Final answer:
To find all solutions to the equation tan(t)=tan(t^3) in the interval 0≤t≤2π, we can estimate the solutions from a graph and find the exact solutions using the properties of tan. The solutions are t = nπ and t = arctan(1).
Step-by-step explanation:
To find all solutions to the equation tan(t)=tan(t^3) in the interval 0≤t≤2π, we can start by estimating the solutions from a graph. By graphing the function y = tan(t) - tan(t^3), we can observe where the curve intersects the x-axis. From the graph, we can see that there are two solutions in the interval 0≤t≤2π.
To find the exact solutions, we can use the fact that tan(t) = tan(t + nπ), where n is an integer. By rearranging the equation tan(t) = tan(t^3), we have tan(t) - tan(t^3) = 0. This can be factorized as tan(t)(1 - tan^2(t^2)) = 0. Therefore, the solutions are t = nπ and t = arctan(1).