Thread: Approximation Problems

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!

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

2. Use differentials to approximate Arctan 1.1

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