What values for theta(0<=theta<=2pi) satisfy the equation?
2 sin theta cos theta + sqrt(3) cos theta = 0
a. pi/2, 4pi/3, 3pi/2, 5pi/3
b. pi/2, 3pi/4, 3pi/2, 5pi/3
c. pi/2, 3pi/4, 3pi/2, 5pi/4
d. pi/2, pi/4, 3pi/2, 5pi/3
Nevermind, i got it, its A.
To find the values of theta that satisfy the equation 2 sin theta cos theta + sqrt(3) cos theta = 0, let's break it down step by step:
Step 1: Factor out common terms
In this equation, we have two terms with cos theta as a common factor. By factoring out cos theta, we get:
cos theta (2 sin theta + sqrt(3)) = 0
Step 2: Set each factor equal to zero
For the equation to be true, either cos theta must be equal to zero, or 2 sin theta + sqrt(3) must be equal to zero. So we have two possible scenarios:
Scenario 1: cos theta = 0
When cos theta is equal to zero, we can find the corresponding values for theta by solving the equation cos theta = 0 in the range 0 <= theta <= 2pi. The values of theta that satisfy this are pi/2 and 3pi/2.
Scenario 2: 2 sin theta + sqrt(3) = 0
To find the values of theta that satisfy this equation, we can isolate sin theta:
2 sin theta = -sqrt(3)
sin theta = -sqrt(3)/2
From the unit circle or trigonometric identities, we know that sin theta = -sqrt(3)/2 when theta is equal to 11pi/6 or 7pi/6.
In summary, the values of theta that satisfy the equation 2 sin theta cos theta + sqrt(3) cos theta = 0 in the range 0 <= theta <= 2pi are:
- pi/2
- 3pi/2
- 11pi/6
- 7pi/6
Thus, the answer is option b: pi/2, 3pi/4, 3pi/2, 5pi/3.