Solve the equation 2 cos(x)+1=0, 0<=x<=2pi. Show all of your work.
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To solve the equation 2 cos(x) + 1 = 0, we can follow these steps:
Step 1: Subtract 1 from both sides:
2 cos(x) = -1
Step 2: Divide both sides by 2:
cos(x) = -1/2
Now, let's find the values of x between 0 and 2π where cosine is equal to -1/2.
The cosine function is negative for angles in the second and third quadrants.
In the second quadrant (π/2 < x < π), the cosine function is negative. We can write:
x = π + arccos(-1/2)
In the third quadrant (π < x < 3π/2), the cosine function is also negative. We can write:
x = 2π - arccos(-1/2)
Step 3: Calculate the values of x:
In the second quadrant:
x = π + arccos(-1/2)
Using the inverse cosine function, we find:
x = π + 2π/3
Simplifying, we get:
x = 3π/3 + 2π/3
x = 5π/3
In the third quadrant:
x = 2π - arccos(-1/2)
Using the inverse cosine function, we find:
x = 2π - 2π/3
Simplifying, we get:
x = 6π/3 - 2π/3
x = 4π/3
Therefore, the solutions to the equation 2 cos(x) + 1 = 0 from 0 to 2π are:
x = 5π/3 and x = 4π/3.
To solve the equation 2cos(x) + 1 = 0, we can follow these steps:
Step 1: Subtract 1 from both sides of the equation:
2cos(x) = -1
Step 2: Divide both sides of the equation by 2:
cos(x) = -1/2
Now, to find the values of x within the given interval (0 <= x <= 2π) that satisfy the equation cos(x) = -1/2, we can refer to the unit circle or the trigonometric ratios of special angles.
The cosine function is negative in the second and third quadrants, and its value is equal to -1/2 at two specific angles: π/3 (60 degrees) and 5π/3 (300 degrees).
Step 3: Express the obtained angles in the given interval (0 <= x <= 2π):
π/3 is approximately 1.047 and 5π/3 is approximately 5.235.
Since the interval is 0 <= x <= 2π, we only need to take the angle π/3 (1.047) as the solution since it falls within the given interval.
Therefore, the solution to the equation 2cos(x) + 1 = 0, 0 <= x <= 2π is:
x = π/3 (approximately 1.047).