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Trigonometry Questions

1. If 2ycosθ = xsinθ and 2xsecθ - ycosecθ = 3, then value of x2 + y2 is




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Correct Ans:4
Explanation:
Given:
2ycosθ = xsinθ ; 2xsecθ - ycosecθ = 3

2xsecθ - ycosecθ = 3
2x/cosθ - y/sinθ = 3
2xsinθ - ycosθ = 3sinθ cosθ ..(i)

2ycosθ = xsinθ
ycosθ = xsinθ/2

Substitute ycosθ in (i),
2xsinθ - (xsinθ/2) = 3sinθ cosθ
(4xsinθ - xsinθ)/2 = 3sinθ cosθ
4xsinθ - xsinθ = 6sinθ cosθ
3xsinθ = 6sinθ cosθ
x = 2cosθ

ycosθ = 2cosθsinθ/2
y = sinθ
x2 + y2 = 4cos2θ + sin2θ
= 4(1) (wkt, sin2θ + cos2θ = 1 )
= 4.
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2. If θ be acute angle and cosθ = 15/17, then the value of cot (90° - θ) is




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Correct Ans:8/15
Explanation:
Given:
Cosθ = 15/17
WKT, sinθ = √(1 - cos2θ)
sinθ = √(1 - (15/17)2)
sinθ = √( 1 - 225/289)
sinθ = √( 64/289)
sinθ = 8/17

Cot (90°- θ) = tanθ
So, tanθ = sinθ/cosθ = (8/17)/(15/17)
= 8/15.
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3. If cotα = 3, then the value of (sin3α + cos3α)/cosα is ?




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Correct Ans:14/15
Explanation:
Given,
cotα = 3
cosα/sinα = 3
cosα = 3sinα
(sin3α + cos3α)/cosα
= (sin3α + 27sin3α)/3sinα
= 28sin3α/3sinα
= 28sin2α/3
WKT, 1/sinα = cosecα
= 28/3cosec2α
As, cosec2α - cot2α = 1
= (28/3)*(1/(1 + cot2α))
= (28/3)*(1/(1 + 32))
= (28/3)*(1/(1 + 9)
= (28/3)*(1/10)
= 14/15.
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4. If 7sinα = 24cosα; 0 < α < π/2, then the value of 14tanα - 75cosα - 7secα is equal to




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Correct Ans:2
Explanation:
Given: 7sinα = 24cosα
sinα /cosα = 24/7
tanα = 24/7
Here, 24 --- opposite side
7 ---- adjacent side
So, Hypotenuse = √[opposite side2 + adjacent side2]
Hypotenuse = √[242 + 72]
= √[576 + 49]
= √625
= 25

cosα = adjacent side/hypotenuse = 7/25
secα = 1/cosα = 25/7

14tanα - 75cosα - 7secα = 14(24/7) - 75(7/25) - 7(25/7)
= 48 - 21 - 25
= 2.
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5. Provided sin (A "“ B) = sinA cosB "“ cosA sinB, then sin 15° will be




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Correct Ans:(√3 - 1)/2√2
Explanation:
sin 15° = sin (45° – 30°)
= sin 45° . cos 30° – cos 45°. sin 30°
= (1/√2) x (√3/2) - (1/√2) x (1/2)
= (√3 /2√2) - (1/2√2)
= (√3 - 1)/2√2.
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6. The angles of elevation of the top of a building from the top and bottom of a tree are x and y respectively. If the height of the tree is h metre, then, in metre, the height of the building is.




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Correct Ans:h cotx/(cotx - coty)
Explanation:


In ∆ABE,
tan x = x/BE
BE = x/tan x ....(i)

In ∆ADC,
tan y = (x + h)/DC
DC = (x + h)/tan y ...(ii)
From (i) & (ii),
Since, DC = BE
tan y = (x + h)/(x/tan x)
tan y(x/tan x) = x + h
x tan y cot x = x + h
x(tan y cot x - 1) = h
x = h/(tan y cot x - 1)

Height of the building = x + h
= {h/(tan y cot x - 1)} + h
= {h/[(cot x/cot y) - 1]} + h
= {h cot y/(cot x - cot y)} + h
= [h cot y + h cot x - h cot y]/(cot x - cot y)
= h cot x/(cot x - cot y).
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7. If sin 21° = x/y, then sec 21° - sin 69° is equal to




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Correct Ans:x2/y√(y2 - x2)
Explanation:
Given:
sin 21° = x/y
sin(90° - 69°) = x/y
cos 69° = x/y
sin 69° = √(1 - cos2 69°)
= √(1 - (x/y)2)
= √(y2 - x2)/y

Similarly,
cos 21° = √(1 - sin2 21°)
= √(1 - (x/y)2)
= √(y2 - x2)/y

sec 21° - sin 69° = (1/cos 21°) - sin 69°
= y/√(y2 - x2) - √(y2 - x2)/y
= (y2 - y2 + x2)/y√(y2 - x2)
= x2/y√(y2 - x2).
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8. If xsin3θ + ycos3θ = sinθ cosθ and x sinθ - y cosθ = 0, then the value of x2 + y2 equals 




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Correct Ans:1
Explanation:
Given:
x sinθ - y cosθ = 0
x sinθ = y cosθ ... (i)

xsin3θ + ycos3θ = sinθ cosθ
y cosθ.sin2θ + ycos3θ = sinθ cosθ
y cosθ(sin2θ + cos2θ) = sinθ cosθ
y cosθ(1) = sinθ cosθ (wkt, sin2θ + cos2θ =1)
y = sinθ

From (i),
x sinθ = y cosθ
x sinθ = sinθ cosθ
x = cosθ
Hence, x2+ y2 = cosθ2 + sinθ 2
= 1
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9. A boy standing in the middle of a field, observes a flying bird in the north at an angle of elevation of 30° and after 2 minutes, he observes the same bird in the south at an angle of elevation of 60°. If the bird flies all along in a straight line at a height of m, then its speed in km/h is 




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Correct Ans:6
Explanation:


When boy observes a bird at point B in north side,
In ΔABD,
tan 30° = BD/AD
1/√3 = BD/50√3
BD = 50√3/√3
BD = 50 m

After 2 minutes, when boy observes a bird at point C in south side,
In ΔACD,
tan 60° = CD/AD
√3 = CD/50√3
CD = 50√3(√3)
CD = 150 m
Total distance travelled by bird, BC = BD + CD
= 50 + 150 = 200 m
Speed of bird = Distance/Time = 200/2 = 100 m/minute
By converting m/min into km/hr,
= [100/1000]/[1/60] = 6 km/hr.
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10. If sin(x + y)/sin(x - y) = (a + b)/(a - b), then the value of tanx/tany is 




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Correct Ans:a/b
Explanation:
Given:
sin(x + y)/sin(x - y) = (a + b)/(a - b)
Using component and divident rule,
[sin(x + y) + sin(x - y)]/[sin(x - y) - sin(x - y)] = [a + b + a - b]/[a + b - a + b]

WKT,
Sin A + Sin B = 2[Sin(A + B)/2 * Cos(A - B)/2]
Sin A - Sin B = 2[Cos(A + B)/2 * Sin(A - B)/2]


2[sin(x + y + x - y)/2*cos(x - y - x + y)/2]/2[cos(x + y + x - y)/2*sin(x - y - x + y)/2] = 2a/2b
2sinx .cosy/2cosx .siny = a/b
tanx.coty = a/b
WKT, coty = 1/tany
tanx/tany = a/b.
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11. If sin3θ sec2θ = 1, then what is the value of (3tan2 (5θ/2) "“ 1)?




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Correct Ans:2
Explanation:
Given: Sin3θ sec2θ = 1

WKT, secθ = 1/cosθ
sin3θ/cos2θ = 1
sin3θ = cos2θ

WKT, cosθ = sin(90° - θ)
sin3θ = sin(90° - 2θ )
3θ = 90° - 2θ
5θ = 90°
45° = 5θ/2

3tan2 (5θ/2) – 1 = 3tan245° - 1
= 3(1) - 1 ..... (wkt, tan45° = 1)
= 2.
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12. Two persons are on either side of a temple, 75 m high, observe the angle of elevation of the top of the temple to be 30° and 60° respectively. The distance between the persons is




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Correct Ans:173.2 m
Explanation:


In ∆ABC,
tan 30° = AC/BC
1/√3 = 75/BC
BC = 75√3

In ∆ACD,
tan 60° = AC/CD
√3 = 75/CD
CD = 75/√3

Distance between two persons = BC + CD = 75√3 + 75/√3
= 75(√3 + 1/√3)
= 300/√3
= 173.2 m.
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13. 2 - cos2θ = 3sinθcosθ, sinθ ≠ cosθ then tanθ is




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Correct Ans:1/2
Explanation:
Given:
2 - cos2θ = 3sinθcosθ
Dividing by cos2θ,
(2/cos2θ) - 1 = 3sinθcosθ/cos2θ
2sec2θ - 1 = 3tanθ
2(1 + tan2θ) - 1 = 3tanθ
2tan2θ + 2 - 1 = 3tanθ
2tan2θ -3tanθ + 1 = 0
2tan2θ - 2tanθ -tanθ + 1
2tanθ(tanθ - 1) - 1(tanθ - 1) = 0
(2tanθ - 1)(tanθ - 1) = 0
tanθ = 1/2 or 1
Here, the given option is 1/2.
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14. An person 1.8m tall is 20√3 away from a tower. The angle of elevation from his eye to the top of the tower is 30°. Find height of the tower is?




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Correct Ans:21.8m
Explanation:
Given:
Height of the person, AB = 1.8m
Angle of elevation = 30°
Distance between tower and person, AC = 20√3



From the diagram we can say,
BE = AC = 20√3
AB = CE = 1.8m

DE/BE = tan30° = 1/√3
DE/(20√3) = 1/√3
DE = 20√3/√3
DE = 20m
Height of the tower, CD = DE + CE
= 20 + 1.8
= 21.8m
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15. Ram and Shyam are 10 km apart. They both see a hot air balloon passing in the sky making an angle of 60° and 30° respectively. What is the height at which the balloon could be flying?




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Correct Ans:Both a and b
Explanation:
Here, the directions are unknown so we have two different cases.
Case 1:



tan 30° = h/x
1/√3 = h/x
h = x/√3 ....(1)
tan 60° = h/(10 - x)
√3 = h/(10 - x)
h = (10 - x)√3 ....(2)
Equating (1) & (2),
(10 - x)√3 = x/√3
(10 - x)3 = x
30 - 3x = x
30 = 4x
x = 15/2
Therefore, h = x/√3 = 15/2√3
h = (5 * 3)/2√3
h = 5√3/2

Case 2:



tan 60° = h/x
√3 = h/x
h = √3 x ....(3)
tan30° = h/(10 - x)
1/√3 = h/(10 - x)
h = (10 - x)/√3 ....(4)
Equate (3) & (4),
√3 x = (10 - x)/√3
x = 10 - x
2x = 10
x = 5
Therefore, h = √3 x = 5√3
Hence, the correct option is both a and b.
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16. If cos θ + sec θ = 2, the value of cos6 θ + sec6 θ is




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Correct Ans:2
Explanation:
cos θ + sec θ = 2
put θ = 0°
cos 0° + sec 0° = 2
(cos 0° = 1 and sec 0° = 1)
1 + 1 = 2
2 = 2
cos6 θ + sec6 θ
= (1)6 + (1)6
= 1 + 1 = 2
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17. If x = cosec𝜃- sin𝜃 and y = sec𝜃 - cos𝜃, then the relation between x and y is




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Correct Ans:x2y2(x2 + y2 + 3) = 1
Explanation:
Given:
x = cosec𝜃 - sin𝜃
y = sec𝜃 - cos𝜃
Put = 45⁰
x = cosec𝜃 - sin𝜃
x = cosec(45⁰) - sin(45⁰)
x = √(2) - (√(2)/2)
x = 1/√(2)

y = sec𝜃 - cos𝜃
y = sec(45⁰)- cos(45⁰)
y = √(2) - (√(2)/2)
y = 1/√(2)
By option,
x²y²(x² + y² + 3) = (1/√(2))² * (1/√(2))² [(1/√(2))² + (1/√(2))² + 3]
=(1/2)(1/2)[(1/2) + (1/2) + 3]
= 1
This equation x²y²(x² + y² + 3) = 1 statisfies.
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18. If is an acute angle and cos𝜃 = 15/17, then the value of cot (90º - 𝜃) is




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Correct Ans:8/15
Explanation:
Given, cos𝜃 = 15/17
Sec𝜃 = 1/cos𝜃 = 17/15
Cot (90º - 𝜃) = tan𝜃
But tan𝜃 = √sec²ðœƒ - 1 (where tan²ðœƒ = sec²ðœƒ - 1)
= √(17/15)² - 1
= √(289/225) - 1
= √(289 - 225)/225
= √64/225
= 8/15
cot (90º - 𝜃) = 8/15
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19. If 0°<θ<90°
2sin²θ + 3cosθ = 3, then the value of θ is ?




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Correct Ans:60°
Explanation:
Given that,
2sin²θ + 3cosθ = 3
2(1 - cos²θ) + 3cosθ - 3 = 0 (where sin²θ = 1 - cos²θ)
2 - 2 cos²θ + 3cosθ - 3 = 0
- 2 cos²θ + 3cosθ - 1 = 0
2 cos²θ - 3cosθ + 1 = 0
By factorization we get,
2 cosθ (cosθ - 1) - 1 (cosθ - 1) = 0
(cosθ - 1) (2cosθ - 1) = 0
If cosθ - 1
Cosθ = 1
Cosθ = Cos 0° (where cos 0° = 1)
θ = 0°
If 2cosθ - 1
2 cosθ = 1
Cosθ = 1/2
Cosθ = Cos 60° (where cos 60° = 1/2)
θ=60°
the value of θ is 60°
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20. What is the value of (2 + tan 60º) ?




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Correct Ans:2 + √3
Explanation:
We know that,
tan 60º = √3
∴ 2 + tan 60º = 2 + √3
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