Single Variable Calculus, Chapter 3, 3.4, Section 3.4, Problem 21

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




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


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) && \text{Substitute value of $x$} \\ \\ m_T =& \sqrt[2]{3} && \end{aligned} \end{equation}$


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


$\begin{equation} \begin{aligned} y - y_1 =& m (x - x_1) \\ \\ y - 2 =& \sqrt[2]{3} \left( x - \frac{\pi}{3} \right) \\ \\ y =& \sqrt[2]{3} x - \frac{2}{3} \sqrt{3} \pi + 2 \end{aligned} \end{equation}$