Small angle approximation

Plot[
{Sin[x],x}, {x,0,2 Pi}
]

Plot[
{Sin[x],x}, {x,0,1}
]

I’d like to know... How big an angle can I have and still have x be within 10% of sin(x)?  Let’s use “Solve” to find the answer and then graph it...

Solve[1.1*Sin[x]==x,x]

Awww, Mathematica is not finding an exact solution.  So let’s try having it solve the problem numerically instead:

NSolve[1.1*Sin[x]==x,x,Reals]

{{x->-0.7489866426973404`},{x->0.`},{x->0.7489866426973404`}}

Plot[
{Sin[x],x, 1.1 Sin[x]}, {x,0,1.0},
PlotLegends->"Expressions",
GridLines->{{0.74899}}
]

0.75 * 180/Pi

0.75 Radians is 42 degrees.