What values for θ (0 ≤ θ ≤ 2π) satisfy the equation? 2 sin θ cos θ + root3 cos θ = 0?

Step-by-step

Given equation:

2sin(theta)cos(theta) + sqrt(3)*cos(theta) = 0  ...........................(1)

Solve for theta for 0<=theta<=2pi.

Factor out cos(theta), we get

cos(theta) * ( 2sin(theta) + sqrt(3) ) = 0

By the zero product theorem, we can conclude

cos(theta) = 0   ...................................(2)

OR  

2sin(theta) + sqrt(3) = 0  ................. (3)

Solving (2)

cos(theta) = 0 has solutions pi/2 or 3pi/2  from the cosing curve.

Solving (3)

2sin(theta) + sqrt(3) = 0  =>

sin(theta) = -sqrt(3)/2

which has solutions  4pi/3 or 5pi/3.

So the solutions to equation (1) are

S={pi/2,  4pi/3, 3pi/2, 5pi/3}

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