Applied biology professionals in Kenya often encounter practical problems involving measurement and spatial relationships, such as determining distances, heights, and angles in environmental surveys, laboratory setups, and agricultural fieldwork. Trigonometry provides essential mathematical tools to solve these problems by relating angles and sides in right-angled triangles. This chapter focuses on applying trigonometric principles, starting with Pythagoras’ theorem and moving to trigonometric ratios, identities, and equations relevant to biological contexts.

6.1 Pythagoras Theorem

Pythagoras theorem is fundamental in applied biology when measuring distances that are not directly accessible, such as estimating the height of trees or the length of a diagonal in a rectangular plot. It applies to right-angled triangles and relates the lengths of the triangle’s sides, enabling calculation of an unknown side when the other two are known.

6.1.1 Statement of Pythagoras Theorem

The theorem states that in a right-angled triangle, the square of the hypotenuse \(c\) (the side opposite the right angle) equals the sum of the squares of the other two sides \(a\) and \(b\). Mathematically:

$$c^2 = a^2 + b^2$$

Worked Examples

Example 1: A biologist measures the base and height of a rectangular garden plot as 30 m and 40 m respectively. Find the length of the diagonal.

Given: \(a = 30\,m\), \(b = 40\,m\)

$$c^2 = a^2 + b^2$$

$$c^2 = 30^2 + 40^2$$

$$c^2 = 900 + 1600$$

$$c^2 = 2500$$

$$c = \sqrt{2500} = 50\,m$$

Answer: 50 m

Example 2: In an environmental survey, a researcher finds a right-angled triangular area where one side measures 24 m and the hypotenuse is 25 m. Find the length of the other side.

Given: \(a = 24\,m\), \(c = 25\,m\)

$$c^2 = a^2 + b^2$$

$$25^2 = 24^2 + b^2$$

$$625 = 576 + b^2$$

$$b^2 = 625 - 576 = 49$$

$$b = \sqrt{49} = 7\,m$$

Answer: 7 m

Example 3: A researcher wants to determine the length of a ladder leaning against a tree. The base of the ladder is 6 m from the tree, and the ladder reaches a height of 8 m. Find the ladder length.

Given: \(a = 6\,m\), \(b = 8\,m\)

$$c^2 = a^2 + b^2$$

$$c^2 = 6^2 + 8^2$$

$$c^2 = 36 + 64 = 100$$

$$c = \sqrt{100} = 10\,m$$

Answer: 10 m

Kenyan Context Example

Example 4: At Egerton University, a biologist needs to measure the diagonal distance across a rectangular experimental plot with sides 20 m and 15 m. Find the diagonal length.

Given: \(a = 20\,m\), \(b = 15\,m\)

$$c^2 = a^2 + b^2$$$$c^2 = 20^2 + 15^2$$$$c^2 = 400 + 225 = 625$$$$c = \sqrt{625} = 25\,m$$

Answer: 25 m

6.1.2 Finding Unknown Sides Using Pythagoras Theorem

To find an unknown side in a right-angled triangle:

• Identify the hypotenuse and the two other sides.
• If the hypotenuse is unknown, use \(c = \sqrt{a^2 + b^2}\).
• If one leg is unknown, use \(a = \sqrt{c^2 - b^2}\) or \(b = \sqrt{c^2 - a^2}\).

Worked Examples

Example 4: A rectangular pond has width 15 m and diagonal 25 m. Find its length.

Given: \(b = 15\,m\), \(c = 25\,m\)

$$a = \sqrt{c^2 - b^2}$$

$$a = \sqrt{25^2 - 15^2}$$

$$a = \sqrt{625 - 225} = \sqrt{400} = 20\,m$$

Answer: 20 m

Example 5: A tree casts a shadow 12 m long. The angle between the ground and the sun’s rays is such that the hypotenuse (sun rays) is 13 m. Find the height of the tree.

Given: \(a = ?\), \(b = 12\,m\), \(c = 13\,m\)

$$a = \sqrt{c^2 - b^2}$$

$$a = \sqrt{13^2 - 12^2}$$

$$a = \sqrt{169 - 144} = \sqrt{25} = 5\,m$$

Answer: 5 m

6.1.3 Application of Pythagoras in Field Measurements

Pythagoras theorem helps in indirect measurement where direct measurement is difficult or impossible:

• Measuring heights of trees or poles using shadows.
• Calculating distances across inaccessible areas such as rivers or ravines.
• Determining lengths of diagonal pathways or plots.
• Estimating lengths of equipment parts in biology labs.
• Mapping rectangular plots in agricultural research.

Worked Examples

Example 6: A biologist maps a rectangular field with sides 50 m and 120 m. Find the shortest distance between two opposite corners.

Given: \(a = 50\,m\), \(b = 120\,m\)

$$c = \sqrt{a^2 + b^2}$$

$$c = \sqrt{50^2 + 120^2}$$

$$c = \sqrt{2500 + 14400} = \sqrt{16900} = 130\,m$$

Answer: 130 m

Example 7: A support beam is placed diagonally across a rectangular frame 3 m by 4 m. Find the beam length.

Given: \(a = 3\,m\), \(b = 4\,m\)

$$c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\,m$$

Answer: 5 m

Chapter Summary

This chapter covered the application of trigonometry in solving mathematical problems relevant to science and engineering. It began by exploring the Pythagoras theorem, which establishes the relationship between the sides of a right-angled triangle and is fundamental in calculating distances and lengths. The chapter then introduced the primary trigonometric ratios: sine, cosine, and tangent, explaining how these ratios relate the angles to the sides of a triangle. Next, trigonometric identities were presented, providing essential formulas that simplify complex trigonometric expressions and solve equations efficiently. The study of trigonometric equations followed, demonstrating methods to find unknown angles or sides by manipulating these identities and ratios. Throughout the chapter, emphasis was placed on practical calculations and the step-by-step approach to problem solving. The knowledge gained enables precise determination of angles and distances in various scientific contexts.