Chapter 4.1, Problem 74E

To determine

To find:the equation of the tangent line to the given curve $x=t-\cos t,y=-1+\sin t$ at $t=0$

Answer to Problem 74E

Option A $y=x$ is the equation of the tangent line to the curve $x=t-\cos t,y=-1+\sin t$ at $t=0$ .

Explanation of Solution

Given information:

  $x=t-\cos t,y=-1+\sin t$ at $t=0$

Calculation:

Finding the value of $x$ and $y$ at $t=0$ ,

  $\begin{array}{l}x=0-\cos 0\\ x=0-1\\ x=-1\\ y=-1+\sin 0\\ y=-1+0\\ y=-1\end{array}$

Finding $\frac{dy}{dx}$ at $t=0$ ,

  $\begin{array}{l}\frac{dy}{dx}=\frac{dy}{dt}\cdot \frac{dt}{dx}\\ \frac{dy}{dx}=\cos t\cdot \frac{1}{1+\sin t}\\ \frac{dy}{dx}=\cos 0\cdot \frac{1}{1+\sin 0}\\ \frac{dy}{dx}=1\times \frac{1}{1+0}\\ \frac{dy}{dx}=1\end{array}$

Substituting the values in point slope form,

  $\begin{array}{l}y-y_1=m\left(x-x_1\right)\\ y-\left(-1\right)=1\left(x-\left(-1\right)\right)\\ y+1=x+1\\ y=x\end{array}$

Thus we can conclude that option A is the right answer.