Let P Q R S be a quadrilateral in a plane, where Q R = 1 , ∠ P Q R = ∠ Q R S = 70 ° , ∠ P Q S = 15 ° and ∠ P R S = 40 ° . If ∠ R P S = θ ° , P Q = α and P S = β , then the interval(s) that contain(s) the value of 4 α β sin θ ° is/are
B. 1 , 2
Given, P Q R S be a quadrilateral in a plane, where Q R = 1 , ∠ P Q R = ∠ Q R S = 70 ° , ∠ P Q S = 15 ° and ∠ P R S = 40 ° . If ∠ R P S = θ ° , P Q = α and P S = β , Now plotting the diagram of given values we get, Applying sine rule in ∆ P Q R we get, α sin 30 ° = 1 sin 80 ° ⇒ α = 1 2 sin 80 °                 . . . i Applying sine rule in ∆ P R S , β sin 40 2 = 1 sin θ ⇒ β sin θ = sin 40 °                 . . . ii From (i) & (ii), 4 α β sin θ = 4 · 1 2 sin 80 ° · sin 40 ° = 1 cos 40 ° Using cos 0 ° = 1 , cos 30 ° = 3 2 , cos 45 ° = 1 2 and cos 60 ° = 1 2 Only Options (A) and (B) are correct.
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