Solve the equation on the interval [0, 2π).
cos x + 2 cos x sin x = 0
would the answer be 7pi/6 , 11pi/6 ?
factor it
cosx(1 + 2sinx) = 0
cosx = 0 or sinx = -1/2
if cosx = 0
x = π/2 , 3π/2
for sinx = -1/2, we know x is in III or IV
we also know that sin π/6 = + 1/2
x = π + π/6 or 7π/6
x = 2π - π/6 = 11π/6
so you had 2 of the 4 answers
To solve the equation cos x + 2 cos x sin x = 0 on the interval [0, 2π), we can factor out cos x from the equation:
cos x (1 + 2 sin x) = 0
Now, we have two possibilities for the equation to be satisfied:
1) cos x = 0
2) 1 + 2 sin x = 0
Let's solve each equation separately:
1) cos x = 0
On the interval [0, 2π), the solutions for this equation are x = π/2 and x = 3π/2.
2) 1 + 2 sin x = 0
Subtracting 1 from both sides, we get 2 sin x = -1.
Then, dividing both sides by 2, we have sin x = -1/2.
On the unit circle, the values of x where sin x = -1/2 are x = 7π/6 and x = 11π/6.
Now, we need to check if these solutions fall within the given interval [0, 2π).
For x = π/2 and x = 3π/2, both of these values are within the interval.
For x = 7π/6 and x = 11π/6, both of these values are also within the interval.
Therefore, the solutions to the equation cos x + 2 cos x sin x = 0 on the interval [0, 2π) are x = π/2, 3π/2, 7π/6, and 11π/6.
To solve the equation cos x + 2 cos x sin x = 0 on the interval [0, 2π), we will need to apply some trigonometric identities and algebraic manipulations.
Step 1: Factor out the common factor cos x:
cos x(1 + 2sin x) = 0
Step 2: Set each factor equal to zero and solve for x:
cos x = 0 ==> x = π/2, 3π/2
1 + 2sin x = 0 ==> sin x = -1/2
Step 3: Solve for x when sin x = -1/2 on the interval [0, 2π):
One way to do this is by finding the reference angles for which sin x = -1/2. The reference angles are π/6 and 5π/6. Since sin function is negative in the 3rd and 4th quadrants, the solutions in the given interval will be:
x = π + π/6 = 7π/6
x = 2π - π/6 = 11π/6
Therefore, the solutions to the equation cos x + 2 cos x sin x = 0 on the interval [0, 2π) are x = π/2, 3π/2, 7π/6, and 11π/6.