# Use the secant method to approximate $\sin^{-1}(0.1)$ correct to 3 decimal

Use the secant method to approximate $\sin^{-1}(0.1)$ correct to $3$ decimal places starting with $x_0 = 0$ and $x_1 = 0.1$.

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Please indicate what you've tried and what you understand about the problem so that people can give relevant help. – user61527 Aug 4 '13 at 15:40

Hint: Note that we are trying to approximate numerically the smallest positive root of the equation $\sin x=0.1$.

Equivalently, we are solving numerically the equation $f(x)=0$, where $$f(x)=\sin x -0.01.$$ Now run the machinery of the secant method, using $x_0=0$, and $x_1=0.1$. Presumably your book or notes tell you how to write down the appropriate recurrence formula for $x_{n+1}$ in terms of $x_n$ and $x_{n-1}$.

If your book or notes fail to do that, there is a good description of the Secant Method in Wikipedia.

As a check on the work you will do, my calculation gave $x_2\approx 0.1001669$. We might as well as well stop there.

Actually, we could have just done nothing, since the $0.1$ we were given is already correct to $3$ decimal places.

If your course expects a formal check that your answer $a$ is close enough, you can evaluate $f(a\pm 0.0005)$. and show there is a sign change.

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thanks, it is a great help for me that i can understand... – YuYu Huang Aug 8 '13 at 11:13
You are welcome. My answer was short because the Wikipedia article is pretty good. If there is detail you need, just ask. – André Nicolas Aug 8 '13 at 13:21