Tangents and normals: Let a polar curve be described by r = f(θ), and let ℓ be the line tangent to the curve at the point P(x,y) = P(r,θ) (see figure).
b. Explain why tan θ = y/x.

Verified step by step guidance

1. Recall that the point P on the polar curve is given by the coordinates \(P(x,y) = P(r, \theta)\), where \(r = f(\theta)\) is the radius and \(\theta\) is the angle measured from the positive x-axis.
2. By definition of polar coordinates, the Cartesian coordinates \((x,y)\) of the point \(P\) are related to \(r\) and \(\theta\) by the formulas: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).
3. To find \(\tan(\theta)\), use the ratio of \(y\) to \(x\): \(\tan(\theta) = \frac{y}{x}\).
4. Substitute the expressions for \(x\) and \(y\) in terms of \(r\) and \(\theta\): \(\tan(\theta) = \frac{r \sin(\theta)}{r \cos(\theta)}\).
5. Since \(r \neq 0\), the \(r\) cancels out, leaving \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), which is the definition of \(\tan(\theta)\). Therefore, \(\tan(\theta) = \frac{y}{x}\).

Key Concepts

Polar to Cartesian Coordinate Conversion

In polar coordinates, a point is represented as (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. The Cartesian coordinates (x, y) can be found using x = r cos θ and y = r sin θ. This conversion is fundamental to relate polar and Cartesian systems.

Definition of the Tangent of an Angle in a Right Triangle

The tangent of an angle θ in a right triangle is defined as the ratio of the length of the side opposite θ to the length of the adjacent side. In the Cartesian plane, for a point (x, y), tan θ = y/x represents the slope of the line from the origin to the point, linking the angle θ to the coordinates.

Geometric Interpretation of Angle θ in Polar Coordinates

Angle θ in polar coordinates is the angle between the positive x-axis and the line segment OP connecting the origin to point P. Since tan θ = opposite/adjacent = y/x, this directly follows from the right triangle formed by dropping a perpendicular from P to the x-axis, confirming why tan θ = y/x.