Determine the value(s) of theta where sin theta =-root 3/2 and 0\< theta\<2pi
To solve the equation sin(theta) = -sqrt(3)/2, we need to find the angles within the given range (0 to 2pi) whose sine is equal to -sqrt(3)/2.
We know that sin(theta) is negative in the third and fourth quadrants. In these quadrants, the value of theta has the same sine as its reference angle (the angle between 0 and pi/2).
The reference angle with the same sine as -sqrt(3)/2 is pi/3.
So, in the third quadrant, theta = pi + pi/3 = 4pi/3.
In the fourth quadrant, theta = 2pi - pi/3 = 5pi/3.
Therefore, the values of theta where sin(theta) = -sqrt(3)/2 and 0 < theta < 2pi are 4pi/3 and 5pi/3.