Question

Find the equation of the lines tangent to the graph at the pole. \(r=\sin (3 \theta) ; \quad[0, \pi]\)

Short answer

The tangent lines at the pole are: \( y = 0 \), \( y = \sqrt{3}x \), and \( y = -\sqrt{3}x \).

Step-by-step solution

Identifying the Points of Interest

To find tangents at the pole (origin), we need points where the curve passes through the pole. In polar coordinates, a point is at the pole if \( r = 0 \). For \( r = \sin(3\theta) \), set \( \sin(3\theta) = 0 \).

Solving for Theta

Set \( \sin(3\theta) = 0 \). This occurs when \( 3\theta = n\pi \), where \( n \) is an integer. Solve for \( \theta \) to get \( \theta = \frac{n\pi}{3} \). Considering the interval \([0, \pi]\), \( \theta \) can be \( 0, \frac{\pi}{3}, \frac{2\pi}{3}, \pi \).

Determine the Tangent Lines

The tangent line at the pole for the polar curve \( r = \sin(3\theta) \) is a line passing through the origin in the direction of \( \theta \). For each \( \theta = \frac{n\pi}{3} \), the direction of the tangent line is given by the line equation in the form of \( y = mx \) with \( m = \tan(\theta) \).

Calculate Slope for Each Theta

For each \( \theta \): \( \tan(\theta) \) determines the slope of the tangent line. Calculate \( \tan(0) = 0 \), \( \tan(\frac{\pi}{3}) = \sqrt{3} \), \( \tan(\frac{2\pi}{3}) = -\sqrt{3} \), \( \tan(\pi) = 0 \).

Write Tangent Line Equations

With the slopes calculated, the tangent lines are as follows: \( y = 0 \) for \( \theta = 0 \) and \( \theta = \pi \); \( y = \sqrt{3}x \) for \( \theta = \frac{\pi}{3} \); \( y = -\sqrt{3}x \) for \( \theta = \frac{2\pi}{3} \).

Key concepts

Tangent Lines

Tangent lines are straight lines that touch a curve at only one point, maintaining the direction of the curve at that contact point. In the context of polar coordinates, these refer specifically to the lines that touch a polar curve exactly at the origin, also known as the pole.
In polar coordinates, a point is defined by its distance from the origin (pole) and the angle it makes with a reference line, typically the positive x-axis. A tangent line to a polar curve at the pole would point in the same direction as the curve does at that point.
To find tangent lines in polar coordinates, you need to first identify the points where the curve crosses the pole. For this exercise, you're working with the equation:

• Given: \( r = \sin(3\theta) \), where \( r = 0 \) indicates the pole.

These points are where \( 3\theta \) is an integer multiple of \( \,\pi \, \) because these are the angles that result in zero for the sine function. Once these are calculated, you find the equation of tangent lines in these directions.

Slope Calculation

Calculating the slope is a critical step in determining the equation of tangent lines. In polar coordinates, the slope of a tangent line is equivalent to the value of a tangent trigonometric function, specifically \( \tan(\theta) \). This value provides the 'steepness' of the line.
Let's calculate the slopes for our key angles. For each angle \( \theta \) found in the original exercise:

• \( \theta = 0 \): The slope is \( \tan(0) = 0 \).
• \( \theta = \frac{\pi}{3} \): The slope is \( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \).
• \( \theta = \frac{2\pi}{3} \): The slope is \( \tan\left(\frac{2\pi}{3}\right) = -\sqrt{3} \).
• \( \theta = \pi \): The slope is \( \tan(\pi) = 0 \).

These calculations provide slopes which are used in developing equations for the tangent lines. Each \( \tan(\theta) \) tells you how much the line rises or falls as it goes from left to right. The slope helps form the equation \( y = mx \) for the tangent line, where \( m \) is the slope.

Trigonometric Functions

Trigonometric functions are mathematical functions of an angle, pivotal when working in polar coordinates. Here, we deal primarily with the sine and tangent functions. These functions relate angles to ratios within a right triangle and are also summed up in unit circles.
For a polar coordinate equation like \( r = \sin(3\theta) \), the sine function takes on a repeating cycle over angles. When we set \( r = 0 \), this boils down to solving for angles where the sine function equals zero, which are integer multiples of \( \,\pi \, \).

• Sine Function: \( \sin(3\theta) \) will be zero at angles like \( 0, \frac{\pi}{3}, \frac{2\pi}{3}, \) and \( \pi \).
• Tangent Function: Calculating \( \tan(\theta) \) supports finding the slope of tangent lines at these angles.

Trigonometric functions serve as a bridge between angles and linear measures, hence making them essential in defining and calculating properties like slopes in polar coordination. Using these functions allows us to handle periodic phenomena and paths traced on circles, especially in coupling with polar curves.