What values for theta (0 <= theta <= 2pi) satisfy the equation?
2 sin theta cos theta + sqrt2 cos theta = 0
a. pi/2, 3pi/4, 3pi/2, 7pi/4
b. pi/2, 3pi/4, 5pi/2, 7pi/4
c. pi/2, 5pi/4, 3pi/2, 7pi/4
d. pi/2, 5pi/4, 3pi/4, 7pi/4
Looks like this is too hard for the bot to handle, it still hasn't
tried it. So let a human do it.
Assuming you meant
2sinθcosθ + √2 cosθ = 0
cosθ(2sinθ + √2) = 0
cosθ = 0, so θ = π /2 or θ = 3π /2
or
sinθ = -√2/2, so θ = 5π/4 or θ = 7π /4
which do you think matches that ?
The answer that matches is c. pi/2, 5pi/4, 3pi/2, 7pi/4.
well, duhhh. I think I was looking for Katya, not you bot!!!
I'm sorry for the misunderstanding. As an AI language model, I am here to assist you with your questions to the best of my abilities. If you have any further questions, feel free to ask and I'll do my best to help you!
To solve the equation 2 sin(theta) cos(theta) + sqrt(2) cos(theta) = 0, we can factor out cos(theta):
cos(theta)(2 sin(theta) + sqrt(2)) = 0
Now we have two possibilities for this equation to be satisfied:
1. cos(theta) = 0, or
2. 2 sin(theta) + sqrt(2) = 0
Now let's solve each possibility:
1. cos(theta) = 0
To find the values of theta that satisfy this equation, we need to find the angles where cos(theta) = 0. These angles occur at pi/2 and 3pi/2.
2. 2 sin(theta) + sqrt(2) = 0
Solving for sin(theta), we have:
2 sin(theta) = -sqrt(2)
sin(theta) = -sqrt(2)/2
The values of theta that satisfy this equation are pi/4 and 7pi/4.
Combining the values from both possibilities, we have:
a. pi/2, 3pi/4, 3pi/2, 7pi/4
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.