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practice question on Trigonometry

Trigonometry

Question no. 1
The minimum value of 2sin2θ + 3cos2θ is
looks_one 3
looks_two 2
looks_3 0
looks_4 1
Option looks_two 2
Solution :
2sin2θ + 3cos2θ = 2sin2θ + 2cos2θ + cos2θ
2(sin2θ + cos2θ) + cos2θ
2 + cos2θ
-1 ≤ cosθ ≤ 1
0 ≤ cos2θ ≤ 1
2 ≤ 2 + cos2θ ≤ 3

Question no. 2
The simplest value of sin2x + 2tan2x–2sec2x + cos2x is
looks_one -1
looks_two 1
looks_3 0
looks_4 2
Option looks_one -1
Solution :
sin2x + 2tan2x–2sec2x + cos2x
sin2x + cos2x + 2 (tan2x - sec2x ) = 1 + 2 (-1) = 1 - 2 = -1

Question no. 3
If cos4θ– sin4θ = 2/3 , then the value of 2cos2θ–1 is
looks_one 0
looks_two 1
looks_3 2/3
looks_4 3/2
option looks_3 2/3
Solution :

Question no. 4
The value of tan4°. tan43°. tan47°. tan86° is
looks_one 0
looks_two 1
looks_3 √3
looks_4 1/√3
Option looks_two 1
Solution :
tan4°. tan43°. tan47°. tan86°
= cot86°. cot47°. tan47°. cot86°.
= cot86°. cot86°.cot47°. tan47° = 1

Question no. 5
If 5tanθ – 4 = 0, then the value of
looks_one 5/3
looks_two 5/6
looks_3 0
looks_4 1/6
Option looks_3 0
Solution :

Question no. 6
The value of sin21°+ sin22° + sin23°+....+ sin289° is
looks_one 44
looks_two 22
looks_3 45/2
looks_4 None of these
Option looks_4 None of these
Solution :
= sin21°+ sin22° + sin23°+.. +sin245°..+ sin289°
= cos289°+ cos288° + cos287°+.. +sin245°..+ sin288°+ sin289°
= cos289° + sin289° + cos288° + sin288° + ............. + cos21° + sin21° + sin245° = 44 + 0.5 = 44.5

Question no. 7
A vertical tree 30m long breaks at a height of 10m. Its two parts and the ground form a triangle. The angle between the ground and the broken part will be
looks_one 60°
looks_two 45°
looks_3 30°
looks_4 None of these
Option looks_3 30°
Solution :

sin θ = 10/20 = 0.5
θ = 30°

Question no. 8
tanθ . sin(π/2 + θ) . cos(π/2 - θ) =
looks_one 1
looks_two-1
looks_3 0.5sin2θ
looks_4 None of these
Option looks_4 None of these
Solution :
tanθ . sin(π/2 + θ) . cos(π/2 - θ) = tanθ . cosθ .sinθ = sin2θ

Question no. 9
If cos2x + 2cosx = 1 then , (2- cos2x)sin2x is equal to
looks_one 1
looks_two -1
looks_3 √5
looks_4 - √5
Option looks_one 1
Solution :
cos2x + 2cosx = 1 ⇒ 2cos2x + 2cosx - 2 = 0
cos2x + cosx - 1 = 0

Question no. 10
2(1-2sin27θ )sin3θ is equal to
looks_one sin 17 θ - sin 11 θ
looks_two sin 11 θ - sin 17 θ
looks_3 cos 17 θ - cos 11 θ
looks_4 cos 17 θ + cos 11 θ
Option looks_one sin 17 θ - sin 11 θ
Solution :
2(1-2sin27θ )sin3θ = 2 (cos14θ)sin3θ
2 cos14θ.sin3θ = sin 17 θ - sin 11 θ

Question no. 11
cos 40° + cos 80° + cos 160° + cos 240°
looks_one 0
looks_two 1
looks_3 1/2
looks_4 -1/2
Option looks_4 -1/2
Solution :
cos 40° + cos 80° + cos 160° + cos 240° = 2cos60°. cos20° + cos160° - cos60°
2.(0.5 cos20° ) + cos160° - cos60° = cos20° + cos160° - 1/2 = 2cos90°cos70° - 1/2 = 0 - 1/2 = 1/2

Question no. 12
If A, B , C , D are the angles of a cyclic quadrilateral , then cos A + cos B + cos C + cos D =
looks_one 4
looks_two 1
looks_3 0
looks_4 -1
Option looks_3 0
Solution :
A + C = 180° ⇒ C = 180° - A
B + D = 180° ⇒ B = 180° - D
cos A + cos B + cos C + cos D = cos A + cos(180° - D ) + cos(180° - A ) + cos D
= cos A - cos D - cos A + cos D = 0

Question no. 13
sin 47° + sin 61° - sin 11° - sin 25° is equal to
looks_one sin 36°
looks_two cos 30°
looks_3 sin 7°
looks_4 cos 7°
Option looks_4 cos 7°
Solution :
sin 47° + sin 61° = 2sin54°cos7°
sin 11° + sin 25° = 2sin 18°cos 7°
sin 47° + sin 61° - sin 11° - sin 25° = 2sin54°cos7° - 2sin 18°cos 7° = 2cos 7° (sin54° - sin 18° )
Solution : sin 47° + sin 61° - sin 11° - sin 25° is equal to

Question no. 14
If tan 69° + tan 66° - tan 69° tan66° = 2k , then k =
looks_one -1
looks_two 1/2
looks_3 -1/2
looks_4 None of these
Option looks_3 -1/2
Solution :
Solution : If tan 69° + tan 66° - tan 69° tan66° = 2k , then k =

Question no. 15
The value of cos (36° - A). cos (36° + A ) + cos (54° + A ).cos (54° - A)
looks_one sin 2A
looks_two cos 2A
looks_3 cos 3A
looks_4 sin 3A
Option looks_two cos 2A
Solution :
Cos(A-B).cos(A + B) = Cos2A - sin2B
cos (36° - A). cos (36° + A ) = cos236° - sin2A
cos (54° + A ).cos (54° - A) = cos254° - sin2A
cos (36° - A). cos (36° + A ) + cos (54° + A ).cos (54° - A) = cos236° - sin2A + cos254° - sin2A
= cos236° - 2sin2A + sin236° = 1 - 2sin2A = cos 2A

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