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The greatest value of cos θ-√3sin θ is Single correct option
The greatest value of
cos θ − 3 sin θ \cos\theta-\sqrt3\sin\theta cos θ − 3 sin θ is
A. 2 2 2 B. 3 \sqrt3 3 C. 1 1 1 D. 4 4 4 Show answer and solution Answer: (A) 2 2 2
Write
cos θ − 3 sin θ = 2 ( 1 2 cos θ − 3 2 sin θ ) = 2 cos ( θ + π 3 ) \cos\theta-\sqrt3\sin\theta=2\left(\dfrac12\cos\theta-\dfrac{\sqrt3}2\sin\theta\right)=2\cos\left(\theta+\dfrac\pi3\right) cos θ − 3 sin θ = 2 ( 2 1 cos θ − 2 3 sin θ ) = 2 cos ( θ + 3 π ) , whose greatest value is
2 2 2 . Equivalently the amplitude is
1 2 + ( 3 ) 2 = 2 \sqrt{1^2+(\sqrt3)^2}=2 1 2 + ( 3 ) 2 = 2 . Adding the coefficients,
1 + 3 1+\sqrt3 1 + 3 , instead of using the square root of the sum of squares is the typical trap.
Original question written in the NDA 2019 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake