Answer
Please see below.
Work Step by Step
The function $g(t)=2 \cos t -3t$ is clearly continuous on the closed interval $[0,1]$, and also we have $f(0)=2 >0$ and $f(1) \approx -1.92<0$. So by applying the Intermediate Value Theorem, there must exist some real number $c$ such that $f(c)=0$.
Looking at the graph, we can approximate the root:$$c \approx 0.6 \, .$$By zooming in repeatedly on the graph we can approximate the root much better:$$c \approx 0.56 \, .$$
Using a root calculator, we can find the root more accurately:$$c \approx 0.5636 \, .$$
