Answer: We can start by using the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero.
So, we can set each factor equal to zero and solve for theta:
Factor 1: cot(theta) - 1 = 0
Using the identity cot(theta) = cos(theta) / sin(theta), we can rewrite this as:
cos(theta) / sin(theta) - 1 = 0
cos(theta) - sin(theta) = 0
cos(theta) = sin(theta)
Dividing both sides by cos(theta), we get:
tan(theta) = 1
The solutions to this equation are theta = 45 degrees and theta = 225 degrees (since tangent is positive in the first and third quadrants).
Factor 2: 2sin(theta) + sqrt(3) = 0
Subtracting sqrt(3) from both sides and dividing by 2, we get:
sin(theta) = -sqrt(3)/2
The solutions to this equation are theta = 240 degrees and theta = 300 degrees (since sine is negative in the third and fourth quadrants, and sin(240) = sin(300) = -sqrt(3)/2).
Therefore, the exact solutions in the interval [0 degrees, 360 degrees) are:
theta = 45 degrees, 225 degrees, 240 degrees, 300 degrees.
Explanation: