73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.


x=cos t+t sin t,y=sin t−t cos t; t=π/4

Verified step by step guidance

1. Identify the parametric equations given: \(x = \cos t + t \sin t\) and \(y = \sin t - t \cos t\).
2. Find the derivatives of \(x\) and \(y\) with respect to \(t\): compute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) using the product rule where necessary.
3. Evaluate \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) at \(t = \frac{\pi}{4}\) to find the slope of the tangent line using \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\).
4. Calculate the coordinates of the point on the curve at \(t = \frac{\pi}{4}\) by substituting \(t\) into the original parametric equations to get \((x_0, y_0)\).
5. Use the point-slope form of the line equation: \(y - y_0 = m(x - x_0)\), where \(m\) is the slope found in step 3, to write the equation of the tangent line.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves. Understanding how to work with these equations is essential for finding derivatives and tangent lines.

Derivative of Parametric Curves

To find the slope of the tangent line to a parametric curve, compute dy/dx by dividing dy/dt by dx/dt, provided dx/dt ≠ 0. This method uses the chain rule and allows determination of the instantaneous rate of change of y with respect to x at a given parameter value.

Equation of a Tangent Line

Once the slope of the tangent line at a point is found, the tangent line's equation can be written using the point-slope form: y - y₀ = m(x - x₀), where (x₀, y₀) is the point on the curve and m is the slope. This equation represents the line that just touches the curve at that point.