Q:

Find the largest angle for which sine and tangent agree to within two significant figures.

I am kind of stuck with the solving process of this question:
\begin{align}
& \text{error } = 0.01 \\
\implies & \tan\theta = \text{ error }+ \sin\theta \\
\implies & \tan\theta = 0.01 + \sin\theta \\
\implies & \tan\theta - \sin\theta = 0.01 \\
\implies & \dfrac {\sin\theta}{\cos\theta}  - \sin\theta = 0.01
\end{align}
And this is where I stop, I don't know where to go from here.

A:

It $\sin(\theta) = x \in (0,\pi/2)$, $\tan(\theta) = x/\sqrt{1-x^2}$.  The equation $\tan(\theta) - \sin(\theta) = r$ then implies
$${x}^{4}+2\,r{x}^{3}+{r}^{2}{x}^{2}-2\,rx-{r}^{2} = 0$$
For $r = 1/100$ this is irreducible.  It does have closed-form solutions in terms of radicals, but they are not pleasant.  Numerical methods can be used.

