# What is an equation of the line tangent to the graph of y=cos(2x) at x=pi/4?

## What is an equation of the line tangent to the graph of $y = \cos \left(2 x\right)$ at $x = \frac{\pi}{4}$?

Dec 18, 2016

#### Answer:

$y = - 2 x + \frac{\pi}{2}$

#### Explanation:

To find the equation of the tangent line to the curve $y = \cos \left(2 x\right)$ at $x = \frac{\pi}{4}$, start by taking the derivative of $y$ (use the chain rule).

$y ' = - 2 \sin \left(2 x\right)$

Now plug in your value for $x$ into $y '$:

$- 2 \sin \left(2 \cdot \frac{\pi}{4}\right) = - 2$

This is the slope of the tangent line at $x = \frac{\pi}{4}$.

To find the equation of the tangent line, we need a value for $y$. Simply plug your $x$ value into the original equation for $y$.

$y = \cos \left(2 \cdot \frac{\pi}{4}\right)$
$y = 0$

Now use point slope form to find the equation of the tangent line:

$y - {y}_{0} = m \left(x - {x}_{0}\right)$

Where ${y}_{0} = 0$, $m = - 2$ and ${x}_{0} = \frac{\pi}{4}$.

This gives us:

$y = - 2 \left(x - \frac{\pi}{4}\right)$

Simplifying,

$y = - 2 x + \frac{\pi}{2}$

Hope that helps!
graph{(y-cos(2x))(y+2x-pi/2)=0 [-2.5, 2.5, -1.25, 1.25]}