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MATHEMATICS
Trigonometric Ratios & Identities
Basic Questions
1.Express tanθ in terms of cosθ.
2.Prove that 2(sin6θ+cos6θ) – 3(sin4θ+cos4θ)+1=0.
3.Prove that (sinθ+cosecθ)2+(cosθ+sec2θ) ≥9.
4.If tan2θ= 1 – e2 then show that sec θ+ tan3 θ cosec θ=(2 – e2)3/2.
Trigonometric Ratios of any Angle
5.Find the general value of θ satisfying both sin θ = -1/2 and tan θ =1/√3.
6.(i)Find the value of tan10 tan20tan30……………………tan890.
(ii)Prove that sin250+sin2100+…………..+sin2900 = 19/2.
Trigonometric Ratios of Compound Angles
7.Prove that tan 700= 2tan500 +tan200.
8.If A+B=450, show that(1+tanA)(1+tanB) =2.
9.(i)Show that cot (π/4+x) cot(π/4 – x)=1
ii) Give n 3 tanαtanβ =1, show that 2cos (α+β) = cos (α – β)
iii)Eliminate θ if tan (θ – α)=a and tan (θ+α)=b.
Trigonometry Ratios of Multiples of an Angle.
10.Find the values of (i) sin 180 (ii)tan 150.
11. Find the values of sin 67 10
2
12.(i)Prove that 2 sin2 θ +4cos(θ+α)sinα.sinθ + cos2(θ+α) is independent of θ.
(ii)If tanθ = b/a, then find the value of acos2θ +bsin2θ.
13.If α,β and γ are in A.P., show that cot β = sinα – sinγ
Cosγ – cosα
0
0
0
14.Show that sin 12 .Sin48 .sin54 = 1/8.
15.If sin x+ siny = √3(cosy – cosx) show that sin3x+sin3y=0.
16.If cos(A – B) + cos(C+D) = 0, then prove that tanA.tanB.tanC.tanD = -1.
Cos(A+B)
cos(C–D)
17.If sin (θ+α) = 1 – m , prove that tan (π/4 – θ) tan(π/4 – α) =m.
Cos(θ – α) 1+m
18.Prove that (cosα – cosβ)2 +(sin α – sinβ)2 = 4sin2 α – β
2
19.Prove that sinα+sin(α+2π/3) + sin (α+4π/3) =0.
20.Find the value of cos 22 10
2
IDENTITIES
1.If A+B+C =π, prove that sin 2A+sin2B+sin2C = 8cosA/2 cosB/2 cosC/2.
Cos A+cosB+cosC–1
2.If A,B and C are the angles of a triangle, show that tan2 A/2 + tan2B/2+tan2C/2 ≥1.
3.Find the maximum and minimum values of a cosθ+ b sinθ.
4.(i)If A+B+C =π, then prove that
Sin(B+2C) + sin(C+2A)+sin(A+2B) = 4sin (B – C) . sin (C – A). sin (A – B).
2
2
2
2
2
2
(ii)If A+B+C =π, then find the minimum value of cot A + cot B+cot C.
SOME USEFUL FORMULE
1.Simplify the product cosAcos2A.cos22A……………cos2n – 1 A.
2.Prove that cosπ/65 cos2π/65 cos4π/65 cos8π/65 cos16π/65 cos32π/65 = 1/64.
3.Prove that cos60 . cos420 .cos660 . cos780 = 1/16.
PROBLEMS
1.Let 0<A,B< π/2 satisfy the equations 3sin2A+2sin2B=1 and 3sin2A – 2sin2B =0. Prove that
A+2B = π/2.
2.Prove that cos 2π/15 . cos4π/15 cos8π/15.cos16π/15 =1/16.
3.If cosA = m cosB, then prove that cot A+B =m+1 tan B – A
2
m–1
2
4.If a cos2θ + b sin2θ =c has α and β as its solutions, then prove that :
tanα + tanβ = 2b , tanαtanβ = c – a
(c+a)
c+a
5.Prove that 5cosθ + 3cos(θ+π/3) +3 lies between -4 and 10.
6.(i)For all θ in [0,π/2], show that cos(sinθ) > sin(cosθ).
ii) Find the smallest positive number p for which the equation cos(p sinx) = sin(p cosx) has a
solution x ϵ [0,2π].
7.If ABC is a triangle and tan A/2, tan B/2, tanC/2 are in H.P., then find the minimum value of
cot B/2.
8.Let cosA+cosB+cosC = 3/2 in a triangle ABC. Show that the triangle is equilateral.
9.Prove that 1+cotθ≤ cotθ/2 for 0<θ<π. Find θ when equality sign holds.
OBJECTIVE
Multiple Choice (Single Correct)
1.If 3sinθ+5cosθ =5, then the value of 5sinθ – 3cosθ is equal to
a) 5
b) 3
c) 4
d) none of these
2.If tanθ = √n for some non – square natural number n, then sec 2θ is
a) a rational number
b) an irrational number
c) a positive number
d) none of these
3.The minimum value of cos(cosx) is
a) 0
b) –cos 1
c) cos1
d) -1
4.If sinx = cos2x, then cos2x(1+cos2x) is equal to
a) 0
b) 1
c) 2
d) none of these
5.The maximum value of 4 sin2x + 3cos2x + sin x/2 + cosx/2 is.
a) 4+√2
b) 3+√2
c) 9
d) 4.
6.If α and β are the solutions of sin2x + a sinx + b =0 as well as that of cos2x +c cosx +d=0, then
sin (α+β) is equal to.
a) 2bd
b) a2+c2
c) b2+d2
d) 2ac
2
2
b +d
2ac
2bd
a2+c2
7.If sinα, sinβ and cosα are in G.P, then roots of the equation x2+2x cotβ +1 =0 are always
a) equal
b) real
c) imaginary
d) greater than 1
2
2
2
8.If in a triangle ABC, sin A +sin B+sin C =2, then the triangle is always
a) isosceles triangle
b) right angled
c) acute angled
d) obtuse angled
9.If θ ≠ (2n+1) π/2, nϵl then the least value of (sinθ+cosecθ)2+ (cosθ+secθ)2 is
a) 2
b) 4
c) 8
d) 9.
10.If 4nα =π, then cotα cot2α cot3α……….cot(2n – 1)α is equal to
a) 0
b) 1
c) n
d) none of these
11.If cos5θ = a cosθ + bcos3θ +c cos5θ+d, then
a) a = 20
b) b = -30
c)a+b+c=2
d) a+b+c+d =1
12.If in a triangle ABC (sinA+sinB+sinC)(sinA+sinB – sinC) = 3 sinAsinB, then angle C is equal to
a) 300
b) 450
c) 600
d) 750
ASSIGNMENTS
Section – I
Part – A
Level – I
1.Prove the following:
i) cos2θ+cos2(θ+1200)+cos2(θ – 1200) = 3/2.
ii)(secA – cosecA)(1+tanA+cotA)=tanAsecA – cotAcosecA.
iii)(cosecθ – secθ)(cotθ – tanθ) = (cosecθ+secθ)(secθcosecθ – 2).
2.In a ∆ABC, prove that (sinA+sinB)(sinB+sinC)(sinC+sinA)> sinAsinBsinC.
3.If tan-1x+tan-1y+tan-1z =π/2 and (x – y)2+(y – z)2+(z – x)2=0,
Then prove that x2+y2+z2=1.
4.Prove that 2cosx – cos3x – cos5x=16cos3x sin2x.
5.Find the value of sin4π/8 + sin4 3π/8+sin4 5π/8+sin4 7π/8.