# How do you find the equation of the tangent and normal line to the curve y=x+cosx at x=1?

Dec 26, 2016

$T a n \ge n t : y = 0.15852 x + 1.3818$ See Socratic graph for the curve, the point of contact and the tangent. Normal : $y = - 6.3084 x + 7.8487$

#### Explanation:

$x = 1 r a \mathrm{di} a n = {57.296}^{o}$

The point of contact is (1, 1.5403)

y' ar x = 1 is $m = 1 - \sin {57.2903}^{o} = 0.15852$, nearly.

So, the equation of the tangent is

$y - 1.5403 = 0.15852 \left(x - 1\right)$ Simplifying,

$y = 0.15852 x + 1.3818$.

The tangent crosses the curve, elsewhere.

The normal to the curve is given by

$y - 1.5403 = - \frac{1}{0.15852} \left(x - 1\right)$, Simplifying,

$y = - 6.3084 x + 7.8487$

graph{(y-0.16x-1.38)(y-x-cos x)(y+6.31x-7.84)=0 [-20, 20, -10, 10]}