Architecture professionals in Kenya frequently encounter problems involving angles and distances when designing and analyzing building elements. Trigonometry and hyperbolic functions are essential mathematical tools for solving such problems, especially when working with sloped roofs, inclined planes, and structural components. This chapter focuses on applying trigonometry to right-angled triangles and triangles with acute and obtuse angles, providing practical calculation techniques relevant to architectural design and construction.

2.1 Right-angled triangles

Right-angled triangles are fundamental in architectural calculations since many building components form or can be decomposed into right triangles. Understanding the relationships between their sides and angles enables precise determination of lengths, slopes, and heights in design and site measurements.

2.1.1 Basic trigonometric ratios in right-angled triangles

The three primary trigonometric ratios, sine, cosine, and tangent, relate the angles of a right-angled triangle to the ratios of its sides. These ratios are crucial for finding unknown sides or angles when at least one side and one angle are known.

$$\sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}}$$

$$\cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}}$$

$$\tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}}$$

Worked Examples

Example 1: A roof truss in a school has a rafter making an angle of \(30^\circ\) with the horizontal. The horizontal span is 6 m. Find the length of the rafter.

Given: \(\theta = 30^\circ\), adjacent side \(= 6\,m\)

$$\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$

$$\cos 30^\circ = \frac{6}{\text{Rafter length}}$$

$$0.866 = \frac{6}{\text{Rafter length}}$$

$$\text{Rafter length} = \frac{6}{0.866}$$

$$\text{Rafter length} = 6.93\,m$$

Answer: 6.93 m

Example 2: A staircase rises at an angle of \(40^\circ\). The length of the staircase is 5 m. Calculate the vertical height of the staircase.

Given: \(\theta = 40^\circ\), hypotenuse \(= 5\,m\)

$$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$

$$\sin 40^\circ = \frac{\text{Height}}{5}$$

$$0.643 = \frac{\text{Height}}{5}$$

$$\text{Height} = 5 \times 0.643$$

$$\text{Height} = 3.215\,m$$

Answer: 3.22 m

Example 3: A scaffolding pole leans against a building making an angle of \(60^\circ\) with the ground. The pole is 8 m long. Calculate the horizontal distance from the base of the building to the bottom of the pole.

Given: \(\theta = 60^\circ\), hypotenuse \(= 8\,m\)

$$\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$

$$\cos 60^\circ = \frac{\text{Distance}}{8}$$

$$0.5 = \frac{\text{Distance}}{8}$$

$$\text{Distance} = 8 \times 0.5$$

$$\text{Distance} = 4\,m$$

Answer: 4 m

2.1.2 Using Pythagoras’ theorem in architectural contexts

Pythagoras’ theorem relates the lengths of the sides in a right-angled triangle and is widely used to determine distances when two sides are known.

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

where \(c\) is the hypotenuse.

Worked Examples

Example 1: A rectangular window has a width of 1.2 m and a height of 1.5 m. Find the length of the diagonal.

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

$$c^2 = 1.2^2 + 1.5^2$$

$$c^2 = 1.44 + 2.25$$

$$c^2 = 3.69$$

$$c = \sqrt{3.69}$$

$$c = 1.92\,m$$

Answer: 1.92 m

Example 2: A ramp must rise 1.8 m in height and the horizontal distance available is 6 m. Find the length of the ramp.

Given: \(a = 1.8\,m\), \(b = 6\,m\)

$$c^2 = 1.8^2 + 6^2$$

$$c^2 = 3.24 + 36$$

$$c^2 = 39.24$$

$$c = \sqrt{39.24}$$

$$c = 6.26\,m$$

Answer: 6.26 m

Example 3: A ladder leans on a wall such that the base is 3 m from the wall and the ladder reaches a height of 4 m on the wall. Find the length of the ladder.

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

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

$$c^2 = 9 + 16$$

$$c^2 = 25$$

$$c = \sqrt{25}$$

$$c = 5\,m$$

Answer: 5 m

2.1.3 Angle of elevation and depression in site measurements

Angles of elevation and depression are used in architecture to determine heights and distances on site when direct measurements are difficult.

• Angle of elevation: The angle between the horizontal line and the line of sight looking upward.
• Angle of depression: The angle between the horizontal line and the line of sight looking downward.

These angles help calculate heights or distances using trigonometric ratios.

Worked Examples

Example 1: From a point 10 m from a building, the angle of elevation to the top is \(25^\circ\). Find the height of the building.

Given: \(d = 10\,m\), \(\theta = 25^\circ\)

$$\tan \theta = \frac{\text{Height}}{d}$$

$$\tan 25^\circ = \frac{h}{10}$$

$$0.466 = \frac{h}{10}$$

$$h = 10 \times 0.466$$

$$h = 4.66\,m$$

Answer: 4.66 m

Example 2: A person stands on a balcony 8 m above the ground. The angle of depression to a point on the ground is \(35^\circ\). Find the horizontal distance from the base of the building to the point.

Given: height \(= 8\,m\), \(\theta = 35^\circ\)

$$\tan \theta = \frac{\text{Height}}{\text{Distance}}$$

$$\tan 35^\circ = \frac{8}{d}$$

$$0.700 = \frac{8}{d}$$

$$d = \frac{8}{0.700}$$

$$d = 11.43\,m$$

Answer: 11.43 m

Example 3: From the top of a tower 20 m high, the angle of depression to a car is \(15^\circ\). Find the horizontal distance from the base of the tower to the car.

Given: height \(= 20\,m\), \(\theta = 15^\circ\)

$$\tan \theta = \frac{20}{d}$$

$$\tan 15^\circ = \frac{20}{d}$$

$$0.268 = \frac{20}{d}$$

$$d = \frac{20}{0.268}$$

$$d = 74.63\,m$$

Answer: 74.63 m

2.1.4 Solving right-angled triangles for unknown sides and angles

Given some sides and/or angles, architects often need to solve the entire triangle to find missing measurements critical for design or construction.

Key steps:

1. Use Pythagoras’ theorem if two sides are known.
2. Use trigonometric ratios to find unknown sides or angles.
3. Use angle sum property: \(90^\circ + \text{other two angles} = 180^\circ\).

Worked Examples

Example 1: A triangle has one angle \(30^\circ\) and hypotenuse 10 m. Find the lengths of the other sides.

Given: \(\theta = 30^\circ\), hypotenuse = 10 m

$$\sin 30^\circ = \frac{\text{Opposite}}{10}$$

$$0.5 = \frac{\text{Opposite}}{10}$$

$$\text{Opposite} = 10 \times 0.5 = 5\,m$$

$$\cos 30^\circ = \frac{\text{Adjacent}}{10}$$

$$0.866 = \frac{\text{Adjacent}}{10}$$

$$\text{Adjacent} = 10 \times 0.866 = 8.66\,m$$

Answer: Opposite = 5 m, Adjacent = 8.66 m

Example 2: A right triangle has adjacent side 7 m and opposite angle \(45^\circ\). Find the hypotenuse and opposite side.

Given: adjacent = 7 m, \(\theta = 45^\circ\)

$$\tan 45^\circ = \frac{\text{Opposite}}{7}$$

$$1 = \frac{\text{Opposite}}{7}$$

$$\text{Opposite} = 7\,m$$

$$\cos 45^\circ = \frac{7}{\text{Hypotenuse}}$$

$$0.707 = \frac{7}{\text{Hypotenuse}}$$

$$\text{Hypotenuse} = \frac{7}{0.707} = 9.9\,m$$

Answer: Hypotenuse = 9.9 m, Opposite = 7 m

Example 3: A ladder leans against a wall at \(70^\circ\) to the ground. The foot of the ladder is 2.5 m from the wall. Find the length of the ladder.

Given: \(\theta = 70^\circ\), adjacent = 2.5 m

$$\cos 70^\circ = \frac{2.5}{\text{Hypotenuse}}$$

$$0.342 = \frac{2.5}{\text{Hypotenuse}}$$

$$\text{Hypotenuse} = \frac{2.5}{0.342} = 7.31\,m$$

Answer: Ladder length = 7.31 m

Chapter Summary

This chapter covered the essential trigonometric principles related to right-angled triangles, focusing on calculating side lengths and angles using sine, cosine, and tangent ratios. It then extended these concepts to acute and obtuse angled triangles, demonstrating how to solve for unknown sides and angles in more complex triangle configurations. Calculations involving angles of elevation and depression were explored to apply trigonometry in practical surveying and architectural contexts. The chapter introduced hyperbolic functions and detailed methods for calculating values of sinh, cosh, tanh, coth, sech, and cosec, highlighting their differences from circular trigonometric functions. Techniques for solving trigonometric equations were also presented, enabling the determination of angle measures from given function values. Finally, the behaviour and characteristics of sine and cosine graphs were examined alongside the graphical representations of hyperbolic functions, providing a comprehensive understanding of their periodicity and amplitude variations relevant to architectural design and analysis.