷, ï, ï 1 - cot ç, 9a, ï, è 4 ø + cos a cot 4a ï, = cosec 4a., ý sec, í, 2, a, p, 2, ö, æ, ï, ï1 + cot 2 ç, ÷, ïþ, ïî, è 4 ø, , (ii), , 1, 1, = cot 2a., tan 3a - tan a cot3a -cota, , (iii), , sec 8 A - 1, tan 8 A, =, sec 4 A - 1, tan 2A, , (iv), , cos A - sin A, cos A + sin A, = 2 tan 2A), –, cos A + sin A, cos A - sin A, Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , E

Page 35 :
JEE(Adv.)-Mathematics, , Trigonometry, , Section (G) : Heights and Distances, G-1. Two pillars of equal height stand on either side of a roadway which is 60 m wide. At a point in the roadway, between the pillars, the angle of elevation of the top of pillars are 60º and 30º. Then find height of pillars G-2. At a point on level ground, the angle of elevation of a vertical tower is found to be such that its tangent is, , 5, ., 12, , 3, . Find the height of the, 4, , On walking 192 metres towards the tower, the tangent of the angle of elevation is, , tower., G-3. A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on, the plane, the angle of elevation of the bottom and the top of the flag staff are a and b respectively. Prove that, the height of tower is, , h tan a, ., tan b - tan a, , G-4. From the top of a cliff 25 m high the angle of elevation of a tower is found to be equal to the angle of depression, of the foot of the tower. Then find height of the tower -, , PART-II : OBJECTIVE QUESTIONS, Section (A) : Allied angle, , A-1., , 5p ö, æ, æ 7p, ö, æ 3p, ö, tan ç x . cos ç, + x ÷ - sin 3 ç - x ÷, ÷, 2 ø, è, è 2, ø, è 2, ø, when simplified reduces to:, pö, æ, æ 3p, ö, cos ç x - ÷ . tan ç + x ÷, 2ø, è, è 2, ø, (B) - sin2 x, , (A) sin x cos x, , A-2., , é, ë, , If sin q = –, , (B) 1, , If x = y cos, (A) – 1, , E, , é 6 æ 3p, ù, ö, 6, êsin ç 2 - a ÷ + sin (3p + a) ú is equal to, è, ø, ë, û, , (C) 3, , (D) sin 4a + sin 6a, , (B) 150°, , (C) 210°, , (D) 120º, , (C) ¥, , (D), , The value of tan 1° tan 2° tan 3° ... tan 89° is, (A) 1, , A-5, , (D) sin2x, , 1, 1, and tan q =, then q is equal to 3, 2, , (A) 30°, A-4., , ù, æp, ö, + a ÷ + sin 4 (5p + a) ú – 2, è2, ø, û, , The expression 3 êsin ç, (A) 0, , A-3, , 4, , (C) - sin x cos x, , (B) 0, , 1, 2, , 2p, 4p, = z cos, , then xy + yz + zx is equal to, 3, 3, , (B) 0, , (C) 1, , (D) 2, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , 35

Page 36 :
JEE(Adv.)-Mathematics, A-6, , If 0° < x < 90° & cosx =, (A) 0, , A-7, , 3, 10, , , then the value of log10 sin x + log10 cos x + log10tan x is, , (B) 1, , Find the value of tan, , (A), , Trigonometry, , 3-2 3, 2, , (C) – 1, , (D) 2, , 11p, 9p 3, p, 17p, - 2 sin, - cosec 2 + 4 cos2, is, 3, 3 4, 4, 6, (B), , 3+2 3, 2, , 3 +1, 2, , (C), , 3 -1, 2, , (D), , Section (B) : Addition/Subtraction of T-Ratio, sum into product/vice versa, , cos 66° cos 6° - sin 6° cos 24°, is, sin 21° cos 39° - sin 39° sin 69°, , B-1., , The value of, , B-2., , (A) - 1, (B) 1, (C) 2, (D) 0, 2, If tan A and tan B are the roots of the quadratic equation x - ax + b = 0, then the value of sin2 (A + B)., (A), , B-3., , a2, a 2 +(1-b)2, , 1- x2, 2x, , a2, (b+c )2, , (D), , a2, b 2 (1-a)2, , 1, 1, – y, x, , (C), , 1, 1, + y, x, , (D), , 1, x+y, , (B), , 1+ x2, 2x, , (B), , 1, 2, , l +1, l -1, , (B), , (C), , 1+ x2, 1- x2, , (D), , 1- x2, 1+ x2, , æ cot B ö, ÷÷ is, çç, è 1 + cot B ø, , (C) 3, , (D), , 1, 3, , tan(A + B), =, tan(A - B), , l -1, l +1, , (C), , l, 2, , (D), , 1, l, , (C), , 44, 125, , (D), , 117, 125, , If cosec A + cotA = 11/2 then tanA is equal to, (A), , 36, , (B), , If sin2A = l sin2B, then value of, , (A), B-7., , a 2 +b 2, , æ cot A ö, ÷÷ ., If A + B = 225°, then the value of çç, è 1 + cot A ø, , (A) 2, , B-6., , (C), , tan 155 ° - tan 115 °, If tan 25° = x, then 1 + tan 155° tan 115 ° is equal to, , (A), , B-5., , a2, , If tan A – tan B = x and cot B – cot A = y, then cot (A – B) is equal to, 1, 1, (A) y –, x, , B-4., , (B), , 111, 44, , (B), , 44, 117, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , E

Page 37 :
JEE(Adv.)-Mathematics, B-8., , If a =, , p, sin 23a – sin 3a, then value of, is equal to :19, sin16a + sin 4a, , (A) 1, B-9., , Trigonometry, , (B), , 1, 2, , (C) –1, , (D) 2, , Value of expression cos2 73º + cos2 47º + cos 73º cos 47º is equal to :(A), , 1, 4, , (B) 2, , (C), , 3, 4, , (D) 1, , B-10. If a cos q + b sin q = 3 & a sin q - b cos q = 4 then a2 + b2 has the value equal to :(A) 25, (B) 14, (C) 7, (D) 10, , Section (C) : Multiple & sub-multiple angle formula, C-1., , If A lies in the third quadrant and 3 tan A – 4 = 0, then 5 sin 2A + 3sinA + 4 cosA is equal to, (A) 0, , C-2., , C-3., , C-4., C-5., , C-6., , 24, 5, , (D), , 48, 5, , (D) 4, , (A) tan q, (B) cos q, (C) cot q, (D) sinq, 2, 2, 2, If tan q = 2 tan f + 1, then the value of cos 2q + sin f is, (A) 1, (B) 2, (C) – 1, (D) Independent of f, The value of tan 3A – tan 2A – tan A is equal to, (A) tan 3A tan 2A tan A, (B) – tan 3A tan 2A tan A, (C) tan A tan 2A – tan 2A tan 3A – tan 3A tan A, (D) none of these, The value of, , 1, 1, –, is, 3 cos 20°, sin 20°, , 2 3, 3, , (B), , 4 3, 3, , (C), , (D) 1, , 3, , 3p ö æ, 7p ö æ, 9p ö, p ö æ, æ, ÷ ç1 + cos, ÷ ç1 + cos, ÷ is, The value of the expression ç1 + cos ÷ ç1 + cos, 10, 10, 10, 10, ø è, ø è, øè, ø, è, , 1, 8, , sin 67, , (A), , E, , (C), , sin 5q + sin 2q - sin q, is equal to cos 5q + 2 cos 3q + 2 cos2 q + cos q, , (A), , C-8., , 24, 5, , If cos A = 3/4, then the value of 16cos2 (A/2) – 32 sin (A/2) sin (5A/2) is, (A) – 4, (B) – 3, (C) 3, , (A), , C-7., , (B) –, , (B), , 1, 16, , (C), , 1, 4, , (C), , 1, 4, , (D) 0, , 1, 1, ° + cos 67 ° is equal to, 2, 2, , 1, 4+2 2, 2, , (B), , 1, 4-2 2, 2, , (, , 4+2 2, , ), , (D), , 1, 4, , (, , 4-2 2, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , ), 37

Page 38 :
JEE(Adv.)-Mathematics, C-9., , Trigonometry, , Find exact value of sin78° – sin66° – sin42° + sin6° is :(A), , 1, 2, , (B), , 3, 4, , (C) –, , 1, 2, , (D) 1, , C-10. If A = cos 6° cos 42° and B = sec 66° sec 78°, then, (A) A = 8B, , (B) A =, , 1, B, 4, , (C) A =, , 1, B, 16, , (D) 3A = 2B, , Section (D) : Trigonometric series and conditional identities, D-1., , D-2., , If A + B + C =, , 3p, , then cos 2A + cos 2B + cos2C is equal to2, , (A) 1 – 4cosA cosB cosC, , (B) 4 sinA sin B sinC, , (C) 1 + 2cosA cosB cosC, , (D) 1 – 4 sinA sinB sinC, , In any triangle ABC, sin A – cos B = cos C, then angle B (where B > C) is, (A) p/2, , D-3., , (B) p/3, , The value of cos, , D-4., , The value of cos, , (A), , D-5., , 10 + 2 5, 64, , The value of cos, (A) 1/2, , (D) p/6, , p, 2p, 3p, 4p, 5p, 6p, + cos, + cos, + cos, + cos, + cos, is, 7, 7, 7, 7, 7, 7, , (B) - 1/2, , (A) 1/2, , (C) p/4, , (C) 0, , (D) 1, , p, 2p, 4p, 8p, 16p, cos, cos, cos, cos, is :, 10, 10, 10, 10, 10, , (B) –, , cos(p / 10 ), 16, , (C), , cos(p / 10 ), 16, , (D) –, , 10 + 2 5, 16, , p, 3p, 5p, 17p, + cos, + cos, +...... + cos, is equal to:, 19, 19, 19, 19, , (B) 0, , (C) 1, , (D) 2, , Section (E) : Range and graph of trigonometric function, E-1., , STATEMENT-1 : sin 2 > sin 3, æp ö, STATEMENT-2 : If x, y Î ç , p ÷ , x < y, then sin x > sin y, è2 ø, , (A) STATEMENT-1 is true, STATEMENT-2 is true and STATEMENT-2 is correct explanation for, STATEMENT-1, (B) STATEMENT-1 is true, STATEMENT-2 is true and STATEMENT-2 is not correct explanation for, STATEMENT-1, (C) STATEMENT-1 is true, STATEMENT-2 is false, (D) STATEMENT-1 is false, STATEMENT-2 is true, , 38, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , E

Page 41 :
JEE(Adv.)-Mathematics, , Trigonometry, , G-4. A round ballon of radius r subtends an angle a at the eye of the observer, while the angle of elevation of its, centre is b. The height of the centre of ballon is(A) r cosec a sin, , b, 2, , (B) r sin a cosec, , b, 2, , (C) r sin, , a, cosec b, 2, , (D) r cosec, , a, sin b, 2, , G-5. A man on the top of a vertical tower observes a car moving at a uniform speed coming directly towards it. If it, takes 12 minutes for the angle of depression to change from 30º to 45º, then the car will reach the tower in, (A) 17 minutes 23 seconds, , (B) 16 minutes 23 seconds, , (C) 16 minutes 18 seconds, , (D) 18 minutes 22 seconds, , PART-III : MATCH THE COLUMN, 1., , If a and b are distinct roots of the equation, a cos q + b sin q = c such that a – b ¹ 2np then match the entries, of column-I with the entries of column-II, Column – I, Column–II, (A), , sina + sin b =, , (P), , 2b, a+b, , (B), , sina · sin b =, , (Q), , c-a, c+a, , (C), , tan, , (R), , 2bc, a + b2, , (D), , a, b, tan ·tan =, 2, 2, , (S), , c2 - a 2, a 2 + b2, , 2., , a, b, + tan =, 2, 2, , Column–I, (A), , 2, , Column–II, , If for some real x, the equation x +, , 1, = 2 cos q holds,, x, , (p), , 2, , (q), , 1, , then cos q is equal to, (B), , If cos q + sec q = 2, then cos2020 q + sec2020q is equal to, , (C), , Maximum value of sin q + cos q is, , (r), , 0, , (D), , Least value of 3 sin2q + 2 cos2q is, , (s), , –1, , (A), , tan 9° - tan 27° - tan 63° + tan 81°, , (p), , 1, , (B), , cosec 10° –, , (q), , 2, , (C), , é sec 5° cos 40°, ù, 2 2 sin10° ê, +, – 2 sin 35°ú, sin 5°, ë 2, û, , (r), , 3, , (s), , 4, , 3., , 4, , Column - I, , (D), , E, , 4, , Column - II, , 3 sec 10°, , 3 (cot 70º + 4 cos 70º), , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , 41

Page 42 :
JEE(Adv.)-Mathematics, , Trigonometry, , PART-I : OBJECTIVE, 1., , If, , sin A, cos A, 3, 5, and, , 0 < A, B < p/2, then tan A + tan B is equal to, =, =, sin B, cos B, 2, 2, , (A), , (B), , 3/ 5, , (D) ( 5 + 3 ) / 5, , (C) 1, , 5/ 3, , b, 2 cos b - 1, a, If cos a =, · cot 2 has the value equal to {where a, b Î (0, p)}, then tan, 2 - cos b, 2, , 2., , (A) 2, , (B), , (C) 3, , 2, , (D), , 3, , tan a + 2 tan 2a + 4 tan 4a + 8 cot 8 a =, , 3., , (A) tan a, 4., , (B) cot a, , (C) cot 16a, , (D) 16 cota, , The value of (cos 1º + cos 2º + cos 3º + ... + cos 179º) –(sin 1º + sin 2º + sin4 3º + .... + sin4 179º) equals, 4, , 4, , 4, , 4, , 4, , 4, , to :(A) 2 cos 1º, 5., , (B) –1, , (C) 2 sin 1º, , (D) 0, , In a right angled triangle the hypotenuse is 2 2 times the perpendicular drawn from the opposite vertex. Then, the other acute angles of the triangle are, (A), , p, p, &, 3, 6, , (B), , p, 3p, &, 8, 8, , (C), , p, p, &, 4, 4, , 6., , If cos a + cos b = a, sin a + sin b = b and a – b = 2q, then, , 7., , (A) a2 + b2 – 2, If x + y = 3 – cos4q, , cos 3q, =, cos q, (B) a2 + b2 – 3, (C) 3 – a2 – b2, and x – y = 4 sin2q then, , (A) x4 + y4 = 9, , (B), , x + y =16, 2, , (C) x3 + y3 = 2(x2 + y2), , (D), , p, 3p, &, 5, 10, , (D) (a2 + b2) /4, (D), , x + y =2, , A, B, C, + sec2 + sec2 is equal to, 2, 2, 2, , 8., , In triangle ABC, the minimum value of sec, , 9., , (A) 3, (B) 4, (C) 5, (D) 6, The number of all possible triplets (a1, a2, a3) such that a1 + a2 cos 2x + a3 sin2x = 0 for all x is, (A) 0, (B) 1, (C) 2, (D) infinite, , 10., , å (-1), , 88, , K +1, , k =1, , (A) tan2º, 11., 12., , 42, , 1, is equal to, sin (k + 1)º - sin 2 1º, 2, , (B) cot 2º, , (C), , sin 2º, cot 2º, , (D), , cot 2º, sin 2º, , Equation kcosx – 3sinx = k + 1 possess a solution iff, (A) k Î (–¥, 4], (B) k Î [4, ¥), (C) k Î (–¥, 6], , (D) k Î (–¥, 6) È (8, ¥), , Number of solution of the equation tan2 a +2 3 tan a = 1 in [0, 2p] is :(A) 3, (B) 5, (C) 4, , (D) 2, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , E

Page 45 :
JEE(Adv.)-Mathematics, , Trigonometry, , 22., , If 0 £ x £ 4p , 0 £ y £ 4p and cos x . sin y = 1, then find the possible number of values of the ordered pair, (x, y), Find the number of values of q satisfying the equation sin3 q = 4sin q. sin 2q. sin 4q in 0 £ q £ 2p, , 23., , Consider the equation for 0 £ q £ 2 p; sin 2q + 3 cos2q, , 21., , (, , (k, p are coprime), then find, , ), , 2, , kp, ö, æp, - 5 = cos ç - 2q ÷ . If greatest value of q is, p, ø, è6, , k, ., p, , 24., 25., , Find the number of solutions of sinq + 2sin2q + 3sin3q + 4sin4q = 10 in(0, 6p)., Find the values of x satisfying the equation 2 sin x = 3 x 2 + 2 x + 3., , 26., , Number of solution of sinx cosx – 4cosx + 6 sinx –25 > 0 in [0, 4p] is :-, , 27., , Number of solution of the equation (1 + sin q), , 1/8, , 1/8, , 1 ö, æ 1, +ç 8 + 7 ÷, è sin q sin q ø, , = 2 9/8 (sin q)1/8 in [0, 2p] is :-, , PART - III : ONE OR MORE THAN ONE CORRECT, 1., , The value of, , (cos17° - sin17°), is, (cos17° + sin17°), , (A) tan 332°, 2., , If, , (B) tan 28°, , (C) cot 242°, , sin 4 x cos 4 x 1, +, = , then which of the following is/are TRUE ?, 5, 4, 9, 4, , (A) cot2x = 5, , (C), , 3., , If, , (B) tan2 x =, , 64, 125, + 6 = 1458, 6, cos x sin x, , (B), , 125, 64, + 6 = 1458, 6, cos x sin x, , 1, 2, , (C) 3, , (D), , 1, 3, , If sin x + sin y = a and cos x + cos y = b, then which of the following may be true., (A) sin (x + y) =, , (C) tan, , E, , (D), , 4, 5, , 1 + cos 2x, xö, æ, + 3 ç 1 + (tan x) tan ÷ sin x = 4, then the value of tanx can be equal to, sin 2x, 2ø, è, , (A) 1, 4., , (D) cot 62°, , x- y, = –, 2, , 2 ab, , (B) tan, , a2 + b2, , 4 - a2 - b2, 2, , a +b, , 2, , x- y, =, 2, , (D) cos (x + y) =, , 4 - a2 - b2, a2 + b2, 2 ab, a2 + b2, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , 45

Page 46 :
JEE(Adv.)-Mathematics, 5., , Trigonometry, , Which of the following is correct ?, (A) sin 2° > sin 2, (B) sin 2° < sin 2, (C) cos 2° > cos 2, (D) cos 2° < cos 2, If cos x + cos y = a, cos 2x + cos 2y = b, cos 3x + cos 3y = c, then, , 6., , (A) cos2 x + cos2 y = 1 +, , b, 2, , (B) cos x · cos y =, , a2 æ b + 2 ö, 2 çè 4 ÷ø, , (C) 2a3 + c = 3a (1 + b), (D) a + b + c = 3abc, If Pn = cosnq + sinnq and Qn = cosnq – sinnq, then which of the following is/are true., (A) Pn – Pn – 2 = – sin2q cos2q Pn – 4, (B) Qn – Qn – 2 = – sin2q cos2q Qn – 4, 2, 2, (C) P4 = 1 – 2 sin q cos q, (D) Q4 = cos2q – sin2q, If 2 cos q + sin q = 1, then the value of 4 cos q + 3 sin q is equal to, , 7., , 8., , (A) 3, , 9., , (B) –5, , If cot x =, , (C), , 7, 5, , (D) – 4, , æ 3p ö, æ 3p, ö, 4, -5, , 2p ÷ then which of the following is(are) correct?, , x Î ç p,, ,yÎ ç, ÷ and tan y =, 3, 12, è 2 ø, è 2, ø, , (A) sin(x + y) =, , -56, 65, , (B) cos (x – y) =, , -33, 65, , (C) sin 3x =, , -117, 125, , (D) cos 2y =, , 119, 169, , If tan2a + 2tana. tan2b = tan2b + 2tanb. tan2a, then, (A) tan2a + 2tana. tan2b = 0, (B) tan a + tan b =0, 2, (C) tan b + 2tanb. tan2a = 1, (D) tan a = tan b, If A, B, C are angle of DABC and tan A tan C = 3, tan B tan C = 6 then :-, , 10., , 11., , (A) A =, , p, 4, , (B) tan (A + B) = –3, , (C) tan (B –A) =, , 1, 3, , (D) cot (C–A) = 2, , If the sides of a right angled triangle are {cos2a + cos2b + 2cos(a + b)} and, {sin2a + sin2b + 2sin(a + b)}, then the length of the hypotenuse is:, , 12., , (A) 2[1+cos(a - b)], , (B) 2[1 - cos(a + b)], , (C) 4 cos2, , a-b, 2, , (D) 4sin2, , a +b, 2, , (a + 2) sin a + (2a – 1) cos a = (2a + 1) if tan a =, , 13., , (A), , 3, 4, , (B), , 4, 3, , (C), , 2a, , (D), , 2, , a +1, , æ, è, , Let S = sec2q sec2f + 4 sec2q cosec2 f + 9 cosec2q where q, f Î ç 0,, , 14., , 2a, 2, , a -1, , pö, ÷ . Which of the following statements is/, 2ø, , are true?, (A) The minimum value of S is 36, (B) The minimum value of S is 18, , 46, , (C) The minimum value of S occurs at q =, , p, , f = tan–1 2, 4, , (D) The minimum value of S occurs at f =, , p, , q = tan–1 2, 4, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , E

Page 47 :
JEE(Adv.)-Mathematics, 15., , Trigonometry, , 2b, , (a ¹ c), a-c, y = a cos2x + 2b sin x cos x + c sin2x, z = a sin2x – 2b sin x cos x + c cos2x, then, (A) y = z, (B) y + z = a + c, , If tan x =, , 5, , If a =, , 17., , (A) 2, (B) 3, (C) 4, The equation sin6x + cos6x = a2 has real solution if, , 1, sin 4 x + cos4 x - sin 2x + 1, 2, , 18., , 1ö, æ, (B) a Î ç - 1, - ÷, 2ø, è, , Let y =, , (B) 1/2, , (C) 1/2 2, , (C) The value of y when x = p/32 is, , 21., 22., , –3, 4, , 25., , 2 -1 ., , (B) The value of y when x = p/16 is 1., (D) The value of y when x = p/48 is 2 + 3 ., , (B), , 1, 4, , (C), , 1, 2, , (D), , 3, 4, , (D), , –1, 2, , If sin x(3 – 2 cos 2x) = 6 sin2x – 1, then (cos 2x + sin x – 1) is equal to :(A) 1, , 24., , (D) –1/ 2, , In DABC if sin A sin (B – C) = sin C sin (A – B), then (where A ¹ B ¹ C), (A) tan A, tan B, tan C are in AP, (B) cot A, cot B, cot C are in AP, (C) cos2A, cos 2B, cos2C are in AP, (D) sin 2A, sin2B, sin 2C are in AP, If x + y = z, then cos2 x + cos2 y + cos2 z – 2 cos x cos y cos z is equal to, (A) cos2 z, (B) sin2 z, (C) cos (x + y – z), (D) 1, The equation sin x + cos (k + x) + cos (k – x) = 2 has real solution(s), then sin k can be :(A), , 23., , æ1 ö, (D) a Î ç , 1÷, è2 ø, , cos x + cos 2 x + cos 3x + cos 4 x + cos 5x + cos 6x + cos 7 x, , then which of the following hold good?, sin x + sin 2x + sin 3x + sin 4x + sin 5x + sin 6 x + sin 7 x, , (A) The value of y when x = p/8 is not defined., , 20., , æ 1 1ö, ÷, (C) a Î ç è 2 2ø, , (D) 5, , If 2 sec2 a – sec4 a – 2 cosec2 a + cosec4 a = 15/4, then tan a is equal to, (A) 1/ 2, , 19., , (D) y – z = (a – c)2 + 4b2, , , then a can be, , 16., , (A) a Î (–1, 1), , E, , (C) y – z = a – c, , (B) –1, , (C), , 3, 2, , If the quardratic equations x 2 + (sin q) x + cosec q = 0 (q Î (0, p)) and 2x2 + x + c = 0 (where c Î R) have a, common root, then:(A) c = 4, (B) c = 2, (C) sum of all values of q is p, (D) number of solution of q is 4, If the equation 2(1 + a2) = sin 2q + 2a (sin q + cosq) has real solution, then which of the following statements, is (are) true ?, (A) Sum of all possible value of 'a' is zero, (B) 'a' can take only two real values., (C) Number of values of q satisfying the equation in [0, 4 p] are 4., (D) Number of values of q satisfying the equation in [0, 4 p] are 2., Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , 47

Page 52 :
JEE(Adv.)-Mathematics, , Trigonometry, , PART - II : AIEEE PROBLEMS (LAST 10 YEARS), 1., , AB is a vertical pole with B at the ground level and A at the top. A man finds that the angle of elevation of the, point A from a certain point C on the ground is 60º. He moves away from the pole along the line BC to a point, D such that CD = 7 m. From D the angle of elevation of the point A is 45º. Then the height of the pole is[AIEEE 2008 (3, –1), 105], (1), , 2., , 7 3æ 1 ö, ç, ÷m, 2 çè 3 + 1 ÷ø, , (2), , 7 3, 2, , æ 1 ö, ç, ÷ m, ç, ÷, è 3 – 1ø, , (3), , 7 3, ( 3 + 1) m, 2, , (4), , Let A and B denote the statements, , 7 3, ( 3 – 1) m, 2, [AIEEE 2009 (4, –1), 144], , A : cos a + cos b + cos g = 0, B : sin a + sin b + sin g = 0, If cos (b – g) + cos (g – a) + cos (a – b) = –, (1) A is false and B is true, , (2) both A and B are true, , (3) both A and B are false, , (4) A is true and B is false, , 4, 5, p, and let sin(a – b) =, , where 0 £ a, b £ . Then tan 2a =, 5, 13, 4, , Let cos(a + b) =, , 3., , 3, , then :, 2, , [AIEEE 2010 (4, –1), 144 JEE Mains-19], (1), 4., , 56, 33, , (2), , 19, 12, , (3), , 20, 7, , (4), , If A = sin2 x + cos4 x, then for all real x :, (1), , 3, £ A £1, 4, , (2), , 13, £ A £1, 16, , 25, 16, , [AIEEE 2011 (4, –1), 120], (3) 1 £ A £ 2, , (4), , 3, 13, £A£, 4, 16, , In a DPQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to :, , 5., , [AIEEE-2012, (4, –1)/120], (1), , 5p, 6, , (2), , p, 6, , (3), , p, 4, , (4), , 3p, 4, , ABCD is a trapezium such that AB and CD are parallel and BC ^ CD. If ÐADB = q , BC = p and CD = q, then, AB is equal to :, [AIEEE - 2013, (4, –¼),360], , 6., , (p 2 + q2 ) sin q, (1), p cos q + q sin q, , 7., , The expression, , 52, , p 2 + q2 cos q, (2), p cos q + q sin q, , (3), , p 2 + q2, p 2 cos q + q2 sin q, , tan A, cot A, +, can be written as :, 1 - cot A 1 - tan A, , (1) sinA cosA + 1, , (2) secA cosecA + 1, , (3) tanA + cotA, , (4) secA + cosecA, , (4), , (p 2 + q2 ) sin q, (p cos q + q sin q)2, , [AIEEE - 2013, (4, –¼),360], , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , E

Page 53 :
JEE(Adv.)-Mathematics, 8., , Let f k (x) =, , Trigonometry, , 1, (sinkx + coskx) where x Î R and k ³ 1. Then f4(x) – f6(x) equals, k, , [JEE(Main) 2014, (4, – ¼), 120], (1), 9., , 1, 4, , 11., , (3), , 1, 6, , (4), , 1, 3, , 3: 2, , (3) 1 : 3, , (4) 2 : 3, , If 0 £ x < 2p, then the number of real values of x, which satisfy the equation, cosx + cos2x + cos3x + cos4x = 0, is :-, , [JEE(Main)-2016, (4, – 1), 120], , (1) 5, , (4) 3, , (2) 7, , (3) 9, , If 5(tan2x – cos2x) = 2cos 2x + 9, then the value of cos4x is :-, , 7, 9, , (2) –, , 3, 5, , (3), , 1, 3, , [JEE(Main)-2017, (4, – 1), 120], (4), , 2, 9, , Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point on the, ground such that AP = 2AB. If ÐBPC = b, then tanb is equal to, (1), , 13., , (2), , 3 :1, , (1) –, 12., , 1, 12, , If the angles of elevation of the top of a tower from three collinear points A, B and C, on a line leading to the foot, of the tower, are 30º, 45º and 60º respectively, then the ratio, AB : BC , is [JEE(Main) 2015, (4, – ¼), 120], (1), , 10., , (2), , 6, 7, , (2), , 1, 4, , (3), , 2, 9, æ, , (4), , 4, 9, , æp, ö, æp, ö 1ö, + x ÷ .cos ç - x ÷ - ÷ = 1 in [0, p] is kp,, è6, ø, è6, ø 2ø, , If sum of all the solutions of the equation 8 cos x· ç cos ç, , è, , then k is equal to :, (1), , 14., , 15., , [JEE(Main) 2018], , 13, 9, , (2), , 8, 9, , (3), , 20, 9, , (4), , 2, 3, , æ p pö, , ÷ , the expression 3(sinq – cosq)4 + 6(sinq + cosq)2 + 4sin6q equals : [JEE(Main)-Jan 19], è4 2ø, , For any q Î ç, , (1) 13 – 4 cos6q, , (2) 13 – 4 cos4q + 2 sin2qcos2q, , (3) 13 – 4 cos2q + 6 cos4q, , (4) 13 – 4 cos2q + 6 sin2qcos2q, , If 0 £ x <, , p, , then the number of values of x for which sin x – sin2x + sin3x = 0, is, 2, [JEE(Main)-Jan 19], , (1) 2, 16., , (2) 1, , (3) 3, , Let a and b be two real roots of the equation (k + 1) tan2x –, numbers. If tan2 (a + b) = 50, then a value of l is ;, (1) 5, , E, , (2) 10, , (4) 4, , 2 . l tanx = (1 – k), where k(¹ –1) and l are real, [JEE(Main)-2020 (Jan)], , (3) 5 2, , (4) 10 2, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , 53

Page 54 :
JEE(Adv.)-Mathematics, , Trigonometry, , SECTION-(E), , Exercise # 1, PART - I, SECTION-(A), A-1., , A-2., A-3., , A-4., , A-6., , E-1., , (ii), , 7p, 6, , 150°, –120°, – 0.5, 14.5, , (ii), (iv), (ii), (iv), , 1440°, 165°, 0, 6, , (i), , æ- 3ö, ÷, ç, ç 2 ÷, ø, è, , (ii), , –, , (iii), , –, , (iv), , 1, , (i), , 5p, 12, , (iii), , 43p, 9, , (i), (iii), (i), (iii), , 1, 3, , (i), , (ii), , 1, 2, , (iii), , 181, 338, , 1 1, ,, 4 4, , E-2., , –, , E-3., , (i), , 3, –5, , (iii), , 243,, , (i), (ii), 2, , ymax = 11; ymin = 1, ymax = 10; ymin = – 4, E-6., , SECTION-(B), B-2., B-3., B-7., , (i), , 3 /2, , (ii), , 3 /4, , k, 2, , E-4., , x2 + y2 = a2 + b2, , E-5., , SECTION-(C), C-5., C-8., , F-1., , (iii), , (ii), 5 +1, 8, , 54, , 1/4, , 3, , – 5 /4, , p, , nÎI, 4, , 2np ±, , (ii), , np +, , (iii), , np + tan–1 (2), n Î I, , (iv), , np + (– 1)n, , (v), , np ±, , SECTION-(D), D-3., , 1, 11, , 1, 243, , (i), , 2 2, , 1, , 1,, , SECTION-(F), , 3 -1, , (i), , (ii), , p, +1, n Î I, 6, p, , n ÎI, 6, , p, , n ÎI, 4, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , E

Page 56 :
JEE(Adv.)-Mathematics, , Trigonometry, 19., 21., 23., 25., 27., , SECTION-(E), E-1., E-3., E-5., , (A), (D), (D), , E-2., E-4., , (C), (A), , 0, 8, 1.58, 0, 1.00, , (C), (A), (A), (B), (B), (B), , F-2., F-4., F-6., F-8., F-10., F-12., , (D), (B), (D), (B), (B), (A), , SECTION-(G), G-1., G-3., G-5., , (D), (A), (B), , G-2., G-4., , (C), (D), , PART - III, (A) ® r; (B) ® s; (C) ® p; (D) ® q, (A) ® (q, s), (B) ® (p), (C) ® (q), (D) ® (p), (A) ® (s), (B) ® (s), (C) ® (s), (D) ® (r), , 1., 2., 3., , Exercise # 2, PART - I, 1., 3., 5., 7., 9., 11., 13., 15., 17., , (D), (B), (B), (D), (D), (A), (C), (A), (B), , 2., 4., 6., 8., 10., 12., 14., 16., 18., , (D), (B), (B), (B), (D), (C), (A), (A), (A), , 2.50, 0.25, 6.41, 42.00, 18.00, 1.66 or 1.67, 0.50, 100.00, 4.00, , 56, , 2., 4., 6., 8., 10., 12., 14., 16., 18., , 1., 3., 5., 7., 9., 11., 13., 15., 17., 19., 21., 23., 25., 27., 29., 31., , (B,C,D), (A, D), (B,C), (A,B,C,D), (B, C, D), (A,B,C,D), (B,D), (B,C), (B,D), (B, D), (C,D), (B,D), (A,B,C), (A,B), (A,C,D), (A,B,D), , 2., 4., 6., 8., 10., 12., 14., 16., 18., 20., 22., 24., 26., 28., 30., , (A, C), (A,B,C), (A, B, C), (A, C), (B,C,D), (A,C), (A, C), (B, C, D), (A,D), (B, C), (B,C), (A,C), (A,C,D), (A,C), (A,B), , PART - II, 1., 3., 5., 7., 9., , (C), (D), (A), (C), (D), , 2., 4., 6., 8., , (B), (B), (A), (D), , Exercise # 3, PART - I, 3., , 2, 3, , 5., 7., 9., 11., , (A,C,D), 8, (C), (B,C), , 1., , PART - II, 1., 3., 5., 7., 9., 11., 13., 15., 17., , 17, 15, 0, 0.00, , PART - III, , SECTION-(F), F-1., F-3., F-5., F-7., F-9., F-11., , 20., 22., 24., 26., , 25.00, 4.00, 0.25, 6.00, 2, 42.00, 0.46, 0.00, 2.50, , 2., 4., 6., 8., 10., 12., , (n = 7), (D), (D), (C), (C), 0.5, , PART - II, 1., 3., 5., 7., 9., 11., 13., 15., , (3), (1), (2), (2), (1), (1), (1), (1), , 2., 4., 6., 8., 10., 12., 14., 16., , (2), (1), (1), (2), (2), (3), (1), (2), , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , E

Page 57 :
JEE(Adv.)-Mathematics, , Trigonometry, , 1., , Let a = 4 sin2 10º + 4 sin2 50º · cos20º + cos 80º and b = cos2, , 2., , Simplify the expression, , 3., , Let a, b, c, d be numbers in the interval [0, p] such that, , p, 2p, 8p, + cos2, + cos2, find (a + b), 5, 15, 15, , sin 4 x + 4 cos 2 x – cos 4 x + 4 sin 2 x, , sin a + 7 sin b = 4(sin c + 2sin d),, cos a + 7 cos b = 4(cos c + 2cos d), Prove that 2 cos (a – d ) = 7 cos (b – c )., 4., , Find the value of a for which the three element set S = {sin a, sin 2a, sin 3a} is equal to the three element set, T = {cos a, cos 2 a, cos3a}, , 5., , If a sinq + b cosq = a cosec q – b secq = 1, prove that a2 + b2 = 1 + b2/3 – b4/3., , 6., , If p (sina - cosa tanq ) secq = q tanq . sec(a – q) then prove that q =, , 7., , If tan a =, , æ q + p cos 2a ö, 1, ÷÷ ., cot-1 çç, 2, è p sin 2a ø, , p, where a = 6 b, a being an acute angle, prove that;, q, , 1, (p cosec 2 b - q sec 2 b) = p 2 + q2 ., 2, , E, , cos2q cos6q cos18q 1, +, +, = [ cot 2q - cot 54q], sin 6q sin18q sin 54q 2, , 8., , Prove that, , 9., , If sin (q + a) = a & sin (q + b) = b (0 < a, b, q < p/2) then find the value of cos2 (a - b) - 4 ab cos(a - b), , 10., , Show that: 4 sin 27° = (5 + 5 )1/ 2 - (3 - 5 )1/ 2, , 11., , If tan b =, , 12., , Prove that in an acute angled triangle ABC,, , 13., , If xy + yz + xz = 1, then prove that, , tan a + tan g, sin 2a + sin 2g, , prove that sin 2b =, ., 1 + tan a. tan g, 1 + sin 2a. sin 2g, , x, 1- x, , 2, , å tanAtanB ³ 9., +, , y, 1- y, , 2, , +, , z, 1- z, , 2, , =, , 4 xyz, 2, , (1 - x )(1 - y 2 )(1 - z 2 ), , ., , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , 57

Page 59 :
JEE(Adv.)-Mathematics, 26., , Trigonometry, , Solve the following system of equations for x and y :, 2, , 5(cosec x–3sec, , 2, , y), , = 1 , 2(2cosecx +, , 3|secy|), , = 64, , 27., , Find number of solution of equation 8 sin x =, , 28., , Solve : sin 2x >, , 29., , Solve for x and y :, , 1, 3, in x Î [0, 2p], +, sin, x, cos x, , 2 sin 2 x + (2 – 2) cos2x, , x cos3 y + 3x cos y sin2y = 14, x sin3y + 3x cos2 y sin y = 13, 30., , E, , Solve for x, the equation, , 2., , 13 - 18 tanx = 6 tan x - 3, where - 2 p < x < 2 p ., , 1., , 4, , 14., , (c), , 21., , np, ±, 4, , 24., , x=, , 26., , x = np + (–1)n, , 28., , np +, , 30., , a - 2 p; a - p, a, a + p, where tan a =, , 1, ; (e), 2, 1+, , cos2 x – sin2x = cos 2x, , 7 ; (f) 21; (g) 35; (h) 5, , n2 p 2, ,nÎI, 16, , p 5p, p 5p, , ,y = ,, 6 6, 3 3, , 4., , np p, + , nÎI, 2 8, , 9., , 15., , 1, , 17., , 22., , x=, , 25., , 2p, 5, , p, p, and y = mp± ; where m, n Î I, 6, 6, , p, p, < x < np + , nÎI, 8, 4, , 29., , p, 2, , 1 - 2a2 - 2b2, 1, 2, , 999, , 23., , f, , 27., , 6.00, , x = ± 5 5 , y = np + tan–1, , 1, ,nÎI, 2, , 2, 3, , Corporate Office : Reliable Institute, A-10 Road No.1, IPIA, Kota-324005 (Rajasthan) INDIA, visit us at: www.reliablekota.com, ( +91-7427056522, 7568756522, 7425906522, , 59