https://file.aiquickdraw.com/mathbot/file/17279207840534gn5n63t.jpg

We are given the parametric equations for the curve:

\[
x = t - \cos t \quad \text{and} \quad y = -1 + \sin t,
\]
and the question asks for the equation of the tangent line to the curve at \( t = 0 \).

### Step 1: Find the derivatives \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \)
First, differentiate both parametric equations with respect to \( t \).

For \( x(t) = t - \cos t \):
\[
\frac{dx}{dt} = 1 + \sin t.
\]

For \( y(t) = -1 + \sin t \):
\[
\frac{dy}{dt} = \cos t.
\]

### Step 2: Slope of the tangent line
The slope of the tangent line at any point \( t \) is given by the ratio of \( \frac{dy}{dt} \) to \( \frac{dx}{dt} \):
\[
\text{slope} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{\cos t}{1 + \sin t}.
\]

Now, substitute \( t = 0 \):
\[
\frac{dy}{dt}\Big|_{t=0} = \cos(0) = 1,
\]
\[
\frac{dx}{dt}\Big|_{t=0} = 1 + \sin(0) = 1.
\]
Thus, the slope of the tangent line at \( t = 0 \) is:
\[
\text{slope} = \frac{1}{1} = 1.
\]

### Step 3: Find the point on the curve at \( t = 0 \)
Substitute \( t = 0 \) into the parametric equations to get the coordinates of the point:
\[
x(0) = 0 - \cos(0) = -1,
\]
\[
y(0) = -1 + \sin(0) = -1.
\]
So, the point of tangency is \( (-1, -1) \).

### Step 4: Equation of the tangent line
The equation of a line with slope \( m \) passing through the point \( (x_1, y_1) \) is:
\[
y - y_1 = m(x - x_1).
\]
Substitute \( m = 1 \), \( x_1 = -1 \), and \( y_1 = -1 \):
\[
y - (-1) = 1(x - (-1)),
\]
\[
y + 1 = x + 1,
\]
\[
y = x.
\]

Thus, the equation of the tangent line at \( t = 0 \) is:
\[
y = x.
\]

---

Would you like more details or have any further questions?

Here are five related questions for further exploration:
1. How do we find the normal line to the curve at \( t = 0 \)?
2. Can we generalize this process for finding tangents at other points \( t \)?
3. How would we interpret the curve if we eliminated the parameter \( t \)?
4. What are the applications of parametric curves in physics or engineering?
5. How do we use parametric equations to describe motion along a curve?

**Tip:** When solving parametric curve problems, always differentiate the parametric equations to find the slope, and evaluate carefully at the specific parameter value for accuracy.

Which of the following is an equation of the tangent line to the curve defined by x = t - cos(t) and y = -1 + sin(t) at t = 0?

Learn how to find the tangent line to the parametric curve defined by x = t - cos(t) and y = -1 + sin(t) at t = 0. This problem covers key calculus concepts, including parametric differentiation and the equation of a tangent line. Suitable for advanced high school or college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation