Question

Find the points at which the following polar curves have a horizontal or vertical tangent line.$r=2\sin (2\theta )$

Short answer

The origin $(0,0)$ has both horizontal and vertical tangent lines.

Step-by-step solution

Convert the Polar Equation to Cartesian Coordinates

First, recall the formulas to convert polar coordinates to cartesian coordinates: $$x=r\cos (\theta )$$$$y=r\sin (\theta )$$Substitute $r=2\sin (2\theta )$ into these formulas:$$x=2\sin (2\theta )\cos (\theta )$$$$y=2\sin (2\theta )\sin (\theta )$$Use the double angle identity $\sin (2\theta )=2\sin (\theta )\cos (\theta )$ to simplify. Thus, $$x=4\sin (\theta ){\cos }^2(\theta )$$ and$$y=4{\sin }^2(\theta )\cos (\theta )$$."},{

Calculate Derivatives

Find $\frac{dy}{d\theta }$ and $\frac{dx}{d\theta }$ using the product rule.$$\frac{dx}{d\theta }=4({\cos }^2(\theta )\cos (\theta )-{\sin }^2(\theta )\cos (\theta ))=4\cos (\theta )({\cos }^2(\theta )-{\sin }^2(\theta ))$$$$\frac{dy}{d\theta }=4(2\sin (\theta ){\cos }^2(\theta ))=8\sin (\theta ){\cos }^2(\theta )$$

Determine Horizontal Tangent Lines

A tangent line is horizontal where $\frac{dy}{d\theta }=0$ and $\frac{dx}{d\theta }eq0$.Set the derivative $\frac{dy}{d\theta }=8\sin (\theta ){\cos }^2(\theta )=0$. This happens when $\sin (\theta )=0$. Therefore, $\theta =n\pi$ for integers $n$. Avoid points where $\cos (\theta )=0$ to ensure non-zero $\frac{dx}{d\theta }$.

Determine Vertical Tangent Lines

A tangent line is vertical where $\frac{dx}{d\theta }=0$ and $\frac{dy}{d\theta }eq0$.Set $\frac{dx}{d\theta }=4\cos (\theta )({\cos }^2(\theta )-{\sin }^2(\theta ))=0$. This occurs when $\cos (\theta )=0$.The angle for this case is $\theta =\frac{\pi }{2}+n\pi$ for integers $n$. Avoid points where $\sin (\theta )=0$ to ensure non-zero $\frac{dy}{d\theta }$.

Identify Points on the Curve

Convert the specific $\theta$ values found in steps 3 and 4 back to points on the polar curve $r=2\sin (2\theta )$. For horizontal tangents: $\theta =n\pi$ gives $r=2\sin (2n\pi )=0$, hence the origin. For vertical tangents: $\theta =\frac{\pi }{2}+n\pi$ gives points $r=2\sin (\pi +2n\pi )=2\sin (\pi )=0$, which are again the origin.Thus, the origin $(0,0)$ is where the tangents are either horizontal or vertical.

Key concepts

Tangent Lines

In mathematics, a tangent line is a straight line that touches a curve at a single point without crossing it. The slope of this line represents how steep the curve is at that point. Understanding tangent lines is crucial for analyzing the behavior of curves, especially in calculus.

When applied to polar coordinates, tangent lines can either be horizontal or vertical, each indicating a unique property of the curve's behavior at certain points.

• **Horizontal Tangent Lines:** These occur when there is no change in the vertical direction, meaning the derivative of the y-coordinate with respect to the angle $\theta$, $\frac{dy}{d\theta }$, equals zero.
• **Vertical Tangent Lines:** These occur when there is no change in the horizontal direction, meaning $\frac{dx}{d\theta }=0$, but $\frac{dy}{d\theta }$ is not zero.

For the curve $r=2\sin (2\theta )$, both vertical and horizontal tangents ultimately indicated that the point of tangency was at the origin $(0,0)$. This implies that the curve makes contact with the origin in a both horizontal or vertical manner, depending on the value of $\theta$.

Cartesian Coordinates

Cartesian coordinates provide a way to describe the position of points in a plane. It uses two numbers: $x$ and $y$. This system is commonly used due to its straightforward representation of space and is essential when dealing with complex equations and analyses.

Converting from polar to Cartesian coordinates involves mathematical transformations because polar coordinates are based on a radius and angle, while Cartesian coordinates are based on x, y positions in a plane. The formulas for conversion are:

• $x=r\cos (\theta )$
• $y=r\sin (\theta )$

In the exercise, the polar equation $r=2\sin (2\theta )$ was converted to Cartesian form using these formulas and trigonometric identities, allowing us to analyze the derivatives of $x$ and $y$ in terms of $\theta$. This step was essential to find the points of horizontal and vertical tangents on the curve.

Derivatives

Derivatives are a cornerstone in calculus, allowing us to analyze how functions change. They provide the rate at which one quantity changes with respect to another, which is essential when understanding the slopes of tangent lines.

In the context of polar and Cartesian transformations, derivatives are used to find the slope of a curve at any given point. By calculating $\frac{dx}{d\theta }$ and $\frac{dy}{d\theta }$ in Cartesian coordinates, one can determine where tangent lines to a polar curve are horizontal or vertical.

• **Product Rule:** This rule is often applied when differentiating products of two functions, as was used in the solution. It states $(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$, where $u$ and $v$ are functions of $\theta$.
• **Trigonometric Derivatives:** Knowing how to differentiate basic trigonometric functions like sine and cosine is crucial. For example, $\frac{d}{d\theta }(\sin (\theta ))=\cos (\theta )$, and $\frac{d}{d\theta }(\cos (\theta ))=-\sin (\theta )$.

These principles were key to deriving the expressions for $\frac{dx}{d\theta }$ and $\frac{dy}{d\theta }$ in order to find where the tangent lines to the curve are horizontal or vertical. This understanding aids greatly in analyzing curve behavior and characteristics.