PU I Year Trigonometry
|
|
- Christiana Norris
- 5 years ago
- Views:
Transcription
1 PU I Year Trigonometry
2 Remember: 1. Angle between Minute hand and Hour hand in X Hr. andy min. is 30X- 11 Y 2
3 2. The maximum value of acosθ+ bsinθ+ c, is c+ a 2 + b 2 and the minimum value is c a + b 2 2
4 2 2 then A+ B = π If asinx + bcosx = c, then 3. If Cos A + Cos B = 1 = Sin 2 A + Sin 2 B bsinx -acosx = ± a 2 + b 2 c 2
5 1. The angle between Hr.hand and Min. hand of a clock when the time is 3 : ) 10 2) 20 3)30 4)22 2
6 11 Solution: Req. = 30(3)- (20) 2 = = 20 Ans : (2)
7 2. The vertical angle of an isosceles triangle is 45 then the base angle in circular measure is (1) (2) (3) 3π (4) 3π 8 16
8 Solution: A+ B+ C= 180 A+ B= 180-C= = 135 B= A 2A = 135 =3π3 4 A =3π3 8 Ans: (3)
9 3.If the length of a chord of a circle is equal to that of the radius of the circle, then the angle subtended in radians at the centre of the circle by chord is π π π 1) 1 2) 3) 4) 2 3 4
10 Solution: OAB is an equilateral triangle = π Ans:(3) AOB 3 o r r A r B
11 4. If SinA + 1 = 5 and A is SinA 2 acute then A is π π 1) 2) 6 4 π 3 ) 4) None of these 3
12 1 1 Solution SinA + = 2+ : SinA 2 1 π SinA = A = Ans: (1)
13 5.If Sec θ + tan θ = 2, then the values of Sec θ & tan θ are respectively ), 2), 3), 4)None
14 Solution : Sec θ - tan θ = 1 2 Q( Sec θ + tanθ)( Sec θ tanθ) = 1 adding and Simplifying 5 3 Sec θ = and tanθ = 4 4 Ans : (2)
15 6. The value of Cos Cos 2 5, is 1 1)0 2) -1 3)1 4) 2
16 Solution : Cos 2 A + Cos 2 B= 1 when A + B = 90 Ans.(3)1
17 7. The maximum value of 4Sin θ + 3Cos θ + 2, is 1)7 2)4 3)6 4)5
18 Solution : Max.value = c + a 2 + b 2 Ans.(1)7 = = 7
19 8. Sin θ + Cos θ = 1, then Sin2θθ = 1)1 2) -1 3)0 4)2
20 Solution : Sq. both sides, 2 2 Sin θ + Cos θ + 2Sin θ Cos θ = 1 = Sin2θ 0 Ans.(3)0
21 9. If Cos θ + Sec θ = 2, then the value of Cos 100 θ Sec 100 θ = 1)0 2)1 3)2 4) -1
22 Solution: Cosθ + 1 = Cosθ Cosθ = 1 = Secθ G.E. = 1-1 = 0
23 10. If Sec θ + tanθ = 4 then Cos θ = ) 2) 3) 4)
24 1 Solution : Sec θ tan θ = 4 1 adding, we get 2Sec θ = Sec θ = 8 8 Cos θ = 17
25 11. The value of tan20 + tan40 + tan tan180, is 1)0 2)1 3)2 4)4
26 Solution : If A +B =180 then tana = -tanb or tana + tanb = 0 tan20+ t tan160+ tan40+ tan tan180 = =0 Ans : (1) 0
27 12.The value of 2 2 α α Sin tan + + Cos 1+ Cot 2 (1+ tan ) 2 α α 2 α 1) -1 2)0 3)1 4)2
28 Solution : Put α = G.E. = (1+1) 2 2 =1 Ans : (3) 1
29 13. A,B,C are the angles of a ABC, then 3A + 2B+C A - C Cos +Cos = 2 2 1)1 2)0 3)CosA 4)CosC
30 Solution : Put A = B = C G.E. = Cos180+Cos0 = -1+1= 0 Ans : (2) 0
31 14. If x =Cos1 and y =Cos1 then 1) x = y 2) x < y 3) x > y 4) 2x = y
32 Sl Solution : Cos θ is decreasing π for 0< θ < 2 Cos1 > Cos1 Ans : (3)x>y
33 15. In a ABC, C=90,then the equation whose roots are tana & tan B is )abx + c x + ab = 0 2)abx + c x - ab = )abx - c x - ab = 0 3)abx - c x +ab = 0
34 Solution : C=90 A+B= (a + b = c ) tanatanb=1 αβ =1 a b α+ β =tana + tanb = + b a a + b -c = - Ans:(4) ab ab ab Vikasana - CET 2012
35 16. If 5Sinx + 4Cosx = 3, then 4Sinx - 5Cosx = 1)4 2)4 2 3)3 2 4) 2
36 Solution : GE G.E. = a + b -c = = 32 = 4 2 Ans : (2)
37 17. If a = Sin1 and b = Sin1 then 1)a = b 2)a < b 3)a > b 4)a = 2b
38 Solution : Sinθ is increasing in 0 < θ < 90 Ans : (2)a<b
39 18. IF CosA = acosb and SinA = bsinb, then (b - a )Sin B = 1) 1+ a 2 2) 2+a 2 3) 1- a 2 4) 2 - a 2
40 Solution : squaring we get Cos A = a Cos B = a (1- Sin B) = a - a Sin B...(1) & Sin A = b Sin B...(2)
41 adding g( (1)&(2) ( ) = a - a Sin B+b Sin B (b 2 - a 2 )Sin 2 B =1- a 2 Ans : (3)
42 19. The maximum value of 2 2 4Sin x + 3Cos x is 1)3 2)4 3)5 4)None
43 Solution : G.E. = Sin x + 3(Sin x +Cos x) 2 = Sin x = 4 Ans : (2)4
44 20. The value of tan100+ tan125 + tan100tan125 = 1 1)2 2)3 3) 4) 1 3
45 Sol : tan225 = tan( ) =1 tan100+ tan125 tan100+ tan125 =1 1- tan100 tan125 G.E. =1 Ans : ( 4) 1
46 21. If ABCD is a cyclic quadrilateral then 1) Sin(A+C) =1 2)Cos(A+C) =-1 3) Sin(B+D) =1 4) Cos(A +C) =1
47 Solution : Sum of opp.angles =180 A+C =180 Cos(A+C)=-1 Ans : (2)
48 1 a b 22. For a ABC, 1 c a = 0 then the value of b c Cos A +Cos B+Cos C = ) 2) 3) 4)
49 Solution : = 0 if any two rows / columns are identical a = b = c (by inspection) A=B=C= G.E. = 3Cos 60 = Ans : (2)
50 Cos 10 +Cos 110 +Cos 130 = ) 2) 3) 4)
51 Solution : If α = 60 or 120 or 240 or θ ( α θ)+ ( α θ) θ then, Cos +Cos + Cos - = Cos x2 8 3 G.E. = Cos (3x10 ) = = Ans : (3) ()
52 24. Cos Cos Cos 2 70 = 1 3 1) 2) 1 3) 4)2 2 2
53 Solution : when α = 60 or 120 or 240 or then Cos θ +Cos ( α - θ)+cos ( α +θ) = 3 2 Ans : (3)
54 1-Cosθ 2x 25. If x =, then, = 1+Cosθ 1- x 2 1) Sin θ 2)Cosθ 3)tan θ 4) Cotθ
55 θ Solution : x = tan 2 G.E. = θ 2tan 2 θ 1- tan 2 2 = tan θ Ans : (3)
56 π π Cos 2 - A - Sin 2 -A = π π Cos 2 + A + Sin 2 + A 4 4 1)Cos2A 2)tan2A 3)Sin2A 4)Cot2A
57 Solution : G.E. = Ans : (3) Cos2θ 1 π = Cos - 2A = Sin2A 2
58 27. If tanβ β =2SinαSinγCosec(α γ + γ), then Cotα, Cotβ, Cotγ are in 1)A.P. 2)G.P. 3)H.P. 4)A.G.P.
59 Solution : Taking reciprocals, Sin(α + γ) ) Cotγ +Cotα Cotβ = = 2SinαSinγγ 2 They are in A.P. Ans : (1)
60 28. If Cos(x-y)+ -3 Cos(y -z)+cos(z-x) ) = 2 then, Cosx = 1)0 2)1 3)2 4)3
61 Solution : Cos(x-y) = -3 2(CosxCosy + SinxSiny) = (CosxCosy + SinxSiny) = 0 i.e.,(cosx +Cosy +Cosz) + (Sinx + Siny + Sinz) = 0 Cosx = 0 = 2 Sinx 2 2
62 2 2 Minimum of (Sin x +Cos x) 29. = x x Maximum m of Cos 2 + Sin ) -1 2)1 3)2 4) -2
63 Min.of 1 Solution : G.E. = Max. of 1 1 = =1 1 Ans : (2)
64 30. 3Sin 2 x + 4Cos 2 x 1) [ 0,3 ] 2) [ 0,4 ] [ ] [ ] 3) 3,4 4) -4,-3-3
65 2 2 2 G.E. = 3(Sin x +Cos x)+cos x 2 [ ] = 3 +Cos x 3,4 2 [ ] QCos x 0,1 Ans :(3)
66 31.The maximum value of 3 5Sinx -12Cosx +19 is ) 1 2) 3) 4) 2 3 4
67 n Sol : G.E. is maximum when Denominator is Minimum and Min.value of Dr. = = GE G.E. = = Ans.(2) Ans(2) 6 2
68 If tanθ = then, Sinθ = ) but not 2) or ) but not 4)None of these 5 5
69 -4 Sol n : tanθ θ = 3 θ II or IV quadrant Sinθ =± 4 5 Ans:(2)
70 33. The value of 3Cosec20 - Sec20 is 1) 2 2) 4 2Sin20 3) 4) Sin40 4Sin20 Sin40
71 n 3 1 Sol : G.E. = - Sin20 Cos20 3Cos20 - Sin20 = Sin20 Cos20
72 = Cos20 - Sin Sin20 Cos20 2 Sin(60-20) =4 Sin40 = 4 1= 4 Ans : (2)
73 34. If A = Cos 2 θ +Sin 4 θ, then for all the values of θ 13 1)1 A 2 2) A ) A 4) A
74 n 2 4 Sol : A =1- Sin θ +Sin θ =1-Sin θ(1- Sin θ)=1-sin θcos θ 2 Sin2θ =1- =1-0 if Sin2θ is least or = 1- =, if Sin2θ is greatest A 1 Ans : (4) 4 4
75 35. tan20 +tan40 + tan40 3tan20 tan40 = tan ) 2) 3 3) 4) 3 3 3
76 n Sol : We have, tan(40+20) = 3 tan40+ tan20 = 3 1- tan40 tan20 G.E. = 3 Ans : (2)
77 If Cos(α +β) =, Sin(α - β) = 5 13 and α and β lies between π 0and,then tan2α α = ) 2) 3) 4)None
78 n Sol : tan2α = tan(α +β + α - β) tan(α +β)+tan(α - β) = 1- tan(α +β) tan(α - β) = = Ans : (2)
79 37. The value of π 3π 5π 7π Sin 2 +Sin 2 +Sin 2 +Sin )1 2)2 3)1 4)2 8 8
80 7π π Sol n : Sin = Sin and 8 8 5π 3π Sin = Sin 8 8 π 3π G.E. = Sin +Sin 8 8 π π = Sin +Cos = 2 1= 2 8 8
81 tan70 - tan = tan50 1) 3 2) 0 3) 1 4) 2
82 n Sol : tana - tanb =1+ tana tanb tan(a - B) tan70 - tan20 G.E. = tan(70-20) =1+ tan70.tan20 =1+ tan70.cot70 t70 =1+1= 2
83 39. If CosACosB+SinASinBSinC =1 then a : b : c = 1) 1:1:1 2) 2 :1:1 3)1: 2 :1 4)1:1: 2
84 n Sol : 1= CosACosB+SinASinBSinC CosA CosB+SinA SinB =Cos(A-B) ( Q SinC 1) 1 Cos(A - B) Cos(A - B) =1
85 A - B=0 A=B By given expression we get C = 90 A =B = 45. Hence a:b :c=sina : SinB : SinC = 1 : 1 :1=1:1: Ans:(4) 2 2
86 40. In a ABC, A > B and if the measures of A and B satisfy 3Sinx - 4Sin 3 x - k = 0, 0 < k <1 then C = 1)30 2)45 3)120 4)None
87 n 3 Sol : 3Sinx - 4Sin x = k Sin3x = k. Given that Sin3A = k and Sin3B = k Sin3A = Sin3B 3A = 3B or 3A =180-3B But A > B means A B 3A =180-3B A +B = 60 C =120 Ans : (3)
88 41. ABC is right angled at C, then tana + tanb = b 2 a 2 c 2 1) 2) a+b 3) 4) ac bc ab
89 a b Sol n : tana =, tanb = b a 2 2 a b a +b GE G.E. = + = b a ab B c 2 = QC=90 ab a C Ans : (4) b c A
90 42. If P,P,P are altitudes of a triangleabc, from the vertices A,B,C, and, the area of a triangle, then P -1 +P -1 +P -1 is ) s a s b s c s 2) 3) 4)
91 1 2 Sol n : = ap P = a a P P -1 = 1 2 b c P -1 = and P -1 = a+b + c 2s s Hence G.E. = = = 2 2 Ans : (4)
92 43. If in a ABC, CosACosB+SinASinBSinC =1, then the triangle is 1) isosceles 2) right angled 3)isosceles right angled 4)equilateral
93 n Sol : By Que.No.39 A =B = 45 and C = 90 Ans : (3)
94 44. If two sides a,b and angle A be such that t 2 triangles are formed then the sum of two values of third side is 1) 2b SinA 2) 2b CosA 3) b CosA 4)(c+b)CosA a
95 a 2 =b 2 +c2-2bccosa c -(2bCosA)c+(b -a )=0 which is Quad. in c, if c &c be two roots then 1 2 c +c =2bCosA 1 2
96 45. log y = log z = log x, then x y z 1)x=y=z 2) x>y>z 3)x=y>z 4)x<y<z
97 k Sol n: y=x, z=y k and x=z xyz=(xyz) k k=1 x = y = z k Ans:(1)
98 46. If log 2, log(2x-1) and log(2x+3) are in A.P. then x = 1) -1 2) 1 3)1 4) None 2 2
99 n Sol : 2log(2x-1) = log2+log(2x+3) +3) (2x-1) 2=2(2x+3) x= 5 or -1 but, when x= log(2x-1) is not difined. Ans(4)
100 47. If x, y, z are in G.P. and y a x =b =c z, then 1) log cb= log ac 2)log ab= log cb 3) logac= log a 4)log a= logcb b b
101 Sol n: a x= b y= c z= k (say) x=log k, y=log k, z=log k a b c x,y,z are in G.P. y = x z y log k log k b = c log a=log b log k log k b c a b Ans:(4)
102 48. If loga + logb= log(a+b) then a= 1) b 2) b 3) b-1 4) b b1 b-1 b b+1
103 Sol n: log(ab) = log(a+b) ab=a+b a = b b-1 Ans:(2)
104 2n is divisible by 1) 10 2) 9 3) 20 4) 24
105 n Sol : By inspection, put n=1 2(1) we get 5-1 = 24 which is divisible by 24 Ans: (4)
106 2 50. If x + bx+c =0 and 2 x + cx+ b = 0, have a common root and b c then b + c = 1) 0 2) -1 3) 2 4) 1
107 n Sol : If α be the common root 2 2 then, α +bα+c=0 and α +cα+b=0 solving we get, α = 1 & α = -b-1 -b-c = 1 b + c = -1
108 51. The value of is 1) 3 2) 2 3) 4 4) 5
109 n Sol : x= = 6+x 2 2 x = 6+x x -x-6=0 x = 3 or -2, but x -2 x = 3 Ans : (1)
110 52. If a,b,c are the roots of 3 2 x - 6x +2x-7=0 then, = ab bc ca 1) 2 2) -7 3) 6 4)
111 Sl Sol n : GE G.E.= a+b+c = abc = 6 7 a abc Ans: (3)
112 53. Remainder when x + x + 1 is divided by x + 1 is 1) 0 2) 1 3) 2 4) -1
113 n Sol : When f(x) is divided by y( (x-a) the remainder = f(a) = f(-1) = = 1
114 2 54. The domain of 4x-x is 1) [0, 4] 2) (0, 4) 3) R-(0,4) 4) R-[0,4]
115 n 2 2 Sol : 4x-x 0 i.e., x -4x 0 Ans: (1) 2 (x-2) 4 x x-2 2, 0 x 4
116 55. The range of the function f(x)= x-2, x 2 is 2-x 1)1 2) -1 3) {1} 4){ -1}
117 ( ) n Sol : f(x)= - x-2 x2 = -1 x-2 Range (f) = { -1} Ans : (4)
118 56. The range of the function Sin([x]π), (where [x] is greatest integer function) is 1) 0 2){0} 3)[-1, 1] 4)(0,1)
119 n Sol : [x]=integer Sin(kπ) =0 k Z Range ={0} Ans: (2)
120 57. A set A has 6 elements. Then the number of possible relations on A is ) 6 2) 2 3)2 4)6
121 n Sol : No. of possible relations mn from A to B is 2 where m=o(a) and n=o(b) 36 Hence Req.=2 Ans : (3)
122 58. The number of functions from a set A containing 7 elements into a set B containing 3 elements is 7 3 1) 3 2) 7 3) 3 4) 7
123 n Sol : No. of functions from a finite set A into a finite set B is [ n(b) ] n(a) 7 Req. No. of functions = 3 Ans : (3)
124 59. If 3x = (x-6)(x+a) (x-6) x+a then a = 1) 4 2) 3 3) 2 4) 1
125 n Sol : Put x = 6 in Nr. of LHS and Dr. of II term 3(6) we get, = 2 a=3 6+a Ans: (2)
126 60. In the expansion of 50 (1 + x), the sum of the coefficients i odd powers of x is ) 0 2) 2 3) 2 4) 2
127 n Sol : Req. Sum = C +C +C +...+C+C = 2 = 2 Ans : (3)
Formulae and Summary
Appendix A Formulae and Summary Note to student: It is not useful to memorise all the formulae, partly because many of the complicated formulae may be obtained from the simpler ones. Rather, you should
More informationLesson-3 TRIGONOMETRIC RATIOS AND IDENTITIES
Lesson- TRIGONOMETRIC RATIOS AND IDENTITIES Angle in trigonometry In trigonometry, the measure of an angle is the amount of rotation from B the direction of one ray of the angle to the other ray. Angle
More information2 Trigonometric functions
Theodore Voronov. Mathematics 1G1. Autumn 014 Trigonometric functions Trigonometry provides methods to relate angles and lengths but the functions we define have many other applications in mathematics..1
More information[STRAIGHT OBJECTIVE TYPE] log 4 2 x 4 log. x x log 2 x 1
[STRAIGHT OBJECTIVE TYPE] Q. The equation, log (x ) + log x. log x x log x + log x log + log / x (A) exactly one real solution (B) two real solutions (C) real solutions (D) no solution. = has : Q. The
More informationC3 Exam Workshop 2. Workbook. 1. (a) Express 7 cos x 24 sin x in the form R cos (x + α) where R > 0 and 0 < α < 2
C3 Exam Workshop 2 Workbook 1. (a) Express 7 cos x 24 sin x in the form R cos (x + α) where R > 0 and 0 < α < 2 π. Give the value of α to 3 decimal places. (b) Hence write down the minimum value of 7 cos
More informationSTUDY PACKAGE. Subject : Mathematics Topic: Trigonometric Equation & Properties & Solution of Triangle ENJOY MATHEMA WITH
fo/u fopkjr Hkh# tu] ugha vkjehks dke] foifr ns[k NksMs rqjar e/;e eu dj ';kea iq#"k flag ladyi dj] lgrs foifr vusd] ^cuk^ u NksMs /;s; dks] j?kqcj jk[ks VsdAA jfpr% ekuo /kez iz.ksrk ln~xq# Jh j.knksmnklth
More information[STRAIGHT OBJECTIVE TYPE] Q.4 The expression cot 9 + cot 27 + cot 63 + cot 81 is equal to (A) 16 (B) 64 (C) 80 (D) none of these
Q. Given a + a + cosec [STRAIGHT OBJECTIVE TYPE] F HG ( a x) I K J = 0 then, which of the following holds good? (A) a = ; x I a = ; x I a R ; x a, x are finite but not possible to find Q. The minimum value
More informationINVERSE TRIGONOMETRY: SA 4 MARKS
INVERSE TRIGONOMETRY: SA MARKS To prove Q. Prove that sin - tan - 7 = π 5 Ans L.H.S = Sin - tan - 7 5 = A- tan - 7 = tan - 7 tan- let A = Sin - 5 Sin A = 5 = tan - ( ( ) ) tan - 7 9 6 tan A = A = tan-
More information(+1) PAPER -A IIT-JEE (2013) (Trigonomtery-1) Solutions
L.K. Gupta (Mathematic Classes) www.pioneermathematics.com MOBILE: 9877, 4677 IIT-JEE () (Trigonomtery-) Solutions (+) PAPER -A TOWARDS IIT- JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL
More informationWORKBOOK. MATH 30. PRE-CALCULUS MATHEMATICS.
WORKBOOK. MATH 30. PRE-CALCULUS MATHEMATICS. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Contributor: U.N.Iyer Department of Mathematics and Computer Science, CP 315, Bronx Community College, University
More informationTrigonometric Identities and Equations
Trigonometric Identities and Equations Art Fortgang, (ArtF) Lori Jordan, (LoriJ) Say Thanks to the Authors Click http://www.ck.org/saythanks (No sign in required) To access a customizable version of this
More informationFrom now on angles will be drawn with their vertex at the. The angle s initial ray will be along the positive. Think of the angle s
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 1 Chapter 8A Angles and Circles From now on angles will be drawn with their vertex at the The angle s initial ray will be along the positive.
More informationAMB121F Trigonometry Notes
AMB11F Trigonometry Notes Trigonometry is a study of measurements of sides of triangles linked to the angles, and the application of this theory. Let ABC be right-angled so that angles A and B are acute
More informationMath 112, Precalculus Mathematics Sample for the Final Exam.
Math 11, Precalculus Mathematics Sample for the Final Exam. Solutions. There is no promise of infallibility. If you get a different solution, do not be discouraged, but do contact me. (1) If the graph
More informationContact hour per week: 04 Contact hour per Semester: 64 ALGEBRA 1 DETERMINANTS 2 2 MATRICES 4 3 BINOMIAL THEOREM 3 4 LOGARITHMS 2 5 VECTOR ALGEBRA 6
BOARD OF TECHNICAL EXAMINATION KARNATAKA SUBJECT: APPLIED MATHEMATICS I For I- semester DIPLOMA COURSES OF ALL BRANCHES Contact hour per week: 04 Contact hour per Semester: 64 UNIT NO. CHAPTER TITLE CONTACT
More informationMathematics, Algebra, and Geometry
Mathematics, Algebra, and Geometry by Satya http://www.thesatya.com/ Contents 1 Algebra 1 1.1 Logarithms............................................ 1. Complex numbers........................................
More informationThe matrix: "Is an organization of some elements written in rows
st term st Sec. Algebra. The matrix: "Is an organization of some elements written in rows Ex: and columns between brackets in the form ( ) ". st column nd rd - st row - nd row 7 rd row * The order of any
More information(This type of questions may be asked in the examination )
34. The slope of a straight line parallel to the line 2x + 4y + 5 = 0 is... a) 2 b) 1 / 2 c) - 1 / 2 d) - 2 35. The angle of inclination of a straight line whose slope is is... a) 0 0 b) 30 0 c) 60 0 d)
More informationChapter 1. Functions 1.3. Trigonometric Functions
1.3 Trigonometric Functions 1 Chapter 1. Functions 1.3. Trigonometric Functions Definition. The number of radians in the central angle A CB within a circle of radius r is defined as the number of radius
More informationSection 6.2 Trigonometric Functions: Unit Circle Approach
Section. Trigonometric Functions: Unit Circle Approach The unit circle is a circle of radius centered at the origin. If we have an angle in standard position superimposed on the unit circle, the terminal
More informationSince 1 revolution = 1 = = Since 1 revolution = 1 = =
Fry Texas A&M University Math 150 Chapter 8A Fall 2015! 207 Since 1 revolution = 1 = = Since 1 revolution = 1 = = Convert to revolutions (or back to degrees and/or radians) a) 45! = b) 120! = c) 450! =
More informationMath 112, Precalculus Mathematics Sample for the Final Exam.
Math 11, Precalculus Mathematics Sample for the Final Exam. Phone use is not allowed on this exam. You may use a standard two sided sheet of note paper and a calculator. The actual final exam consists
More information1.3 Basic Trigonometric Functions
www.ck1.org Chapter 1. Right Triangles and an Introduction to Trigonometry 1. Basic Trigonometric Functions Learning Objectives Find the values of the six trigonometric functions for angles in right triangles.
More information4. The G.C.D of 15x 4 y 3 z 5,12x 2 y 7 z 2... a) 15x 4 y 7 z 5 b)3x 2 y 3 z 2 c)12x 2 y 3 z 2 d)3x 4 y 7 z 5
4. The slope of a straight line parallel to the line x + 4y + 5 = 0 is... a) b) 1 / c) - 1 / d) - 5. The angle of inclination of a straight line whose slope is is... a) 0 0 b) 0 0 c) 60 0 d) 90 0 6. The
More informationTransweb Educational Services Pvt. Ltd Tel:
. An aeroplane flying at a constant speed, parallel to the horizontal ground, km above it, is observed at an elevation of 6º from a point on the ground. If, after five seconds, its elevation from the same
More informationSETS. set of natural numbers by N set of integers by Z set of rational numbers by Q set of irrational numbers by T
Chapter SETS. Overview This chapter deals with the concept of a set, operations on sets.concept of sets will be useful in studying the relations and functions... Set and their representations A set is
More informationCore Mathematics 3 Trigonometry
Edexcel past paper questions Core Mathematics 3 Trigonometry Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Maths 3 Trigonometry Page 1 C3 Trigonometry In C you were introduced to radian measure
More informationTrigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters
Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters α( alpha), β ( beta), θ ( theta) as well as upper case letters A,B,
More informationMath 112, Precalculus Mathematics Solutions to Sample for the Final Exam.
Math 11, Precalculus Mathematics Solutions to Sample for the Final Exam. Phone and calculator use is not allowed on this exam. You may use a standard one sided sheet of note paper. The actual final exam
More informationMath Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
More informationSpherical trigonometry
Spherical trigonometry 1 The spherical Pythagorean theorem Proposition 1.1 On a sphere of radius, any right triangle AC with C being the right angle satisfies cos(c/) = cos(a/) cos(b/). (1) Proof: Let
More informationMath Analysis Chapter 5 Notes: Analytic Trigonometric
Math Analysis Chapter 5 Notes: Analytic Trigonometric Day 9: Section 5.1-Verifying Trigonometric Identities Fundamental Trig Identities Reciprocal Identities: 1 1 1 sin u = cos u = tan u = cscu secu cot
More informationA-Level Mathematics TRIGONOMETRY. G. David Boswell - R2S Explore 2019
A-Level Mathematics TRIGONOMETRY G. David Boswell - R2S Explore 2019 1. Graphs the functions sin kx, cos kx, tan kx, where k R; In these forms, the value of k determines the periodicity of the trig functions.
More informationweebly.com/ Core Mathematics 3 Trigonometry
http://kumarmaths. weebly.com/ Core Mathematics 3 Trigonometry Core Maths 3 Trigonometry Page 1 C3 Trigonometry In C you were introduced to radian measure and had to find areas of sectors and segments.
More informationoo ks. co m w w w.s ur ab For Order : orders@surabooks.com Ph: 960075757 / 84000 http://www.trbtnpsc.com/07/08/th-eam-model-question-papers-download.html Model Question Papers Based on Scheme of Eamination
More informationD. 6. Correct to the nearest tenth, the perimeter of the shaded portion of the rectangle is:
Trigonometry PART 1 Machine Scored Answers are on the back page Full, worked out solutions can be found at MATH 0-1 PRACTICE EXAM 1. An angle in standard position θ has reference angle of 0 with sinθ
More information3.1 Fundamental Identities
www.ck.org Chapter. Trigonometric Identities and Equations. Fundamental Identities Introduction We now enter into the proof portion of trigonometry. Starting with the basic definitions of sine, cosine,
More informationDESIGN OF THE QUESTION PAPER Mathematics Class X NCERT. Time : 3 Hours Maximum Marks : 80
DESIGN OF THE QUESTION PAPER Mathematics Class X Weightage and the distribution of marks over different dimensions of the question shall be as follows: (A) Weightage to Content/ Subject Units : S.No. Content
More informationMATHEMATICS. 1. The line parallel to x- axis and passing through the intersection of the lines ax + 2by + 3b =0 And bx -2ay -3a = 0 where (a, b) 0.
1. The line parallel to x- axis and passing through the intersection of the lines ax + 2by + 3b =0 And bx -2ay -3a = 0 where (a, b) 0. a) Below x- axis at a distance 2/3 from it b) Below x- axis at a distance
More informationA List of Definitions and Theorems
Metropolitan Community College Definition 1. Two angles are called complements if the sum of their measures is 90. Two angles are called supplements if the sum of their measures is 180. Definition 2. One
More informationWORKBOOK. MATH 06. BASIC CONCEPTS OF MATHEMATICS II. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
WORKBOOK. MATH 06. BASIC CONCEPTS OF MATHEMATICS II. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Contributors: N. Apostolakis, M. Bates, S. Forman, U. Iyer, A. Kheyfits, R. Kossak, A. McInerney, and
More informationADDITONAL MATHEMATICS
ADDITONAL MATHEMATICS 00 0 CLASSIFIED TRIGONOMETRY Compiled & Edited B Dr. Eltaeb Abdul Rhman www.drtaeb.tk First Edition 0 5 Show that cosθ + + cosθ = cosec θ. [3] 0606//M/J/ 5 (i) 6 5 4 3 0 3 4 45 90
More informationMATH 127 SAMPLE FINAL EXAM I II III TOTAL
MATH 17 SAMPLE FINAL EXAM Name: Section: Do not write on this page below this line Part I II III TOTAL Score Part I. Multiple choice answer exercises with exactly one correct answer. Each correct answer
More informationDual-Enrollment Final Exam Preparation
Dual-Enrollment Final Exam Preparation Dates: May 7 th and 8 th : Part 1 (75 minutes) 20-25 questions covering 1 st Semester Material May 9 th and 10 th Part 2 (75 minutes) 35-40 Questions covering 2 nd
More informationMath Trigonometry Final Exam
Math 1613 - Trigonometry Final Exam Name: Instructions: Please show all of your work. If you need more room than the problem allows, use a new plain white sheet of paper with the problem number printed
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Trigonometry
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH0000 SEMESTER 1 017/018 DR. ANTHONY BROWN 5. Trigonometry 5.1. Parity and Co-Function Identities. In Section 4.6 of Chapter 4 we looked
More informationSum and Difference Identities
Sum and Difference Identities By: OpenStaxCollege Mount McKinley, in Denali National Park, Alaska, rises 20,237 feet (6,168 m) above sea level. It is the highest peak in North America. (credit: Daniel
More information*n23494b0220* C3 past-paper questions on trigonometry. 1. (a) Given that sin 2 θ + cos 2 θ 1, show that 1 + tan 2 θ sec 2 θ. (2)
C3 past-paper questions on trigonometry physicsandmathstutor.com June 005 1. (a) Given that sin θ + cos θ 1, show that 1 + tan θ sec θ. (b) Solve, for 0 θ < 360, the equation tan θ + secθ = 1, giving your
More informationThese items need to be included in the notebook. Follow the order listed.
* Use the provided sheets. * This notebook should be your best written work. Quality counts in this project. Proper notation and terminology is important. We will follow the order used in class. Anyone
More information( )( ) PR PQ = QR. Mathematics Class X TOPPER SAMPLE PAPER-1 SOLUTIONS. HCF x LCM = Product of the 2 numbers 126 x LCM = 252 x 378
Mathematics Class X TOPPER SAMPLE PAPER- SOLUTIONS Ans HCF x LCM Product of the numbers 6 x LCM 5 x 378 LCM 756 ( Mark) Ans The zeroes are, 4 p( x) x + x 4 x 3x 4 ( Mark) Ans3 For intersecting lines: a
More informationSeptember [KV 806] Sub. Code: 3806
September - 2009 [KV 806] Sub. Code: 3806 (Regulations 2008-2009) (Candidates admitted from 2008-2009 onwards) Paper VI REMEDIAL MATHEMATICS Time : Three hours Maximum : 70 marks Answer All questions I.
More informationTHE COMPOUND ANGLE IDENTITIES
TRIGONOMETRY THE COMPOUND ANGLE IDENTITIES Question 1 Prove the validity of each of the following trigonometric identities. a) sin x + cos x 4 4 b) cos x + + 3 sin x + 2cos x 3 3 c) cos 2x + + cos 2x cos
More informationFormulas to remember
Complex numbers Let z = x + iy be a complex number The conjugate z = x iy Formulas to remember The real part Re(z) = x = z+z The imaginary part Im(z) = y = z z i The norm z = zz = x + y The reciprocal
More informationCBSE QUESTION PAPER CLASS-X MATHS
CBSE QUESTION PAPER CLASS-X MATHS SECTION - A Question 1:If sin α = 1 2, then the value of 4 cos3 α 3 cos α is (a)0 (b)1 (c) 1 (d)2 Question 2: If cos 2θ = sin(θ 12 ), where2θ and (θ 12 ) are both acute
More informationPart r A A A 1 Mark Part r B B B 2 Marks Mark P t ar t t C C 5 Mar M ks Part r E 4 Marks Mark Tot To a t l
Part Part P t Part Part Total A B C E 1 Mark 2 Marks 5 Marks M k 4 Marks CIRCLES 12 Marks approximately Definition ; A circle is defined as the locus of a point which moves such that its distance from
More informationTime: 3 Hrs. M.M. 90
Class: X Subject: Mathematics Topic: SA1 No. of Questions: 34 Time: 3 Hrs. M.M. 90 General Instructions: 1. All questions are compulsory. 2. The questions paper consists of 34 questions divided into four
More information3.1 Fundamental Identities
www.ck.org Chapter. Trigonometric Identities and Equations. Fundamental Identities Introduction We now enter into the proof portion of trigonometry. Starting with the basic definitions of sine, cosine,
More informationSolving equations UNCORRECTED PAGE PROOFS
1 Solving equations 1.1 Kick off with CAS 1. Polynomials 1.3 Trigonometric symmetry properties 1.4 Trigonometric equations and general solutions 1.5 Literal equations and simultaneous equations 1.6 Review
More informationAll In One Multi-Color Printing A way to 100% PLUS ONE MATHEMATICS REFERENCE BOOK-CHAPTER 1 f(x) = 5x + For CBSE Schools/ State Boards with NCERT Syllabus Theory explained with simple examples NCERT Exercises
More informationChapter 5: Trigonometric Functions of Angles Homework Solutions
Chapter : Trigonometric Functions of Angles Homework Solutions Section.1 1. D = ( ( 1)) + ( ( )) = + 8 = 100 = 10. D + ( ( )) + ( ( )) = + = 1. (x + ) + (y ) =. (x ) + (y + 7) = r To find the radius, we
More information10 th MATHS SPECIAL TEST I. Geometry, Graph and One Mark (Unit: 2,3,5,6,7) , then the 13th term of the A.P is A) = 3 2 C) 0 D) 1
Time: Hour ] 0 th MATHS SPECIAL TEST I Geometry, Graph and One Mark (Unit:,3,5,6,7) [ Marks: 50 I. Answer all the questions: ( 30 x = 30). If a, b, c, l, m are in A.P. then the value of a b + 6c l + m
More informationMockTime.com. (a) 36 (b) 33 (c) 20 (d) 6
185 NDA Mathematics Practice Set 1. Which of the following statements is not correct for the relation R defined by arb if and only if b lives within one kilometer from a? R is reflexive R is symmetric
More informationChapter 4 Trigonometric Functions
Chapter 4 Trigonometric Functions Overview: 4.1 Radian and Degree Measure 4.2 Trigonometric Functions: The Unit Circle 4.3 Right Triangle Trigonometry 4.4 Trigonometric Functions of Any Angle 4.5 Graphs
More informationFor a semi-circle with radius r, its circumfrence is πr, so the radian measure of a semi-circle (a straight line) is
Radian Measure Given any circle with radius r, if θ is a central angle of the circle and s is the length of the arc sustained by θ, we define the radian measure of θ by: θ = s r For a semi-circle with
More informationTrigonometry - Part 1 (12 pages; 4/9/16) fmng.uk
Trigonometry - Part 1 (12 pages; 4/9/16) (1) Sin, cos & tan of 30, 60 & 45 sin30 = 1 2 ; sin60 = 3 2 cos30 = 3 2 ; cos60 = 1 2 cos45 = sin45 = 1 2 = 2 2 tan45 = 1 tan30 = 1 ; tan60 = 3 3 Graphs of y =
More informationGeometry The Unit Circle
Geometry The Unit Circle Day Date Class Homework F 3/10 N: Area & Circumference M 3/13 Trig Test T 3/14 N: Sketching Angles (Degrees) WKS: Angles (Degrees) W 3/15 N: Arc Length & Converting Measures WKS:
More informationMockTime.com. (b) (c) (d)
373 NDA Mathematics Practice Set 1. If A, B and C are any three arbitrary events then which one of the following expressions shows that both A and B occur but not C? 2. Which one of the following is an
More informationTRIGONOMETRY. Units: π radians rad = 180 degrees = 180 full (complete) circle = 2π = 360
TRIGONOMETRY Units: π radians 3.14159265 rad 180 degrees 180 full (complete) circle 2π 360 Special Values: 0 30 (π/6) 45 (π/4) 60 (π/3) 90 (π/2) sin(θ) 0 ½ 1/ 2 3/2 1 cos(θ) 1 3/2 1/ 2 ½ 0 tan(θ) 0 1/
More informationIIT JEE (2013) (Trigonometry and Algebra)
L.K. Gupta (Mathematic Classes) www.pioneermathematics.com MOBILE: 985577, 4677 PAPER B IIT JEE () (Trigonometry and Algebra) TOWARDS IIT JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL SURVIVE
More informationCollege Trigonometry
College Trigonometry George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 11 George Voutsadakis (LSSU) Trigonometry January 015 1 / 8 Outline 1 Trigonometric
More informationand sinθ = cosb =, and we know a and b are acute angles, find cos( a+ b) Trigonometry Topics Accuplacer Review revised July 2016 sin.
Trigonometry Topics Accuplacer Revie revised July 0 You ill not be alloed to use a calculator on the Accuplacer Trigonometry test For more information, see the JCCC Testing Services ebsite at http://jcccedu/testing/
More informationCONTENTS. FOREWORD iii PREFACE v CHAPTER 1 Sets 1. CHAPTER 2 Relations and Functions 19. CHAPTER 3 Trigonometric Functions 34
CONTENTS FOREWORD iii PREFACE v CHAPTER Sets CHAPTER Relations and Functions 9 CHAPTER 3 Trigonometric Functions 34 CHAPTER 4 Principle of Mathematical Induction 6 CHAPTER 5 Complex Numbers and Quadratic
More informationSolutions for Trigonometric Functions of Any Angle
Solutions for Trigonometric Functions of Any Angle I. Souldatos Answers Problem... Consider the following triangle with AB = and AC =.. Find the hypotenuse.. Find all trigonometric numbers of angle B..
More informationThe American School of Marrakesh. AP Calculus AB Summer Preparation Packet
The American School of Marrakesh AP Calculus AB Summer Preparation Packet Summer 2016 SKILLS NEEDED FOR CALCULUS I. Algebra: *A. Exponents (operations with integer, fractional, and negative exponents)
More information2 Recollection of elementary functions. II
Recollection of elementary functions. II Last updated: October 5, 08. In this section we continue recollection of elementary functions. In particular, we consider exponential, trigonometric and hyperbolic
More informationCK- 12 Algebra II with Trigonometry Concepts 1
14.1 Graphing Sine and Cosine 1. A.,1 B. (, 1) C. 3,0 D. 11 1, 6 E. (, 1) F. G. H. 11, 4 7, 1 11, 3. 3. 5 9,,,,,,, 4 4 4 4 3 5 3, and, 3 3 CK- 1 Algebra II with Trigonometry Concepts 1 4.ans-1401-01 5.
More information(c) n (d) n 2. (a) (b) (c) (d) (a) Null set (b) {P} (c) {P, Q, R} (d) {Q, R} (a) 2k (b) 7 (c) 2 (d) K (a) 1 (b) 3 (c) 3xyz (d) 27xyz
318 NDA Mathematics Practice Set 1. (1001)2 (101)2 (110)2 (100)2 2. z 1/z 2z z/2 3. The multiplication of the number (10101)2 by (1101)2 yields which one of the following? (100011001)2 (100010001)2 (110010011)2
More informationHIGHER SECONDARY FIRST YEAR MATHEMATICS. TRIGONOMETRY Creative Questions Time : 1.15 Hrs Marks : 45 Part - I Choose the correct answer 10 1 = 10
HIGHER SECONDARY FIRST YEAR MATHEMATICS TRIGONOMETRY Creative Questions Time :. Hrs Marks : Part - I Choose the correct answer. If cos x, then x a) n b). The domin of the function y x a) (n ) c) y b),
More informationTrig Identities, Solving Trig Equations Answer Section
Trig Identities, Solving Trig Equations Answer Section MULTIPLE CHOICE. ANS: B PTS: REF: Knowledge and Understanding OBJ: 7. - Compound Angle Formulas. ANS: A PTS: REF: Knowledge and Understanding OBJ:
More informationREQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS
REQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS The Department of Applied Mathematics administers a Math Placement test to assess fundamental skills in mathematics that are necessary to begin the study
More informationCK-12 Trigonometry - Second Edition, Solution Key
CK-1 Trigonometry - Second Edition, Solution Key CK-1 Foundation Say Thanks to the Authors Click http://www.ck1.org/saythanks (No sign in required) www.ck1.org To access a customizable version of this
More informationsecθ 1 cosθ The pythagorean identities can also be expressed as radicals
Basic Identities Section Objectives: Students will know how to use fundamental trigonometric identities to evaluate trigonometric functions and simplify trigonometric expressions. We use trig. identities
More informationπ π π π Trigonometry Homework Booklet 1. Convert 5.3 radians to degrees. A B C D Determine the period of 15
Trigonometry Homework Booklet 1. Convert 5.3 radians to degrees. A. 0.09 B. 0.18 C. 151.83 D. 303.67. Determine the period of y = 6cos x + 8. 15 15 A. B. C. 15 D. 30 15 3. Determine the exact value of
More informationAP Calculus Summer Packet
AP Calculus Summer Packet Writing The Equation Of A Line Example: Find the equation of a line that passes through ( 1, 2) and (5, 7). ü Things to remember: Slope formula, point-slope form, slopeintercept
More informationPreview from Notesale.co.uk Page 2 of 42
. CONCEPTS & FORMULAS. INTRODUCTION Radian The angle subtended at centre of a circle by an arc of length equal to the radius of the circle is radian r o = o radian r r o radian = o = 6 Positive & Negative
More informationDISCUSSION CLASS OF DAX IS ON 22ND MARCH, TIME : 9-12 BRING ALL YOUR DOUBTS [STRAIGHT OBJECTIVE TYPE]
DISCUSSION CLASS OF DAX IS ON ND MARCH, TIME : 9- BRING ALL YOUR DOUBTS [STRAIGHT OBJECTIVE TYPE] Q. Let y = cos x (cos x cos x). Then y is (A) 0 only when x 0 (B) 0 for all real x (C) 0 for all real x
More information11) If tan = and tan =2-, show that ( - )= ". 13) 14) 15) 19)
TRIGONOMETRIC RATIOS OF COMPOUND ANGES 1) Show that tan( )tan( ) 2) Prove that 3) Prove that cot (or) tan 4) If AB,show that (1tanA)(1tanB)2 5) Prove that cos cos sin sin 6) ) 7) - 8) 9) - 10) - 11) If
More informationCO-ORDINATE GEOMETRY
CO-ORDINATE GEOMETRY 1 To change from Cartesian coordinates to polar coordinates, for X write r cos θ and for y write r sin θ. 2 To change from polar coordinates to cartesian coordinates, for r 2 write
More informationPre- Calculus Mathematics Trigonometric Identities and Equations
Pre- Calculus Mathematics 12 6.1 Trigonometric Identities and Equations Goal: 1. Identify the Fundamental Trigonometric Identities 2. Simplify a Trigonometric Expression 3. Determine the restrictions on
More informationLone Star College-CyFair Formula Sheet
Lone Star College-CyFair Formula Sheet The following formulas are critical for success in the indicated course. Student CANNOT bring these formulas on a formula sheet or card to tests and instructors MUST
More information1 Arithmetic calculations (calculator is not allowed)
1 ARITHMETIC CALCULATIONS (CALCULATOR IS NOT ALLOWED) 1 Arithmetic calculations (calculator is not allowed) 1.1 Check the result Problem 1.1. Problem 1.2. Problem 1.3. Problem 1.4. 78 5 6 + 24 3 4 99 1
More informationChapter 1: Analytic Trigonometry
Chapter 1: Analytic Trigonometry Chapter 1 Overview Trigonometry is, literally, the study of triangle measures. Geometry investigated the special significance of the relationships between the angles and
More informationUsing this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained.
Angle in Standard Position With the Cartesian plane, we define an angle in Standard Position if it has its vertex on the origin and one of its sides ( called the initial side ) is always on the positive
More information2017 FAMAT State Convention. Alpha Trigonometry
017 FAMAT State Convention Alpha Trigonometry 1 On this test, select the best answer choice for each question. If you believe that the correct answer is not among the choices, or that the question is flawed,
More informationRecall from Geometry the following facts about trigonometry: SOHCAHTOA: adjacent hypotenuse. cosa =
Chapter 1 Overview Trigonometry is, literally, the study of triangle measures. Geometry investigated the special significance of the relationships between the angles and sides of a triangle, especially
More informationAS and A-level Mathematics Teaching Guidance
ΑΒ AS and A-level Mathematics Teaching Guidance AS 7356 and A-level 7357 For teaching from September 017 For AS and A-level exams from June 018 Version 1.0, May 017 Our specification is published on our
More information2Trigonometry UNCORRECTED PAGE PROOFS. 2.1 Kick off with CAS
. Kick off with CAS Trigonometr. Reciprocal trigonometric functions. Trigonometric identities using reciprocal trigonometric functions. Compound-angle formulas.5 Double-angle formulas. Inverse trigonometric
More informationISI, CMI, Olympiads Study Material
B. Stat. (Hons.) Admission Test : 2009 Group A (Each of the following questions has exactly one correct option and you have to identify it) 1. If k times the sum of the first n natural numbers is equal
More informationSpecial Mathematics Notes
Special Mathematics Notes Tetbook: Classroom Mathematics Stds 9 & 10 CHAPTER 6 Trigonometr Trigonometr is a stud of measurements of sides of triangles as related to the angles, and the application of this
More informationTRIGONOMETRY INTRODUCTION. Objectives. SESSION 1-5 ANGLES A positive angle measures a rotation in an anticlockwise direction.
TRIGONOMETRY INTRODUCTION s the title of the unit suggests, it deals with the calculation of angles or the length of their sides. In this unit, the trigonometric ratios of acute angles, general angles
More information