Question

Find the slope of the tangent line to the graph of the polar equation at the point specified by the value of θ. r=cos( θ​/3 ),θ=π

Answer 1

To find the slope of the tangent line to the graph of a polar equation at a certain value of θ, we convert the polar equation to Cartesian coordinates and calculate the derivative. The slope at θ = π is found by applying the appropriate formula for the slope of a tangent line in polar coordinates.

Step-by-step explanation:

To find the slope of the tangent line to the graph of the polar equation at a given point, we need to accurately calculate the derivative of the polar function with respect to θ (θ = pi in this case). First, we convert the polar equation r = cos( θ /3 ) to Cartesian coordinates using the relationships x = r × cos(θ) and y = r × sin(θ). To find the slope at θ = π, we'll need to calculate dr/dθ and use the formula for the slope of the tangent line in polar coordinates.

Steps to find the slope at θ = π:

1. Calculate dr/dθ of the polar function.
2. Calculate the slope at θ = π using the formula dy/dx = (dr/dθ × sin(θ) + r × cos(θ))/(dr/dθ × cos(θ) - r × sin(θ)).
3. Substitute the value of θ = π into the slope formula to find the specific slope of the tangent line at that point.

Answer 2

The slope of the tangent line to the graph of the polar equation r=cos(θ​/3), θ=π is -√3/2.

Explanation:

To find the slope of the tangent line at a given point on a polar graph, we can use the formula for slope in polar coordinates: m=dy/dθ / dx/dθ. First, we need to find the derivative of r with respect to θ (dr/dθ) and the derivative of θ with respect to θ (dθ/dθ). In this case, dr/dθ=-sin(θ/3) and dθ/dθ=1. We can then substitute these values into the slope formula to get m=-sin(θ/3)/1=-sin(π/3)/1=-√3/2. This is the slope of the tangent line at any point on the graph, but since we are specifically looking at the point where θ=π, we can plug that value in to get the final answer of -√3/2.

In polar coordinates, the slope of a tangent line is defined as the rate of change of the radial distance (r) with respect to the angular position (θ). This means that as we move along the graph in the direction of increasing θ, the change in r is given by the slope of the tangent line at that point. In this case, the graph of r=cos(θ/3) is a circle with a radius of 1. When θ=π, the point on the circle is at the bottom, or 180 degrees.