Problem 92 - Algebra and Trigonometry

Question

Prove that the area of the triangle with vertices \((0,0),\left(r_{1}, \theta_{1}\right),\) and \(\left(r_{2}, \theta_{2}\right), 0 \leq \theta_{1}<\theta_{2} \leq \pi,\) is $$ K=\frac{1}{2} r_{1} r_{2} \sin \left(\theta_{2}-\theta_{1}\right) $$

Step-by-Step Solution

Answer

The area is \frac{1}{2} r_{1} r_{2} \sin(\theta_{2} - \theta_{1})\.

Step 1: Understand the Coordinates System

The vertices of the triangle are given in polar coordinates. The vertices are \(0,0\), \(r_{1}, \theta_{1}\), and \(r_{2}, \theta_{2}\). In this system, a point is described by its distance from the origin (r) and the angle (θ) from the positive x-axis.

Step 2: Convert Polar Coordinates to Cartesian Coordinates

To use the formula for the area of a triangle, convert the polar coordinates to Cartesian coordinates. The conversions are \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). The points in Cartesian coordinates are: \(A(0,0)\), \(B(r_{1} \cos(\theta_{1}), r_{1} \sin(\theta_{1}))\), and \(C(r_{2} \cos(\theta_{2}), r_{2} \sin(\theta_{2}))\).

Step 3: Use the Determinant Formula for Area

The area K of a triangle with vertices at \(A(x_{1}, y_{1})\), \(B(x_{2}, y_{2})\), and \(C(x_{3}, y_{3})\) can be calculated using the determinant formula: \ K = \frac{1}{2} \left| x_{1}(y_{2} - y_{3}) + x_{2}(y_{3} - y_{1}) + x_{3}(y_{1} - y_{2}) \right|\. Since \(A = (0, 0)\), simplify to \ K = \frac{1}{2} \left| x_{2}y_{3} - x_{3}y_{2} \right| \.

Step 4: Substitute the Values

Substitute the Cartesian coordinates of points B and C in the formula. \( x_{2} = r_{1} \cos(\theta_{1})\), \( y_{2} = r_{1} \sin(\theta_{1})\), \( x_{3} = r_{2} \cos(\theta_{2})\), and \( y_{3} = r_{2} \sin(\theta_{2})\). The area K becomes: \ K = \frac{1}{2} \left| r_{1} \cos(\theta_{1}) \cdot r_{2} \sin(\theta_{2}) - r_{2} \cos(\theta_{2}) \cdot r_{1} \sin(\theta_{1}) \right| \.

Step 5: Simplify the Expression

Factor out \(r_{1}r_{2}\) from the expression and use the trigonometric identity for the sine of angle difference, \ \sin(\theta_{2} - \theta_{1}) = \sin(\theta_{2}) \cos(\theta_{1}) - \cos(\theta_{2}) \sin(\theta_{1})\. The area K simplifies to: \ K = \frac{1}{2} r_{1} r_{2} \left| \sin(\theta_{2} - \theta_{1}) \right| \.

Step 6: Finalize the Area Expression

Since \0 \leq \theta_{1} < \theta_{2} \leq \pi\, \(\theta_{2} - \theta_{1}\) is within the range for which \(\sin(\theta_{2} - \theta_{1})\) is non-negative. Therefore, the absolute value is not needed. Thus, the area of the triangle K is: \ K = \frac{1}{2} r_{1} r_{2} \sin(\theta_{2} - \theta_{1})\.

Key Concepts

Polar Coordinates

Polar coordinates provide a way to uniquely specify the location of a point in a two-dimensional plane. They are expressed as \( (r, \theta) \), where \( r \) represents the radius or distance from the origin \( (0,0) \) and \( \theta \) indicates the angle from the positive x-axis. This system is particularly useful when dealing with problems involving circular or rotational symmetry.
Converting a point from polar coordinates to Cartesian coordinates uses the formulas:

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

These formulas help in translating polar points into Cartesian points, which are more commonly used in standard geometry calculations.

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry that simplify complex expressions and solve equations involving trigonometric functions. One such important identity used in the exercise is:

• \( \sin(\theta_2 - \theta_1) = \sin(\theta_2) \cos(\theta_1) - \cos(\theta_2) \sin(\theta_1) \)

This identity transforms the original problem into a more manageable form, allowing us to derive the area of the triangle effectively. Understanding and practicing these identities can significantly simplify solving trigonometric problems.
Always keep these identities handy as they reveal hidden patterns and relationships within trigonometric functions, making problem-solving more intuitive.

Triangle Area Formula

The formula for the area of a triangle using Cartesian coordinates reads:
\[ K = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]For a triangle with one vertex at the origin, it simplifies to:
\[ K = \frac{1}{2} \left| x_2 y_3 - x_3 y_2 \right| \]This further reduces the complexity involved in conversions. This formula relies on the determinant of a matrix, providing a versatile method to calculate area regardless of the triangle's orientation. Be sure to practice applying this formula to various geometric scenarios to master its use.

Coordinate Conversion

Coordinate conversion is the process of translating geometrical points from one coordinate system to another. In this exercise, we convert polar coordinates to Cartesian coordinates to employ the triangle area formula. The steps are:

• Convert each vertex using \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \)
• Apply these coordinates within established geometric formulas, like the determinant-based area formula

Such conversions enhance our ability to solve geometric problems by leveraging the strengths of different coordinate systems. Practice converting various geometric figures to become adept at moving between systems effortlessly.