Apply trigonometry is crucial for Science Laboratory Technology professionals in Kenya, especially when precise measurements and spatial calculations are required in laboratory setups, instrumentation alignment, and data interpretation. Understanding the relationships between angles and sides in right-angled triangles enables accurate calibration and positioning of equipment. This chapter covers fundamental trigonometric concepts, starting with the Pythagoras theorem and extending to the primary trigonometric ratios: sine, cosine, and tangent.

4.1 Pythagoras theorem

Pythagoras theorem is foundational in determining distances and lengths in laboratory environments where direct measurement is challenging. It applies to right-angled triangles, which frequently appear in laboratory apparatus setup and spatial analysis of samples or instruments. Mastery of this theorem allows technicians to calculate unknown side lengths when two sides of a right triangle are known.

4.1.1 Concept of Pythagoras theorem

The Pythagoras theorem relates the lengths of the sides of a right-angled triangle. It states that the square of the hypotenuse (the side opposite the right angle) equals 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 triangular support frame with legs measuring 3 m and 4 m. Find the length of the diagonal support.

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

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

$$ c^{2} = 3^{2} + 4^{2} $$

$$ c^{2} = 9 + 16 = 25 $$

$$ c = \sqrt{25} = 5\,m $$

Answer: 5 m

Example 2: A pipette stand forms a right triangle where one side is 5 cm and the hypotenuse is 13 cm. Calculate the length of the other side.

Given: \(a = 5\,cm\), \(c = 13\,cm\)

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

$$ b^{2} = 13^{2} - 5^{2} $$

$$ b^{2} = 169 - 25 = 144 $$

$$ b = \sqrt{144} = 12\,cm $$

Answer: 12 cm

Example 3: A laser beam path inside a laboratory forms a right triangle with one leg measuring 7.2 m and the hypotenuse is 9 m. Find the length of the other leg.

Given: \(a = 7.2\,m\), \(c = 9\,m\)

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

$$ b^{2} = 9^{2} - 7.2^{2} $$

$$ b^{2} = 81 - 51.84 = 29.16 $$

$$ b = \sqrt{29.16} = 5.4\,m $$

Answer: 5.4 m

Example 4: A rectangular lab floor has a length of 12 m and a width of 5 m. Find the length of the diagonal.

Given: \(a = 12\,m\), \(b = 5\,m\)

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

$$ c^{2} = 12^{2} + 5^{2} $$

$$ c^{2} = 144 + 25 = 169 $$

$$ c = \sqrt{169} = 13\,m $$

Answer: 13 m

Example 5: An inclined plane in the lab is set such that the vertical height is 2.5 m and the base length is 6 m. Find the length of the incline.

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

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

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

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

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

Answer: 6.5 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 how each ratio relates an angle to the lengths of specific sides in a right triangle. The chapter progressed to trigonometric operations, covering essential identities that simplify expressions and equations used to find unknown angles or sides. It also detailed the sine, cosine, and tangent rules, which extend trigonometry to non-right-angled triangles for solving various geometric problems. Finally, the chapter examined angles of elevation and depression, demonstrating how to apply trigonometric principles to calculate heights and distances in practical scenarios. Throughout, the focus remained on applying these concepts to solve real-world mathematical problems effectively.