Approximation Problems

**reiward** · September 17th 2010, 04:47 AM

Help me please, I have no idea how to do these:
1. Approximate the value of sin(32degrees)

2. Use differentials to approximate Arctan 1.1

Thank you!

**Failure** · September 17th 2010, 05:10 AM

Originally Posted by reiward

Help me please, I have no idea how to do these:
1. Approximate the value of sin(32degrees)

What about using $\sin(32^\circ)=\sin(30^\circ + 2^\circ)=\sin(30^\circ)\cos(2^\circ)+\cos(30^\circ )\sin(2^\circ)$ ?
You know the exact values of $\sin(30^\circ)$ and $\cos(30^\circ)$ , and perhaps you also know that for small angles (expressed in radians) you have $\sin(x)\approx x$ and $\cos(x)\approx 1-x^2/2$ .
So set $x := 2^\circ/180^\circ \cdot \pi$ and plug those two approximations into the above expression for $\sin(32^\circ)$ .

2. Use differentials to approximate Arctan 1.1

Just use $f(x)\approx f(x_0)+f'(x_0)\cdot (x-x_0)$ , with $x := 1.1, x_0 := 1$ , for $f(x) := \arctan(x)$ .

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