Question

The equation to the normal to the curve y = sin x at (0, 0) is

(a) x = 0
(b) y = 0
(c) x + y = 0
(d) x − y = 0

Answer 1

(c) x + y = 0

$$\begin{gathered} \mathrm{Given}: \\ y=\sin x \\ \mathrm{On}\mathrm{d}\text{ifferentiating both sides w.r.t.}x,\mathrm{we}\mathrm{get} \\ \frac{dy}{dx}=\cos x \\ \text{Slope of the tangent =}{\left(\frac{dy}{dx}\right)}_{\left(0,0\right)}\text{= cos 0 =1} \\ \text{Slope of the normal},\mathit{\text{m}}\text{=}\frac{-1}{1}\text{=-1} \\ \mathrm{Given}: \\ \left(x_1,y_1\right)=\left(0,0\right) \\ \therefore \text{Equation of the normal} \\ =y-y_1=m\left(x-x_1\right) \\ \Rightarrow y-0=-1\left(x-0\right) \\ \Rightarrow y=-x \\ \Rightarrow x+y=0 \end{gathered}$$