Question

Find all t in the interval 0, 2piu satisfying (cos t)² cos t - 2= 0?

Answer 1

To find all t in the interval (0, 2π) satisfying the equation (cos t)² cos t - 2 = 0, factor out cos t and set each factor equal to zero. The solutions for t in the interval (0, 2π) are t = π/2, 3π/2, 2π/3, 4π/3.

Step-by-step explanation:

To find all t in the interval (0, 2π) satisfying the equation (cos t)² cos t - 2 = 0, we can solve it step by step.

1. Start by factoring out cos t from the equation: cos t ((cos t)² - 2) = 0.
2. Set each factor equal to zero to find the solutions: cos t = 0 or (cos t)² - 2 = 0.
3. Solving cos t = 0 gives us two solutions in the interval (0, 2π): t = π/2 and t = 3π/2.
4. Solving (cos t)² - 2 = 0 gives us another two solutions in the interval (0, 2π): t = 2π/3 and t = 4π/3.
5. Therefore, the solutions for t in the interval (0, 2π) satisfying the equation are: t = π/2, 3π/2, 2π/3, 4π/3.