Coordinate geometry is essential across many Kenyan professional fields, including construction, agriculture, and urban planning. It allows precise location and measurement of points, lines, and shapes in a plane. This chapter focuses on polar coordinates and polar equations, which provide an alternative to Cartesian coordinates for representing points and curves, useful in contexts such as surveying, navigation, and design layouts at hospitals, farms, and county offices.

4.1 Polar equations

Polar coordinates describe points in a plane using a distance from the origin and an angle from a reference direction, offering a natural way to handle circular and angular relationships. Polar equations express curves by relating the radius and angle, enabling representation of complex shapes encountered in real-world engineering and design tasks.

4.1.1 Definition of polar coordinates

Polar coordinates locate a point by its distance \(r\) from the origin and the angle \(\theta\) it makes with the positive x-axis. The relationship to Cartesian coordinates \((x,y)\) is given by

$$ x = r \cos \theta, \quad y = r \sin \theta $$

Conversely,

$$ r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1} (\frac{y}{x}) $$

Worked Examples

Example 1: A surveyor at a county government office marks a point 50 m from the origin at an angle of 30°. Find the Cartesian coordinates of the point.

Given: \(r=50\,m\), \(\theta=30^\circ\)

$$ x = r \cos \theta $$

$$ x = 50 \times \cos 30^\circ $$

$$ x = 50 \times 0.866 $$

$$ x = 43.3\,m $$

$$ y = r \sin \theta $$

$$ y = 50 \times \sin 30^\circ $$

$$ y = 50 \times 0.5 $$

$$ y = 25\,m $$

Answer: (43.3 m, 25 m)

Example 2: At a farm, a point has Cartesian coordinates \((36, 48)\) m. Find the polar coordinates \((r, \theta)\).

Given: \(x=36\,m\), \(y=48\,m\)

$$ r = \sqrt{x^2 + y^2} $$

$$ r = \sqrt{36^2 + 48^2} $$

$$ r = \sqrt{1296 + 2304} $$

$$ r = \sqrt{3600} $$

$$ r = 60\,m $$

$$ \theta = \tan^{-1} (\frac{y}{x}) $$

$$ \theta = \tan^{-1} (\frac{48}{36}) $$

$$ \theta = \tan^{-1} (1.333) $$

$$ \theta = 53.13^\circ $$

Answer: \(r = 60\,m, \theta = 53.13^\circ\)

Example 3: A point in a retail business layout has polar coordinates \(r=100\,m\), \(\theta=120^\circ\). Convert to Cartesian coordinates.

Given: \(r=100\,m\), \(\theta=120^\circ\)

$$ x = 100 \times \cos 120^\circ $$

$$ x = 100 \times (-0.5) $$

$$ x = -50\,m $$

$$ y = 100 \times \sin 120^\circ $$

$$ y = 100 \times 0.866 $$

$$ y = 86.6\,m $$

Answer: (-50 m, 86.6 m)

4.1.2 Definition and examples of polar equations

A polar equation expresses the radius \(r\) as a function of the angle \(\theta\), defining curves in polar coordinates. Common forms include circles, spirals, and rose curves, used in design and layout tasks such as garden planning at hotels or space allocation in hospitals.

Examples:

• Circle: \(r = a\) (constant radius)
• Spiral: \(r = a \theta\)
• Rose curve: \(r = a \sin n \theta\)

Worked Examples

Example 1: Find the radius at \(\theta = 45^\circ\) for the circle given by \(r = 20\) m.

Given: \(r=20\), \(\theta=45^\circ\)

Since \(r\) is constant,

Answer: \(r = 20\,m\)

Example 2: For the spiral \(r = 2 \theta\), find \(r\) when \(\theta = 90^\circ\).

Given: \(r = 2 \theta\), \(\theta = 90^\circ = \frac{\pi}{2}\) radians

Convert degrees to radians:

$$ \theta = \frac{90 \pi}{180} = \frac{\pi}{2} $$

$$ r = 2 \times \frac{\pi}{2} = \pi \approx 3.142\,m $$

Answer: \(r = 3.142\,m\)

Example 3: For the rose curve \(r = 5 \sin 3 \theta\), find \(r\) at \(\theta = 60^\circ\).

Given: \(r = 5 \sin 3 \theta\), \(\theta = 60^\circ\)

Calculate \(3 \theta\):

$$ 3 \times 60^\circ = 180^\circ $$

$$ r = 5 \times \sin 180^\circ = 5 \times 0 = 0 $$

Answer: \(r = 0\,m\)

Example 4: A hotel garden design uses \(r = 4 + 3 \cos \theta\). Find \(r\) at \(\theta = 90^\circ\).

Given: \(r = 4 + 3 \cos \theta\), \(\theta=90^\circ\)

$$ r = 4 + 3 \times \cos 90^\circ = 4 + 3 \times 0 = 4 $$

Answer: \(r = 4\,m\)

4.1.3 Representing curves using polar equations

Polar equations graph curves by plotting \(r\) for various \(\theta\) values, useful for curves with symmetry or rotational features. This technique aids spatial planning in county offices or farm layouts where circular or radial patterns occur.

Worked Examples

Example 1: Plot points for \(r = 3 + 2 \sin \theta\) at \(\theta = 0^\circ, 90^\circ, 180^\circ, 270^\circ\).

Given: \(r = 3 + 2 \sin \theta\)

At \(\theta=0^\circ\):

$$ r = 3 + 2 \times \sin 0^\circ = 3 + 0 = 3 $$

At \(\theta=90^\circ\):

$$ r = 3 + 2 \times \sin 90^\circ = 3 + 2 = 5 $$

At \(\theta=180^\circ\):

$$ r = 3 + 2 \times \sin 180^\circ = 3 + 0 = 3 $$

At \(\theta=270^\circ\):

$$ r = 3 + 2 \times \sin 270^\circ = 3 + 2 \times (-1) = 3 - 2 = 1 $$

Answer: Points (3,0°), (5,90°), (3,180°), (1,270°)

Example 2: For \(r = 5 \cos 2 \theta\), find \(r\) at \(\theta = 0^\circ, 45^\circ, 90^\circ, 135^\circ\).

Given: \(r = 5 \cos 2 \theta\)

At \(\theta=0^\circ\):

$$ r = 5 \times \cos 0^\circ = 5 \times 1 = 5 $$

At \(\theta=45^\circ\):

$$ r = 5 \times \cos 90^\circ = 5 \times 0 = 0 $$

At \(\theta=90^\circ\):

$$ r = 5 \times \cos 180^\circ = 5 \times (-1) = -5 $$

At \(\theta=135^\circ\):

$$ r = 5 \times \cos 270^\circ = 5 \times 0 = 0 $$

Answer: (5,0°), (0,45°), (-5,90°), (0,135°)

Example 3: A SACCO office layout uses \(r = 6 - 4 \sin \theta\). Find \(r\) at \(\theta=30^\circ, 60^\circ, 90^\circ, 120^\circ, 150^\circ\).

Given: \(r = 6 - 4 \sin \theta\)

At \(\theta=30^\circ\):

$$ r = 6 - 4 \times \sin 30^\circ = 6 - 4 \times 0.5 = 6 - 2 = 4 $$

At \(\theta=60^\circ\):

$$ r = 6 - 4 \times \sin 60^\circ = 6 - 4 \times 0.866 = 6 - 3.464 = 2.536 $$

At \(\theta=90^\circ\):

$$ r = 6 - 4 \times \sin 90^\circ = 6 - 4 \times 1 = 6 - 4 = 2 $$

At \(\theta=120^\circ\):

$$ r = 6 - 4 \times \sin 120^\circ = 6 - 4 \times 0.866 = 6 - 3.464 = 2.536 $$

At \(\theta=150^\circ\):

$$ r = 6 - 4 \times \sin 150^\circ = 6 - 4 \times 0.5 = 6 - 2 = 4 $$

Answer: \(r = 4, 2.536, 2, 2.536, 4\,m\)

Chapter Summary

This chapter explored the fundamentals of coordinate geometry, beginning with polar coordinates and their equations, which provide an alternative way to represent points and curves using radius and angle. It then examined Cartesian coordinates, defining points in a plane and illustrating how Cartesian equations describe various geometric shapes such as lines, circles, parabolas, ellipses, and hyperbolas. The chapter also detailed the relationship and conversion methods between polar and Cartesian systems, enabling flexible analysis of geometric figures. Graphing polar equations was covered extensively, including techniques for plotting points, utilizing symmetry and periodicity, and analyzing key features like intercepts and extrema. The concepts of tangents and normals to curves were introduced, with methods to find their slopes and equations in both Cartesian and polar forms. Finally, the chapter discussed loci, emphasizing their definition and significance in engineering, particularly through loci related to circles and their applications in mechanical design and motion analysis. Overall, the chapter provided a comprehensive foundation for understanding and applying coordinate geometry in engineering contexts.