TRIGONOMETRY
BOOSTER - A
1) Change into degree system (30'). (1/2)°
2) 30°30' change into degree. (61/2)°
3) 50' change into grade system. (1/2)ᵍ
4) 340° change into radian system. (17π/9)ᶜ
5) Show: (secx + tanx -1)(secx - tanx +1) - 2tanx =0.
6) If tanx = -4/3, then sinx is
a) -4/5 but not 4/5
b) -4/5 or 4/5
c) 4/5 but not -4/5 d) none. b
7) Find sinx and tanx if cosx = -12/13 and x lies in the third quadrant.
8) Find the value of the following:
a) cos(-45°). 1/√2
b) sin(-30°). -1/2
c) cot(-60°). -1/√3
9) Find the values of the following trigonometry ratios:
a) sin315. -1/√2
b) cos210. -1/2
c) cos(-480).
10) Show: {cosec(2π+x) cos(2π+x) tan(π/2+ x)}/{sec(π/2+ x) cosx cot(π+ x)}. 1
11) Convert each of the following products into the sum of different of sines and cosines:
a) 2sin5x cosx. Sin6x+ sin4x
b) 2cos4x cos3x. Cos7x + cosx
c) sin75° cos15°.
12) Express each of the following as a product:
a) sin4x + sin2x. 2sin3x cosx
b) cos6x - cos8x. 2sin7x sinx
13) Show: (cosx + cosy)² + (sinx + siny)²= 4 cos²{(x+ y)/2}
14) Show: √[2+ √{2+ √(2+ 2 cos8x)= 2cosx.
15) If A+ B + C =π, show that sin2A+ sin2B + sin2C = 4 sinA sinB sinC.
16) If A+ B + C =π, show that cos²A+ cos²B + cos²C = 1- 2 cosA cosB cosC.
17) If A+ B + C =π, show that tanA+ tanB + tanC = tanA tanB tanC.
18) Find the maximum value of 3sinx + 4cosx. 5
19) Show: 5 cosx + 3cos(x + π/3)+ 3, lies between -4 and 10.
20) If 3sinx + 5cosx =5, show that 5sinx - 3cosx = ±3.
21) If cosecx - sinx = m and secx - cosx = n, eliminate x. ³√(m²n)²+ ³√(mn²)²=1
BOOSTER - B
1) If A,B,C,D are angles of a cyclic quadrilateral, show that cosA+ cosB + cosC + cosD = 0.
2) If sin(120° - x)= sin(120° - y), 0 < x, y <π, then find the relation between x and y. x= y or x+ y = 60°
3) Show: cos²y + cos²(x + y) - 2 cosx cosy cos(x + y) is independent of y.
4) If 3 tanx tany =1, show that 2cos(x + y)= cos(x - y).
5) Show: {(cosx+ cosy)/(sinx- siny)}ⁿ + {(sinx + siny)/(cosx - cosy)}ⁿ = 2 cotⁿ{(x -y)/2} or 0 according as n is even or odd.
6) If x+ y=90°, find the maximum value of sinx siny.
7) Show: (sec8x -1)/(sec4x -1)= tan8x/tan2y.
8) Find the value of cos(2π/7)+ cos(4π/7)+ cos(6π/7). -1/2
9) Show: sin6 sin42 sin66 sin78= 1/16
10) In ∆ ABC, show that tanA+ tanB + tanC ≥ 3√3 where A, B, C are acute angles.
11) In ∆ ABC, show that cosA+ cosB + cosC ≤ 3/2.
12) If tanx = a - 1/4a then secx - tanx is equal to
a) -2a, 1/2a b) -1/2a, 2a c) 2a d) 1/2a, 2a a
13) cos³x sin2x = ⁿₘ₌₁∑ aₘ sinmx is an identity in x. Then
a) a₃= 3/8, a₂ =0
b) n= 6, a₁ = 1/2
c) n=5, a₁= 1/4
d) ∑ aₘ = 3/4. c
14) The value of cos(π/15)cos(2π/15) cos(3π/15) cos(4π/15) cos(5π/15) cos(6π/15) cos(π/15) is
a) 1/2⁶ b) 1/2⁷ c) 1/2⁸ d) none. c
15) If cosec x = (a²- b²)/(a²+ b²) where a, b are two unequal non-zero real numbers then show that x has no real value.
16) show: sin²(x + y)+ sin²(x + z) - 2sin(x + y) sin(x + z) cos(y - z) = sin²(y - z).
17) If tan(x + y+ z)/tan(x - y+ z)= tany/tanz, show sin(y - z)=0 or sin2x + sin2y + sin2x)=0
18) show: 4sin27°= √(5+ √5).
19) show: 2ˢᶦⁿˣ + 2ᶜᵒˢˣ ≥ ₂(1- 2/√2) for all x.
20) Show: ⁿᵣ₌₁∑ tan rα. tan(r +1)α = cotα tan(n +1)α - n -1.
21) If sinx = -1/2 and cosx = √3/2, then x lies in
a) 1st quadrant
b) 2nd quadrant
c) 3rd quadrant
d) 4th quadrant
BOOSTER- C
1) If tanx = +1/√5 and x lies in the 1st quadrant, then cosx is
a) 1/√6 b) -1/√6 c) √5/√6 d) - √5/√6
2) If sinx = 3/4 and tanx = 9/2 then cosx is
a) 1/6 b) 8/27 c) 27/8 d) 15/4
3) If tanx = a/b, then the value of (a sinx + b cosx)/(a sinx - b cosx) is
a) (a²+b²)/(a²-b²)
b) (a²- b²)/(a² + b²)
c) a/√(a²+b²)
d) b/√/(a²+ b²)
4) If (x/a) cosα + (y/b) sinα =1 and (x/a) sinα - (y/b) cosα = - 1, then x²/a²+ y²/b² is equal to
a) 0 b) 2 c) -1 d) 1
5) The minimum value of sinx cosx is
a) 1 b) 0 c) -1/2 d) 1/2
6) The maximum value of 12sinx - 9 sin²x is
a) 3 b) 4 c) 5 d) none
7) The minimum value of sinx - cosx is
a) -√2 b) √2 c) -2√2 d) 0
8) sin50 - sin70 + sin10 is equal to
a) 1 b) 0 c) 1/2 d) 2
9) If sinA = cosA, 0°< A < 90°, then A is equal to
a) 15 b) 30 c) 45 d) 60
10) If sinx = √3 cosx, then x is equal to (0< x< 90)
a) 45 b) 30 c) 75 d) 60
11) The value of the expression 3cosx + 4 sinx lie between
a) -7 and 7
b) -25 and 25
c) -1 and 1
d) -5 and 5
12) The value of the expression tan1° tan2°tan3°tan4°......tan87°tan88°tan89° is
a) 0 b) 1 c) 2 d) ∞
13) The value of sin²(π/8) + sin²(3π/8)+ sin²(5π/8) + sin²(7π/8) is
a) 1 b) 2 c) 9/8 d) 17/8
14) If sin(120- x)= sin(120- y) and 0< x,y < π, then all value of xy, are given by
a) x = y b) x = y or x+ y =π/2 c) x+y=0 d) x+ y=π/3
15) If sin²⁰x + cos⁴⁸x, then for all values of x:
a) A≥ 1 b) 0 < A≤ 1 c) 1 <A < 3 d) none
16) If sinx + sin²x =1, then cos²x + cos⁴x is equal to
a) 1 b) -1 c) 0 d) 2
1c 2a 3a 4b 5c 6b 7a 8b 9c 10d 11d 12b 13b 14b 15b 16a
BOOSTER - D
1) The value of cos1° cos2° cos3°.....cos179° is
a) 1/√2 b) 0 c) 1 d) 2
2) Which of the following is correct?
a) sin1°> sin 1
b) sin1°< sin 1
c) sin1°= sin 1
d) sin1°= πsin 1/180
3) The value of cot5° cot10°.....cot85° is
a) -1 b) 1 c) 1/2 d) 0
4) The maximum value of 5 cosx + 3 cos(x + π/3)+ 3, is
a) 5 b) 10 c) 11 d) -1
5) In a triangle ABC secA(cosB cosC - sinB sinC) is equal to
a) -1 b) 0 c) 1 d) none
6) If Pₙ = cosⁿx + sinⁿx, then 2P₆ - 3P₄ +1=
a) 2 b) 3 c) 0 d) 1
7) The value of sin(π/14) sin(3π/14) sin(5π/14) sin(7π/14) is
a) 1 b) 1/4 c) 1/8 d) √2/7
1b 2b 3b 4b 5a 6c 7c
BOOSTER - E
1) If cosx = (2cosy -1)/(2- cosy), then the value of tan(x/2) is equal to
a) √3tan(y/2)
b) tan(y/2)
c) (1/√3) tan(y/2)
d) 3tan(y/2)
2) If in a triangle PQR , sinP, sinQ, sinR are in AP, then
a) the attitude are in AP
b) the altitude are in HP.
c) the median are in GP
d) the medians are in AP.
3) The least value of 2sin²x + 3cos²x is
a) 1 b) 2 c) 3 d) 5
4) log tan1°+ log tan2°+ log tan3°+.... log tan89° is
a) 1 b) 1/√2 c) 0 d) -1
5) If αcos²3θ+ βcos⁴θ= 16 cos⁶θ+ 9 cos²θ is an identity, then
a) α = 1, β = 18
b) α = 1, β = 24
c) α = 3, β = 24
d) α = 4, β = 2
6) The value of cos²(π/12)+ cos²(π/4)+ cos²(5π/12)
a) 2/3 b) 2/(3 +√3) c) 3/2 d) (3+√6)/2
7) The value of cos(π/65) cos(2π/65) cos(4π/65) cos(8π/65) cos(16π/65) cos(32π/65) =
a) 1/8 b) 1/16 c) 1/32 d) 1/64
8) The radius of the circle whose arc of the length 15πcm an angle of 3π/4 radian at the centre is
a) 10cm b) 20cm c) 45/4 cm d) 45/2 cm
9) cos2(x + y)+ 4 cos(x + y) sinx siny + 2 sin²y=
a) cos2x b) cos3x c) sin2x d) sin3x
10) If A+ B+ C=180°, then the value of (cotB + cotC)(cotC + cotA)(cotA+ cotB) will be
a) secA secB secC
b) cosecA cosecB cosecC
c) tanA tanB tanC d) 1
1a 2b 3b 4c 5b 6c 7d 8b 9a 10b
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