find all values of x in (o, Pi) that satify the given equation
cos x/3=1/2
1.5
cos(pi/3) = 1/2
cos(5pi/3) = 1/2
so, x = pi or 5pi
oops only in [0,pi], so x=pi
However, you gave the interval as (0,pi), which excludes pi, so no solution.
To find all values of x in the interval (0, π) that satisfy the equation cos(x/3) = 1/2, we can follow these steps:
Step 1: Identify the equation and the value to which the right side is set.
- The equation is cos(x/3) = 1/2.
Step 2: Understand the meaning of the equation.
- The equation cos(x/3) = 1/2 means that the cosine of x/3 is equal to 1/2.
Step 3: Find the general solution.
- The general solution can be obtained by finding the angles whose cosine is equal to 1/2. In this case, we need to find the angles θ such that cos(θ) = 1/2.
- The angles θ for which cos(θ) = 1/2 lie in the first and fourth quadrants of the unit circle.
Step 4: Determine the reference angle.
- The reference angle is the acute angle between the terminal side of an angle and the x-axis.
Step 5: Solve for x in the interval (0, π).
- Since the original equation is cos(x/3), we need to multiply the reference angle by 3 to find the values of x.
- The reference angle for cos(θ) = 1/2 is π/3.
- Multiplying π/3 by 3 gives us π, so x = π is one solution in the interval (0, π).
Step 6: Find all the solutions in the interval (0, π).
- To find the other solutions, consider that a full rotation around the unit circle is 2π.
- Since x/3 represents one-third of a complete rotation around the unit circle, we need to add or subtract multiples of 2π to the value of x.
- The other possible solution is x = 5π, which is obtained by adding 2π to the first solution, π.
- However, x = 5π exceeds the interval (0, π), so it is not a valid solution.
Step 7: Finalize the solutions.
- The only solution that satisfies the equation cos(x/3) = 1/2 in the interval (0, π) is x = π.