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