Tangent line to $$$y = - 2 \sin{\left(x \right)}$$$ at $$$x = 34 \pi$$$

Calculate the tangent line to $$$y = - 2 \sin{\left(x \right)}$$$ at $$$x = 34 \pi$$$.

We are given that $$$f{\left(x \right)} = - 2 \sin{\left(x \right)}$$$ and $$$x_{0} = 34 \pi$$$.

Find the value of the function at the given point: $$$y_{0} = f{\left(34 \pi \right)} = 0$$$.

The slope of the tangent line at $$$x = x_{0}$$$ is the derivative of the function, evaluated at $$$x = x_{0}$$$: $$$M{\left(x_{0} \right)} = f^{\prime }\left(x_{0}\right)$$$.

Find the derivative: $$$f^{\prime }\left(x\right) = \left(- 2 \sin{\left(x \right)}\right)^{\prime } = - 2 \cos{\left(x \right)}$$$ (for steps, see derivative calculator).

Hence, $$$M{\left(x_{0} \right)} = f^{\prime }\left(x_{0}\right) = - 2 \cos{\left(x_{0} \right)}$$$.

Next, find the slope at the given point.

$$$m = M{\left(34 \pi \right)} = -2$$$

Finally, the equation of the tangent line is $$$y - y_{0} = m \left(x - x_{0}\right)$$$.

Plugging the found values, we get that $$$y - 0 = - 2 \left(x - 34 \pi\right)$$$.

Or, more simply: $$$y = - 2 x + 68 \pi$$$.

The equation of the tangent line is $$$y = - 2 x + 68 \pi\approx 213.62830044410594 - 2 x$$$A.