cos 2x = 1/2
0 ≤ 2x ≤ 4pi
determine all the possible answers for x
To solve the equation cos(2x) = 1/2, you can use the inverse cosine function.
cos(2x) = 1/2 implies that 2x = arccos(1/2).
The inverse cosine function has a range of [0, π], so we can write: 2x = π/3 or 2x = 5π/3.
Now, let's solve each equation separately:
1) For 2x = π/3:
Divide both sides by 2: x = π/6.
2) For 2x = 5π/3:
Divide both sides by 2: x = 5π/6.
Since the given constraint is 0 ≤ 2x ≤ 4π, we need to check if the solutions x = π/6 and x = 5π/6 are within this range:
For x = π/6:
0 ≤ 2(π/6) ≤ 4π
0 ≤ π/3 ≤ 4π
This solution satisfies the given constraint.
For x = 5π/6:
0 ≤ 2(5π/6) ≤ 4π
0 ≤ 5π/3 ≤ 4π
This solution also satisfies the given constraint.
Therefore, the possible answers for x are x = π/6 and x = 5π/6.
To solve the equation cos 2x = 1/2, we will use the inverse cosine function.
Step 1: Rewrite the equation using the inverse cosine function.
2x = arccos(1/2)
Step 2: Find the values for arccos(1/2).
The cosine function has a value of 1/2 at two angles: π/3 and 5π/3.
Step 3: Solve for x.
Since we are given that 0 ≤ 2x ≤ 4π, we need to find all possible angles for x that satisfy this condition.
For π/3:
0 ≤ 2x ≤ 4π
0 ≤ 2(π/3) ≤ 4π
0 ≤ 2π/3 ≤ 4π
π/3 ≤ x ≤ 2π/3
For 5π/3:
0 ≤ 2x ≤ 4π
0 ≤ 2(5π/3) ≤ 4π
0 ≤ 10π/3 ≤ 4π
5π/3 ≤ x ≤ 10π/3
Therefore, the possible values for x are:
x = π/3
x = 2π/3
x = 5π/3
x = 10π/3