Using graphing software, we draw the curve of y = 2 cos2x − sin x − 1 in the region 0 ≤ θ < 2π. Wherever the curve cuts the x-axis will be the solution for our equation.
We see from the graph that the solutions are approximately:
x = 0.5
x = 2.6
x = 4.7
For more accurate solutions, we would just zoom in on the graph.
In this next graph, I have zoomed in to the second root (the one near x = 2.6). We can see that this root is x = 2.618 correct to 3 decimal places.
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We could continue to zoom in as close as we like to get the accuracy that we require.