Find all solutions of the equation in the interval [0,2pi)
2 cos^2 x-cos x = 0
-2cos^2 + cosx + 0
(x+1/2) (x+0/2)
(2x+1) (x+0)
-1/2,0
2Pi/3, 4pi/3, pi/2, 3pi/2
my teacher circled pi/2 and 3pi/2
What did I do wrong? I don't understand...
Hard to figure out what you are doing.
2 cos^2 x-cos x = 0
cosx(2cosx - 1) = 0
cosx = 0 or cosx = 1/2
if cosx = 0,
x = π/2 or x = 3π/2
if cosx = 1/2
x = π/3 or x = 2π - π/3 = 5π/3
so you have 4 answers:
π/2, 3π/2, π/3, and 5π/3
(in degrees: 90°, 270°, 60°, 300° , they all work)
The equation you started with is correct: 2 cos^2(x) - cos(x) = 0.
To solve this quadratic equation, you need to factor it or use the quadratic formula.
Let's factor the equation:
cos(x) (2 cos(x) - 1) = 0
Now, set each factor equal to zero and solve for x:
cos(x) = 0 --> Solution: x = pi/2, 3pi/2
2 cos(x) - 1 = 0 --> Solution: x = pi/3, 5pi/3
You correctly found the solutions x = pi/2, 3pi/2, but missed two more solutions: x = pi/3, 5pi/3.
So the correct solutions in the interval [0, 2pi) are:
x = pi/2, 3pi/2, pi/3, 5pi/3.
Your teacher circled pi/2 and 3pi/2 because those are the solutions within the given interval [0, 2pi).
To find the solutions to the equation 2cos^2 x - cos x = 0 in the interval [0, 2pi), you can follow a few steps:
Step 1: Rewrite the equation in a quadratic form, where cos x is treated as a variable. This will allow you to factorize and solve for the values of cos x.
2cos^2 x - cos x = 0
Step 2: Factor out the common factor cos x from the equation:
cos x(2cos x - 1) = 0
Step 3: Set each factor equal to zero and solve for x:
cos x = 0
2cos x - 1 = 0
Step 4: Solve for x in both equations:
cos x = 0:
In the interval [0, 2pi), the solutions are x = pi/2 and x = 3pi/2. These correspond to the values of x where the cosine function equals zero.
2cos x - 1 = 0:
Solving for x, we get cos x = 1/2, which has solutions x = pi/3 and x = 5pi/3 in the interval [0, 2pi).
Therefore, the solutions to the equation 2cos^2 x - cos x = 0 in the interval [0, 2pi) are x = pi/2, x = 3pi/2, x = pi/3, and x = 5pi/3.
In your answer, you correctly identified x = pi/2 and x = 3pi/2, which are solutions for cos x = 0. However, you missed the other solutions x = pi/3 and x = 5pi/3, where cos x = 1/2. Make sure to check your calculations and consider all possible solutions within the given interval.