Chapter 4 - Applications of the Derivative - Chapter Review Exercises - Page 221: 11

$$L(\theta) =3\theta -\pi $$

Given $$P(\theta)=\sin (3 \theta+\pi), \quad a=\frac{\pi}{3}$$ Since \begin{align*} P'(\theta)&=3\cos (3 \theta+\pi)\\ P'(\pi/3)&= 3 \end{align*} Then the linear approximation is given by \begin{align*} L(\theta)&=P^{\prime}(a)(\theta-a)+P(a)\\ &=3(\theta-\pi/3) \\ &=3\theta -\pi \end{align*}