Single Variable Calculus, Chapter 3, 3.4, Section 3.4, Problem 26

a.) Determine the equation of the tangent line to the curve $y = \sec x - 2 \cos x$ at the point $\displaystyle \left( \frac{\pi}{3}, 1 \right)$

Solving for the derivative of $y = \sec x - 2 \cos x$


$\begin{equation} \begin{aligned} \qquad y' =& \frac{d}{dx} (\sec x) -2 \frac{d}{dx} (\cos x) && \\ \\ \qquad y' =& \sec x \tan x + 2 \sin x && \\ \\ \end{aligned} \end{equation}$




Let $y' = m_T$ (slope of the tangent line)


$\begin{equation} \begin{aligned} y' = m_T =& \sec \left( \frac{\pi}{3} \right) \tan \left( \frac{\pi}{3} \right) + 2 \sin \left( \frac{\pi}{3} \right) && \\ \\ m_T =& \sqrt[3]{3} && \end{aligned} \end{equation}$


Using Point Slope Form substitute the values of $x, y$ and $m_T$


$\begin{equation} \begin{aligned} y - y_1 =& m (x - x_1) && \\ \\ y - 1 =& \sqrt[3]{3} \left( x = \frac{\pi}{3} \right) && \\ \\ y - 1 =& \sqrt[3]{3} x - \sqrt{3} \pi && \\ \\ y =& \sqrt[3]{3} x - \sqrt{3} \pi + 1 && \text{Equation of the tangent line at $\large \left( \frac{\pi}{3}, 1 \right)$} \end{aligned} \end{equation}$



b.) Graph the curve and the tangent line in part (a) on the same screen