1.Find the linear approximation Lo(x) to the function f(x)=sin(x) at x=0
2.use linear approximation to estimate sin(0.03)
The equation of a non-vertical line is passing through point $\displaystyle (x_0,y_0)$ and slope $\displaystyle m$ is,
$\displaystyle \boxed{y-y_0=m(x-x_0)}$
You need to find the equation of tangent line at point $\displaystyle (x_0,y_0)=(0,\sin 0)=(0,0)$.
And the slope is the derivative at the point,
$\displaystyle \cos x$ at $\displaystyle x=1$ yields $\displaystyle x=0$.
Thus,
$\displaystyle y-y_0=1(x-0)$
Thus,
$\displaystyle y=x$
Is the linearazation line.
Thus,
$\displaystyle \sin (.03)\approx .03$ (in radians ).