To solve the trigonometric equation 2 sin(theta) cos(theta) + sqrt(2) cos(theta) = 0, we can start by factoring out the common term cos(theta):
cos(theta) (2 sin(theta) + sqrt(2)) = 0
Now we have two possible cases:
1. The product of the two terms cos(theta) and (2 sin(theta) + sqrt(2)) equals 0:
cos(theta) = 0 or (2 sin(theta) + sqrt(2)) = 0
2. One of the terms equals 0, and the other term is not defined (division by zero is not allowed):
cos(theta) = 0 and (2 sin(theta) + sqrt(2)) is undefined.
For case 1, when cos(theta) = 0, it means that theta is either pi/2 (90 degrees) or 3pi/2 (270 degrees), since these are the angles where the cosine function equals 0.
For case 1, when (2 sin(theta) + sqrt(2)) = 0, we can solve for theta as follows:
2 sin(theta) + sqrt(2) = 0
2 sin(theta) = -sqrt(2)
sin(theta) = -sqrt(2) / 2
The value -sqrt(2) / 2 is associated with two special angles: -pi/4 and -3pi/4 (in the fourth quadrant and third quadrant, respectively). These angles can be written as 7pi/4 and 5pi/4 in the range 0 <= theta <= 2pi.
Therefore, the values for theta that satisfy the equation are:
a. pi/2, 3pi/4, 3pi/2, 7pi/4.