Problem 5 - Calculus

Question

Find the polar coordinates \(r \geqslant 0\) and \(0 \leqslant 0<2 \pi\) of these points. $$ (x, y)=(3,4) $$

Step-by-Step Solution

Answer

The polar coordinates are \((5, 0.927)\).

Step 1: Convert Cartesian to Polar

Given a point in Cartesian coordinates, \( x = 3 \) and \( y = 4 \). The polar coordinates \( (r, \theta) \) can be found using the formulas: \( r = \sqrt{x^2 + y^2} \) Therefore, \( r = \sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5 \).

Step 2: Calculate Angle \( \theta \)

The angle \( \theta \) in polar coordinates is given by \( \theta = \arctan\left(\frac{y}{x}\right) \). Substituting the values, we have \( \theta = \arctan\left(\frac{4}{3}\right) \). Using an arctan calculator or table, we find that \( \theta \approx 0.927 \) radians.

Step 3: Conclusion

The polar coordinates of the point \((3,4)\) are \((5, 0.927)\) where \(r = 5\) and \(\theta \approx 0.927\) radians.

Key Concepts

Cartesian coordinates

Cartesian coordinates describe a point in a 2D space using a pair of numerical values. These values, typically labeled as \(x\) and \(y\), represent the point's precise location relative to two perpendicular lines known as axes: the x-axis (horizontal) and y-axis (vertical). For instance, the point (3,4) indicates that, starting from the origin (0,0), you move 3 units along the x-axis and 4 units up along the y-axis.

This system is incredibly versatile and is widely used in various fields, such as mathematics, physics, engineering, and computer graphics for plotting and visualizing data. This coordinate system is particularly useful for understanding geometric shapes and performing algebraic operations.

conversion from Cartesian to Polar

Conversion from Cartesian coordinates to polar coordinates involves transforming the coordinates of a point from the \( (x, y) \) format to the \( (r, \theta) \) format.

This is an important concept in mathematics as polar coordinates are often more convenient for problems involving angles and curves. Here's how the conversion is executed:

• Calculate the radial distance, \( r \): This is the distance from the origin to the point, determined using the Pythagorean theorem as \( r = \sqrt{x^2 + y^2} \).
• Determine the angle, \( \theta \): This angle is formed between the positive x-axis and the line connecting the origin with the point. It is calculated using \( \theta = \arctan\left(\frac{y}{x}\right) \).

For example, transforming the Cartesian point (3,4) to polar coordinates, as per the calculations, gives \( r = 5 \) and \( \theta \approx 0.927 \) radians.

This represents a point 5 units away from the origin at an angle of 0.927 radians.

angle calculation in polar coordinates

Angle calculation in polar coordinates is crucial for understanding the orientation of a point in a plane. In the polar system, the angle \( \theta \) specifies how far the point deviates from the positive x-axis.

To calculate this angle \( \theta \), we use the arctangent function: \( \theta = \arctan\left(\frac{y}{x}\right) \). Here's a breakdown of the calculation:

• The function \( \arctan \) (or \( \tan^{-1} \)) is used to find the angle whose tangent is a given number. This is derived from the relationship of opposite over adjacent sides in a right triangle.
• First, divide \( y \) by \( x \) to get the tangent value. In our example, \( y = 4 \) and \( x = 3 \), hence \( \frac{y}{x} = \frac{4}{3} \).
• Second, compute \( \arctan(\frac{4}{3}) \), which results in an angle measurement in radians, approximately \( 0.927 \) radians in this case.

Understanding this process is important for fields such as physics and engineering, where direction and angles play significant roles.

The angle, along with the radial distance, forms the basis of describing points in polar coordinates.