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1 EMCET MATHEMATICS TRIGONOMETRY UPTO TRANSFORMATIONS 1. α + β = and β + γ = α then tanα is 1) tan β + tan γ) ) tan β + tan γ 3) tan β + tan γ ) tan β + tan γ 1 1. cos x + cos y =, sin x + -sin y = then sin x + y) = ) ) 3) ) If sin A + sin B = 3 cos B cos A) then sin 3A + sin 3B = 1) 1 ) 1 3) 0 ). sin 55 sin 19 + sin 53 sin 17 = 1) sin1 ) cos1 3) sin1 ) cos1 5. tan tan tan 0 = 1) 3 ) 0 3) 3 ) 3 6. If sin θ + cos θ = p and tan θ + cot θ = q then qp 1) = 1 1) ) 3) 1 ) cos 10 + cos 0 + cos 30 = 1) cos 5 cos 10 cos 15 ) cos 10 cos 0 cos 30 3) sin 5 sin 10 sin 15 ) sin 10 sin 0 sin 30 cos3θ 8. If cos α + cos β = a, sin α + sin β = b and α β= θ, then = cosθ a + b 1) a + b ) a + b 3 3) 3 a b ) cosθ 1 + θ ) 9. If tan θ 1 = k cot θ, then = cosθ 1 θ ) 1 + k 1 k k + 1 k 1 1) ) 3) ) 1 k 1 + k k 1 k If 3 sec α 5 tan α = k and 6 sec α + k tan α = 5 then k = 1) 0 ) 5 3) 3 ) If cosα + β) = and sin α β) =, where 0 α, β. Then, tan α = ) ) 3) ) R

2 P Q 1. In a PQR, R =, if tan ) and tan ) are the roots of ax + bx + c = 0, where a 0, then P Q PQR apple R =, tan ), tan ) ax + bx + c = 0 ßÁ é\ «Å ûë 1) b = a + c ) b = c 3) c = a + b ) a = b + c 13. The number of integral values of k for which the equation 7 cos x + 5 sin x = k + 1 has a solution, is 7cos x + 5sin x = k + 1 ÆæO -é- Ω-ù«-EéÀ ƒüµ ÆæN A Öçú - çõ k ßÁ é\ æ Ωg N - áeo? 1) ) 8 3) 10 ) 1 TRIGONOMETRIC EQUATIONS 1. The equation 3 sin x + cos x = k + 1 is solvable only if k 3 sin x + cos x = k + 1 ÆæO -é- Ω-ù«-EéÀ ƒüµ Öçú - çõ k Åç-ûª Ωç 1) 3, 1) ), 3) 3) [, 3] ) [ 3, 1] 15. The general solution of x for the equation 9 cos x.3 cos x + 1 = 0 is 9 cos x.3 cos x + 1 = 0 Ææ-O -é- Ωù ƒüµ ÆæN A n n + 1) 1) n ) 3) n ) 16. The number of solutions of sec x cos 5 x +1 = 0 in the interval [0, ] is [0, ] Åç-ûª- Ωç apple sec x cos 5x + 1 = 0 ÆæO é Ωù ƒüµ - Ææçêu 1) 5 ) 8 3) 10 ) If 3 tan 5 θ = cos α 3 cos α and 3 cos θ = 1 then the general value of α is 3 tan 5 θ = cos α 3cos α & 3 cos θ = 1 Å ûë α ßÁ é\ ƒüµ - Ωù ƒüµ 1) n, n Z ) n ±, n Z 3 ) n ±, n Z ) n ±, n R 3 tan If α, β, γ, δ satisfy the equation x + ) = 3 tan 3x then tan α tan β tan γ tan δ = = 3 tan 3x ÆæO -é- Ω-ù«Eo ûª% œh æjêæh tan α tan β tan γ tan δ = α, β, γ, δ tan x + ) ) ) 3) 1 ) If 3 cos θ + sin θ = λ has no solution then find the range of λ 3 cos θ + sin θ = λ Ææ-O -é- Ω-ù«-EéÀ ƒüµ é ÚûË λ N ÖçúË Åçûª Ωç 1) [ 5, 5] ), 5) 5, ) 3) 5, 5) ) 0, 5) 0. The set of values of a such that a 6 sin x 5a 0, x is a 6 sin x 5a 0, x ÆæO -é- Ω-ù«Eo ûª% œh- æjîë a N- 1) [ 1, 6] ) [, 3] 3) 1, 6) ) None of these

3 1. The number of solutions of the equation sin x cos x + sin x = in the interval [0, 5] is [0, 5] Åçûª- Ωç apple sin x cos x + sin x = ÆæO -é- Ωù ƒüµ - Ææçêu 1) 3 ) 5 3) ) 6. The number of values of x in [0, 3] such that sin x + 5 sinx 3 = 0 is [0, 3] Åçûª- Ωç apple sin x + 5 sin x 3 = 0 ÆæO é- Ωù ƒüµ - Ææçêu 1) 1 ) 3) ) 6 INVERSE TRIGONOMETRIC FUNCTIONS 3. If u = cot 1 cos θ tan 1 cos θ then sin u = θ θ θ θ 1) tan ) ) tan ) 3) cot ) ) cot ). The number of real solutions of tan 1 xx + 1) + sin 1 x + x + 1 = is tan 1 xx + 1) + sin 1 x + x + 1 = ÆæO -é- Ωù Ææh «Ææçêu 1) 0 ) 1 3) ) infinite Å çûªç) 5 cot ) 1 5. Value of = 0 tan ) ) ) 3) 1 ) 3 6. The value of sin 1 sin 10) is 1) 10 ) ) 3 10 ) The no. of positive integral solutions of the equation y 3 tan 1 x + cos 1 = sin 1 ) 1 y 10 y 3 tan 1 x + cos 1 = sin 1 ) 1 y 10 ÆæO -é- Ω-ùç ßÁ é\ üµ - æ Ωg ƒüµ - Ææçêu 1) 1 ) 3) 3 ) n 8. If cot 1 >, n N then the maximum value of n is 6 n cot 1 6 >, n N Å ûë n ßÁ é\ í J æe N 1) ) 5 3) ) If sin 1 x + sin 1 y + sin 1 z =, then the x y 101 )x 0 + y 0 ) value of is x y 303 )x 0 + y 0 ) 1) 0 ) 1 3) ) 3

4 30. Let x 0, 1). The set of all x such that sin 1 x > cos 1 x, is the interval x 0, 1) Å ûë sin 1 x > cos 1 x ÅßË u Nüµ çí x N - ÆæN A ), ) ) ), 1 3) 0, 1 ) 0, ) y 31. If cos 1 x cos 1 = α,then x xy cos α + y is equal to 1) ) sin α 3) sin α ) sin α HYPERBOLIC FUNCTIONS 3. sec h 1 sin θ) = θ θ θ θ 1) log sec ) ) log sin ) 3) log cos ) ) log cot ) then sin hx) = 33. If x = log [ cot + θ )] 1) tan θ) ) cot θ 3) tan θ ) cot θ 3. If cos h x = 199, then cot hx = ) ) 3) ) If sin x cos hy = cos θ, cos x sin hy = sin θ then sin h y = 1) cos h x ) cos x 3) cos h 3 x) ) sin h x) If cos h x) =, then cos h3x) = ) ) 3) ) PROPERTIES OF TRIANGLES 37. In a triangle ABC, if c = a b) sec θ, then tan θ is equal to ABC apple c = a b) sec θ, Å ûë tan θ N ab C ab C 1) sin ) ) tan a b a b ) ab C ab C a b a b 3) cos ) ) sec ) sin B 38. If the median AD of a triangle ABC divides the BAC in the ratio 1 : then is equal to sin C sin B sin C ABC apple AD üµ u-í ûª Í ê BAC E 1 : E æpah apple N µº->êæh N A 1 A 1 A A 1) cos ) sec 3) sin ) cosec In a triangle ABC, if cos A cos B + sin A sin B sin C = 1 then a : b : c = 1) 1 : : 3 ) 1 : 1 : 3) 1 : 1: ) 1 : 1 : 3

5 sin A sin A B) 0. If in a triangle ABC, = then a, b, c are in sin C sin B C) sin A sin A B) va µº ïç ABC apple =, Å ûë a, b, c apple Öçö«sin C sin B C) 1) A.P. Åçé-v ÏùÀ ) G.P. í ù-v ÏùÀ 3) H.P. æ«-ûªté v ÏùÀ ) None of these àd-é ü 1. If a + b + c = c a + b ), then acute value of C is equal to 1) 30 ) 60 3) 5 ) 75. In a triangle, a + b + c = ca + ab 3. Then the triangle is a + b + c = ca + ab 3. Å ûë va µº -ïç 1) equilateral Ææ - «æ ) right angled and isosceles ç Ææ -Cy- «æ 3) right angled with A = 90, B = 60, C = 30 ç éóù va µº ïç A= 90, B = 60, C = 30 ) None of these àd-é ü 3. If in a triangle, r 1 = r = 3r 3 and D is the middle point of BC, then cos ADC is equal to ADC apple r 1 = r = 3r 3, BC µº ïç üµ u- Gç-ü D Å ûë cos ADC = ) ) 3) ) KEY 1-3; -; 3-3; -; 5-; 6-3; 7-1; 8-; 9-; 10-1; 11-; 1-3; 13-; 1-; 15-; 16-; 17-; 18-; 19-; 0-; 1-1; -3; 3-; -3; 5-; 6-3; 7-; 8-; 9-; 30-; 31-; 3-; 33-3; 3-; 35-; 36-3; 37-1; 38-; 39-3; 0-1; 1-3; -3; 3-. Writer: D. Sankara Rao