40–41. {Use of Tech} Slopes of tangent lines
b. Find the slope of the lines tangent to the curve at the origin (when relevant).
r = 1 −sin θ

Verified step by step guidance

1. Recognize that the given curve is in polar form: \(r = 1 - \sin \theta\). To find the slope of the tangent line at a point, we need to convert this polar equation into Cartesian coordinates or use the formula for the slope of a tangent line in polar coordinates.
2. Recall that the slope of the tangent line in Cartesian coordinates is \(\frac{dy}{dx}\). For polar curves, \(x = r \cos \theta\) and \(y = r \sin \theta\). Therefore, \(\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\).
3. Compute \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\) using the product and chain rules: \(\frac{dx}{d\theta} = \frac{d}{d\theta}(r \cos \theta) = \frac{dr}{d\theta} \cos \theta - r \sin \theta\), \(\frac{dy}{d\theta} = \frac{d}{d\theta}(r \sin \theta) = \frac{dr}{d\theta} \sin \theta + r \cos \theta\).
4. Find \(\frac{dr}{d\theta}\) by differentiating \(r = 1 - \sin \theta\) with respect to \(\theta\), which gives \(\frac{dr}{d\theta} = -\cos \theta\).
5. Evaluate \(\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\) at the origin. Since the origin corresponds to \(r=0\), find the value(s) of \(\theta\) where \(r=0\) (i.e., solve \$1 - \sin \theta = 0\(), then substitute those \)\theta\( values into the expressions for \)\frac{dy}{d\theta}\( and \)\frac{dx}{d\theta}$ to find the slope(s) of the tangent line(s) at the origin.

Key Concepts

Polar Coordinates and Curves

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian (x, y). Understanding how to interpret and manipulate polar equations like r = 1 − sin θ is essential for analyzing curves defined in this system.

Slope of Tangent Lines in Polar Coordinates

The slope of a tangent line to a polar curve at a given point can be found by converting the polar equation to Cartesian form or using the formula dy/dx = (dr/dθ sin θ + r cos θ) / (dr/dθ cos θ − r sin θ). This allows calculation of the slope at specific θ values.

Finding Tangent Lines at the Origin

When the curve passes through the origin (r = 0), special attention is needed to find the tangent slope, often by evaluating limits or using derivatives with respect to θ. This helps determine the direction of the tangent line at the origin.