Question

use an appropriate local linear approximation to estimate the value of the given quantity.answer q3 only.

Answer 1

In this case, we know that

$cos30=\frac{\sqrt[\placeholder{⬚}]{3}}{2}$

so we can use this value to estimate cosine of 32 degrees.

The local linear approximation is given by

$y-y_1=m\lparen x-x_1)$

where, in our case,

$\begin{gathered} x=32 \\ x_1=30 \\ y=cos32 \\ y_1=cos30=\frac{\sqrt{3}}{2} \end{gathered}$

and m is the derivative of the function

$cos\theta$

evaluated at

$\theta=30\text{ degrees}$

In this regard, the derivative of cosine of thetat is given by

$\frac{d}{d\theta}cos\theta=-sin\theta$

then the slope m is given as

$m=\frac{d}{d\theta}cos\theta_{\theta=30}=-s\imaginaryI n30=-\frac{1}{2}$

Then by substituting this value and the above ones on the local linear approximation, we have

$y-y_1=m\operatorname{\lparen}x-x_1)\Rightarrow cos32-\frac{\sqrt[]{3}}{2}=-\frac{1}{2}\left(32-30\right)$

which gives

$\begin{gathered} cos32-\frac{\sqrt{3}}{2}=-\frac{1}{2}\left(2\right) \\ cos32-\frac{\sqrt{3}}{2}=-1 \end{gathered}$

then by moving square root of 3 over 2 to the right hand side, we get

$cos32=-1+\frac{\sqrt[\placeholder{⬚}]{3}}{2}$