If cos2 theta = 1/3 and 0 is less than or equal to 2 theta and 2 theta is less than equal to pi, find cos theta

To find cos(theta), we need to use the given information that cos(2theta) = 1/3 and 0 ≤ 2theta ≤ π.

1. Start by using the double angle identity for cos(2theta): cos(2theta) = 2cos^2(theta) - 1.

2. Plug in the given value of cos(2theta) = 1/3: 1/3 = 2cos^2(theta) - 1.

3. Add 1 to both sides of the equation: 4/3 = 2cos^2(theta).

4. Divide both sides of the equation by 2: 2/3 = cos^2(theta).

5. Take the square root of both sides of the equation to solve for cos(theta): ±√(2/3) = cos(theta).

Since we know that 0 ≤ 2theta ≤ π, this means that 0 ≤ theta ≤ π/2 because 2theta is limited to the first quadrant. Therefore, cos(theta) must be positive.

Thus, the final answer is cos(theta) = √(2/3).