2cosx + 1 = 0
solve the equation for solutions in the interval : 0 < x < 2pie
those are less than or equal too signs
cosx=-1/2
Doesnit it go negative in the second quarter?
To solve the equation 2cos(x) + 1 = 0 for solutions in the given interval 0 < x < 2π, we can follow these steps:
Step 1: Subtract 1 from both sides of the equation to isolate the cosine term:
2cos(x) = -1
Step 2: Divide both sides of the equation by 2 to get cos(x) by itself:
cos(x) = -1/2
Now, to find the solutions for cos(x) = -1/2 in the given interval:
Step 3: Determine the reference angle for which cos(x) = -1/2, which can be found by considering the unit circle or trigonometric values. The reference angle for cos(x) = -1/2 is π/3 (or 60 degrees).
Step 4: Since the cosine function is negative in the second and third quadrants, we need to find the angles in those quadrants with cosine equal to -1/2.
In the second quadrant (π < x < 3π/2), the angle with cosine = -1/2 is (π - π/3) = 2π/3.
In the third quadrant (3π/2 < x < 2π), the angle with cosine = -1/2 is (2π - π/3) = 5π/3.
Therefore, the solutions within the given interval are x = 2π/3 and x = 5π/3.