Problem 28 - College Trigonometry

Question

Convert the point from polar coordinates into rectangular coordinates. $$ \text { (10, } \arctan (3)) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \((\sqrt{10}, 3\sqrt{10})\).

Step 1: Understand Polar Coordinates

Polar coordinates are represented as \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle from the positive x-axis.

Step 2: Polar to Rectangular Conversion Formulas

To convert polar coordinates \((r, \theta)\) into rectangular coordinates \((x, y)\), use the formulas: \(x = r \cdot \cos(\theta)\) and \(y = r \cdot \sin(\theta)\).

Step 3: Identify Given Values

The given polar coordinates are \((10, \arctan(3))\), where \(r = 10\) and \(\theta = \arctan(3)\).

Step 4: Calculate \(\cos(\theta)\) and \(\sin(\theta)\)

Since \(\theta = \arctan(3)\), we know that the tangent of this angle is 3, which corresponds to \(\tan(\theta) = \frac{3}{1}\). Use a right triangle with sides 3, 1, and hypotenuse \(\sqrt{3^2 + 1^2} = \sqrt{10}\). Thus, \(\cos(\theta) = \frac{1}{\sqrt{10}}\) and \(\sin(\theta) = \frac{3}{\sqrt{10}}\).

Step 5: Compute \(x\) and \(y\)

Substitute these values into the formulas: \(x = 10 \cdot \frac{1}{\sqrt{10}} = \frac{10}{\sqrt{10}} = \sqrt{10}\). Similarly, \(y = 10 \cdot \frac{3}{\sqrt{10}} = \frac{30}{\sqrt{10}} = 3\sqrt{10}\).

Step 6: Conclusion

The rectangular coordinates corresponding to the given polar coordinates are \((\sqrt{10}, 3\sqrt{10})\).

Key Concepts

Polar Coordinates

Polar coordinates provide a way to represent points on a plane using a distance and an angle. It is a different system compared to the usual Cartesian coordinate system. In polar coordinates, any point is described with two values:

• \( r \) - the radial distance from the origin, which tells us how far away the point is from the center. It is always a non-negative number.
• \( \theta \) - the angle measured from the positive x-axis in a counter-clockwise direction. This angle is usually in radians or degrees.

The idea is similar to finding a destination using a pair of compass directions and distance, which can be very intuitive when dealing with circular or rotational movements. Using the polar coordinates system can simplify the mathematics involved in many problems related to circles and periodic functions.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use two perpendicular axes to pinpoint a location on a plane. A point is represented as \((x, y)\), with:

• \( x \) - the horizontal distance from the origin, either to the left or right on the x-axis.
• \( y \) - the vertical distance from the origin, either above or below on the y-axis.

This coordinate system is typically more practical for adding, subtracting, and analyzing straight-line motion. Furthermore, it's closely tied to our intuitive perception of geometry and algebra, making it easier to perform a wide range of computations. The key challenge often lies in transforming polar coordinates — which are more intuitive for circular paths — into these straightforward rectangular coordinates.

Trigonometric Functions

Trigonometric functions like sine and cosine play a critical role in converting between polar and rectangular coordinates. These functions relate the angles of a right triangle to the lengths of its sides, which makes them perfect for the conversion formulas:

• \( x = r \cdot \cos(\theta) \) computes the horizontal component or the x-coordinate.
• \( y = r \cdot \sin(\theta) \) computes the vertical component or the y-coordinate.

For the point \((10, \arctan(3))\), the angle \( \theta = \arctan(3)\) corresponds to a situation where the tangent of \( \theta \) is 3 (that is, the ratio of opposite to adjacent in a right triangle is 3/1). To find \( \cos(\theta) \) and \( \sin(\theta) \), you can consider this as a right triangle where the opposite side is 3 units long, the adjacent side is 1 unit, and the hypotenuse is \( \sqrt{10} \), allowing you to determine that:

• \( \cos(\theta) = \frac{1}{\sqrt{10}} \)
• \( \sin(\theta) = \frac{3}{\sqrt{10}} \)

This understanding of trigonometric functions is critical in translating between the two coordinate systems.