By the end of this chapter, you will be able to:
Mastering these skills will help you solve real-world engineering problems involving angles and curves with accuracy and confidence.
Engineering mathematics is foundational for quantity surveyors in Kenya, enabling precise measurement, cost estimation, and analysis of construction projects. Mastery of trigonometry and hyperbolic functions allows quantity surveyors to solve problems involving angles, distances, and structural geometry encountered in site layouts, building measurements, and material quantities. This chapter focuses on applying trigonometric ratios, identities, and rules in practical quantity surveying contexts to enhance accuracy and efficiency.
Trigonometric ratios and identities form the mathematical backbone for solving problems involving angles and lengths in right-angled and oblique triangles. Quantity surveyors use these ratios to calculate heights, slopes, and distances on construction sites, especially when direct measurement is impractical. Understanding these concepts allows precise determination of quantities and helps in site planning, setting out, and verifying structural dimensions.
Trigonometric ratios relate the angles of a right-angled triangle to the ratios of its sides. The primary ratios are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), defined as follows:
$$\sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}}$$
$$\cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}}$$
$$\tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}}$$
Example 1: A surveyor measures the angle of elevation to the top of a water tank as 30°, and the distance from the tank base is 20 m. Calculate the height of the tank.
Given: \(\theta = 30^\circ\), adjacent side \(= 20\,m\)
$$\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$$
$$\tan 30^\circ = \frac{h}{20}$$
$$h = 20 \times \tan 30^\circ$$
$$h = 20 \times 0.5774$$
Answer: \(h = 11.55\,m\)
Example 2: A ladder leans against a wall forming a 60° angle with the ground. If the ladder length is 10 m, find the height it reaches on the wall.
Given: \(\theta = 60^\circ\), hypotenuse \(= 10\,m\)
$$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$
$$\sin 60^\circ = \frac{h}{10}$$
$$h = 10 \times \sin 60^\circ$$
$$h = 10 \times 0.8660$$
Answer: \(h = 8.66\,m\)
Example 3: A trench is dug with a slope angle of 45°, and the vertical depth is 5 m. Find the length of the slope surface.
Given: \(\theta = 45^\circ\), opposite side \(= 5\,m\)
$$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$
$$\sin 45^\circ = \frac{5}{l}$$
$$l = \frac{5}{\sin 45^\circ}$$
$$l = \frac{5}{0.7071}$$
Answer: \(l = 7.07\,m\)
Reciprocal identities express the trigonometric functions in terms of their reciprocals, while quotient identities relate tangent and cotangent to sine and cosine. These are useful for transforming expressions and solving equations in surveying calculations.
Reciprocal identities:
$$\csc \theta = \frac{1}{\sin \theta}$$
$$\sec \theta = \frac{1}{\cos \theta}$$
$$\cot \theta = \frac{1}{\tan \theta}$$
Quotient identities:
$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$
$$\cot \theta = \frac{\cos \theta}{\sin \theta}$$
Example 1: Find \(\sec 45^\circ\) given \(\cos 45^\circ = 0.7071\).
Given: \(\cos 45^\circ = 0.7071\)
$$\sec 45^\circ = \frac{1}{\cos 45^\circ}$$
$$\sec 45^\circ = \frac{1}{0.7071}$$
Answer: \(\sec 45^\circ = 1.414\)
Example 2: Calculate \(\cot 60^\circ\) using sine and cosine values.
Given: \(\sin 60^\circ = 0.8660\), \(\cos 60^\circ = 0.5\)
$$\cot 60^\circ = \frac{\cos 60^\circ}{\sin 60^\circ}$$
$$\cot 60^\circ = \frac{0.5}{0.8660}$$
Answer: \(\cot 60^\circ = 0.5774\)
Example 3: Evaluate \(\csc 30^\circ\) knowing \(\sin 30^\circ = 0.5\).
Given: \(\sin 30^\circ = 0.5\)
$$\csc 30^\circ = \frac{1}{\sin 30^\circ}$$
$$\csc 30^\circ = \frac{1}{0.5}$$
Answer: \(\csc 30^\circ = 2\)
These identities are equations involving trigonometric functions that are true for all values of the variable where the functions are defined. They simplify calculations in quantity surveying tasks such as calculating distances or angles indirectly.
Key identities include:
Pythagorean identity:
$$\sin^2 \theta + \cos^2 \theta = 1$$
Derived identities:
$$1 + \tan^2 \theta = \sec^2 \theta$$
$$1 + \cot^2 \theta = \csc^2 \theta$$
Example 1: Verify the Pythagorean identity for \(\theta = 45^\circ\).
Given: \(\sin 45^\circ = 0.7071\), \(\cos 45^\circ = 0.7071\)
Calculate:
$$\sin^2 45^\circ + \cos^2 45^\circ = (0.7071)^2 + (0.7071)^2$$
$$= 0.5 + 0.5$$
Answer: \(1\) (identity verified)
Example 2: Find \(\sec 60^\circ\) using the identity \(1 + \tan^2 \theta = \sec^2 \theta\).
Given: \(\tan 60^\circ = 1.732\)
Calculate:
$$1 + (1.732)^2 = \sec^2 60^\circ$$
$$1 + 3 = \sec^2 60^\circ$$
$$\sec^2 60^\circ = 4$$
$$\sec 60^\circ = \sqrt{4} = 2$$
Answer: \(\sec 60^\circ = 2\)
Example 3: Evaluate \(\csc 45^\circ\) using \(1 + \cot^2 \theta = \csc^2 \theta\).
Given: \(\cot 45^\circ = 1\)
Calculate:
$$1 + 1^2 = \csc^2 45^\circ$$
$$1 + 1 = 2$$
$$\csc 45^\circ = \sqrt{2} = 1.414$$
Answer: \(\csc 45^\circ = 1.414\)
Hyperbolic functions, analogous to trigonometric functions but based on hyperbolas, appear in advanced structural analysis and curve calculations relevant to quantity surveyors. The basic hyperbolic functions include hyperbolic sine (\(\sinh\)), cosine (\(\cosh\)), and tangent (\(\tanh\)).
Definitions:
$$\sinh x = \frac{e^x, e^{-x}}{2}$$
$$\cosh x = \frac{e^x + e^{-x}}{2}$$
$$\tanh x = \frac{\sinh x}{\cosh x}$$
Example 1: Calculate \(\sinh 1\).
Given: \(x = 1\)
$$\sinh 1 = \frac{e^1 - e^{-1}}{2}$$
$$= \frac{2.7183 - 0.3679}{2}$$
$$= \frac{2.3504}{2} = 1.1752$$
Answer: \(\sinh 1 = 1.1752\)
Example 2: Find \(\cosh 0.5\).
Given: \(x = 0.5\)
$$\cosh 0.5 = \frac{e^{0.5} + e^{-0.5}}{2}$$
$$= \frac{1.6487 + 0.6065}{2}$$
$$= \frac{2.2552}{2} = 1.1276$$
Answer: \(\cosh 0.5 = 1.1276\)
Example 3: Calculate \(\tanh 1\).
Given: \(x = 1\)
$$\tanh 1 = \frac{\sinh 1}{\cosh 1}$$
Calculate \(\cosh 1\):
$$\cosh 1 = \frac{2.7183 + 0.3679}{2} = \frac{3.0862}{2} = 1.5431$$
Then:
$$\tanh 1 = \frac{1.1752}{1.5431} = 0.7616$$
Answer: \(\tanh 1 = 0.7616\)
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Create a free accountThis chapter introduced the fundamental trigonometric ratios and key identities essential for solving problems involving angles and lengths in right-angled triangles. It then covered the main trigonometric rules, including the sine rule and cosine rule, which allow calculation of unknown sides and angles in any triangle. Using these ratios and identities, practical calculations were demonstrated to solve engineering problems related to angles, distances, and gradients. The chapter proceeded to explain hyperbolic functions, their definitions, and properties, highlighting their similarities and differences compared to circular trigonometric functions. Finally, methods for performing calculations using hyperbolic functions were explored, enabling solutions to problems involving hyperbolic angles and engineering applications where these functions arise. Throughout, the emphasis was on applying these mathematical tools to civil engineering contexts.
A surveyor measures a right-angled triangle on a construction site where the angle adjacent to the base is \(30^\circ\) and the base length is 50 m. Calculate the height of the triangle. (2 marks)
Calculate the value of \(\sin 45^\circ \times \cos 30^\circ + \tan 60^\circ\). (2 marks)
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