Trigonometry is essential for Science Laboratory Technology professionals in Kenya, especially when precise measurements and spatial calculations are involved in laboratory setup, instrument calibration, and chemical analysis. Understanding the relationships between sides and angles of triangles enables accurate determination of distances, heights, and depths that cannot be measured directly. This chapter develops skills in applying Pythagoras theorem and trigonometric ratios, which are fundamental in solving practical problems encountered in laboratory environments.

4.1 Pythagoras theorem

Pythagoras theorem is a fundamental principle in trigonometry that relates the lengths of the sides in a right-angled triangle. In laboratory settings, this theorem helps determine unknown distances or lengths when direct measurement is difficult, such as calculating the height of a liquid column or the length of a pipette from its projection on a surface. Mastery of Pythagoras theorem enhances accuracy in spatial problem-solving.

4.1.1 Definition and formula

Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The formula is:

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

where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.

Worked Examples

Example 1: A laboratory bench has a rectangular surface with length 1.5 m and width 1.2 m. Find the diagonal length.

Given: \(a = 1.5 \text{ m}\), \(b = 1.2 \text{ m}\)

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

$$ c^{2} = (1.5)^{2} + (1.2)^{2} $$

$$ c^{2} = 2.25 + 1.44 = 3.69 $$

$$ c = \sqrt{3.69} = 1.92 \text{ m} $$

Answer: 1.92 m

Example 2: A pipette is held at a height of 0.8 m above the bench and the horizontal distance from its base is 0.6 m. Calculate the length of the pipette.

Given: \(a = 0.8 \text{ m}\), \(b = 0.6 \text{ m}\)

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

$$ c^{2} = (0.8)^{2} + (0.6)^{2} $$

$$ c^{2} = 0.64 + 0.36 = 1.00 $$

$$ c = \sqrt{1.00} = 1.00 \text{ m} $$

Answer: 1.00 m

Example 3: A laboratory ladder leans against a wall such that the foot of the ladder is 2.5 m from the wall and the ladder reaches a height of 6 m. Find the length of the ladder.

Given: \(a = 6 \text{ m}\), \(b = 2.5 \text{ m}\)

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

$$ c^{2} = (6)^{2} + (2.5)^{2} $$

$$ c^{2} = 36 + 6.25 = 42.25 $$

$$ c = \sqrt{42.25} = 6.5 \text{ m} $$

Answer: 6.5 m

Example 4: A rectangular chemical tank has a length of 3 m and a diagonal of 5 m. Find the width of the tank.

Given: \(c = 5 \text{ m}\), \(a = 3 \text{ m}\)

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

$$ b^{2} = c^{2} - a^{2} = (5)^{2} - (3)^{2} = 25 - 9 = 16 $$

$$ b = \sqrt{16} = 4 \text{ m} $$

Answer: 4 m

Example 5: A triangular laboratory shelf has sides \(a = 2.4 \text{ m}\), \(b = 1.8 \text{ m}\), and hypotenuse \(c\). Calculate the hypotenuse length.

Given: \(a = 2.4 \text{ m}\), \(b = 1.8 \text{ m}\)

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

$$ c^{2} = (2.4)^{2} + (1.8)^{2} = 5.76 + 3.24 = 9.00 $$

$$ c = \sqrt{9.00} = 3.0 \text{ m} $$

Answer: 3.0 m

Chapter Summary

This chapter began by exploring the Pythagoras theorem, which establishes the fundamental relationship between the sides of a right-angled triangle. It then introduced the primary trigonometric ratios: sine, cosine, and tangent, explaining their definitions and applications in calculating unknown sides and angles in triangles. The chapter progressed to trigonometric operations, covering essential identities that simplify expressions, methods for solving trigonometric equations, and the application of the sine, cosine, and tangent rules for solving non-right-angled triangles. Finally, the chapter addressed angles of elevation and depression, demonstrating how trigonometry can be applied to real-world problems involving heights and distances. Together, these topics provide a comprehensive foundation for using trigonometry in various scientific and engineering contexts.