Question

Find a forμla for the area of the triangle with vertices (in polar coordinates) ((r_1, θ_1)), ((r_2, θ_2)), and ((r_3, θ_3)).

a) (A = 12 left|r_1^2 sin(θ_2 - θ_3) + r_2^2 sin(θ_3 - θ_1) + r_3^2 sin(θ_1 - θ_2)right|)
b) (A = 12 left|r_1 cos(θ_2 - θ_3) + r_2 cos(θ_3 - θ_1) + r_3 cos(θ_1 - θ_2)right|)
c) (A = 12 left|r_1^2 cos(θ_2 - θ_3) + r_2^2 cos(θ_3 - θ_1) + r_3^2 cos(θ_1 - θ_2)right|)
d) (A = 12 left|r_1 sin(θ_2 - θ_3) + r_2 sin(θ_3 - θ_1) + r_3 sin(θ_1 - θ_2)right|)

Answer 1

The formula for the area
`(\(A\))` of the triangle with vertices in polar coordinates
`\((r_1, \theta_1)\), \((r_2, \theta_2)\),` and
`\((r_3, \theta_3)\)` is given by option c)
`\[A = (1)/(2) \left| r_1^2 \cos(\theta_2 - \theta_3) + r_2^2 \cos(\theta_3 - \theta_1) + r_3^2 \cos(\theta_1 - \theta_2) \right|\].`

Step-by-step explanation:

The formula for the area of a triangle given its vertices in polar coordinates is derived from the formula for the area of a triangle using Cartesian coordinates. In polar coordinates, the vertices
`\((r_1, \theta_1)\), \((r_2, \theta_2)\), and \((r_3, \theta_3)\)`are treated as points in a plane. The formula involves the summation of the product of half the product of the radii with the cosine of the angular differences between the vertices.

Mathematically, this is expressed as:

`\[ A = (1)/(2) \left| r_1^2 \cos(\theta_2 - \theta_3) + r_2^2 \cos(\theta_3 - \theta_1) + r_3^2 \cos(\theta_1 - \theta_2) \right| \]`

This formula accounts for both the radial distances
`(\(r_i\))` and the angular differences between the vertices, providing an accurate representation of the area of the triangle in polar coordinates.

Understanding how to adapt formulas from Cartesian coordinates to polar coordinates is crucial in dealing with problems involving circular or radial symmetry. In this case, the selected formula (option c) reflects the appropriate conversion of the Cartesian formula to the polar coordinate system for calculating the area of the triangle.