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Welding professionals in Kenya frequently encounter situations requiring precise angle measurements to ensure accurate weld joints, correct positioning, and structural integrity. Trigonometric functions provide essential mathematical tools for calculating lengths, angles, and positions in welding applications such as beveling, fillet weld sizing, and setting up welding jigs. Mastery of angles and their trigonometric properties enables welders to interpret blueprints and execute welding tasks with high precision and safety. This chapter focuses on applying trigonometric functions to various types of angles relevant to welding contexts.
Trigonometry in welding is vital for determining the relationships between angles and sides of triangles formed by welding components. Understanding how to apply trigonometric functions to different angle types, acute, obtuse, reflex, and right angles, allows welders to calculate dimensions that are not directly measurable. This section breaks down the application of trigonometric functions to these angle categories, with worked examples tailored to welding scenarios in Kenya.
Acute angles are angles less than 90°, commonly encountered in welding when setting bevel angles or fillet welds. Calculating side lengths or heights in triangular weld preparations often involves sine, cosine, and tangent functions of acute angles.
The fundamental trigonometric ratios for an acute angle \( \theta \) in a right triangle are:
$$\sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}}$$
$$\cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}}$$
$$\tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}}$$
Example 1: A welder needs to prepare a bevel with a 30° angle. The hypotenuse length of the bevel cut is 10 cm. Find the height of the bevel cut (opposite side).
Given: \( \theta = 30^\circ \), hypotenuse \( h = 10 \) cm
$$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$
$$\sin 30^\circ = \frac{\text{Opposite}}{10}$$
$$0.5 = \frac{\text{Opposite}}{10}$$
$$\text{Opposite} = 0.5 \times 10$$
$$\text{Opposite} = 5 \text{ cm}$$
Answer: 5 cm
Example 2: A welding jig forms a right triangle with an acute angle of 45°. The adjacent side length is 7 cm. Calculate the hypotenuse.
Given: \( \theta = 45^\circ \), adjacent \( a = 7 \) cm
$$\cos \theta = \frac{a}{\text{Hypotenuse}}$$
$$\cos 45^\circ = \frac{7}{\text{Hypotenuse}}$$
$$\frac{\sqrt{2}}{2} = \frac{7}{\text{Hypotenuse}}$$
$$\text{Hypotenuse} = \frac{7}{\sqrt{2}/2}$$
$$\text{Hypotenuse} = 7 \times \frac{2}{\sqrt{2}} = \frac{14}{\sqrt{2}}$$
$$\text{Hypotenuse} = \frac{14 \sqrt{2}}{2} = 7 \sqrt{2} \approx 9.9 \text{ cm}$$
Answer: 9.9 cm
Example 3: A fillet weld forms an angle of 60° with the base plate. If the length of the adjacent side (base plate) is 12 cm, find the length of the opposite side (vertical leg of the weld).
Given: \( \theta = 60^\circ \), adjacent \( a = 12 \) cm
$$\tan \theta = \frac{\text{Opposite}}{a}$$
$$\tan 60^\circ = \frac{\text{Opposite}}{12}$$
$$\sqrt{3} = \frac{\text{Opposite}}{12}$$
$$\text{Opposite} = 12 \times \sqrt{3} = 12 \times 1.732 = 20.78 \text{ cm}$$
Answer: 20.78 cm
Example 4: A welding angle plate forms a 15° acute angle. If the opposite side length is 3 cm, find the hypotenuse.
Given: \( \theta = 15^\circ \), opposite \( o = 3 \) cm
$$\sin \theta = \frac{o}{\text{Hypotenuse}}$$
$$\sin 15^\circ = \frac{3}{\text{Hypotenuse}}$$
$$0.2588 = \frac{3}{\text{Hypotenuse}}$$
$$\text{Hypotenuse} = \frac{3}{0.2588} = 11.59 \text{ cm}$$
Answer: 11.59 cm
Example 5: A metal plate is cut forming a 75° acute angle with the horizontal. The hypotenuse is 20 cm. Find the length of the adjacent side.
Given: \( \theta = 75^\circ \), hypotenuse \( h = 20 \) cm
$$\cos \theta = \frac{\text{Adjacent}}{h}$$
$$\cos 75^\circ = \frac{\text{Adjacent}}{20}$$
$$0.2588 = \frac{\text{Adjacent}}{20}$$
$$\text{Adjacent} = 20 \times 0.2588 = 5.176 \text{ cm}$$
Answer: 5.18 cm
Obtuse angles are angles greater than 90° but less than 180°. Although the basic sine, cosine, and tangent functions are defined for obtuse angles, welders mostly use their properties in combination with supplementary angles to calculate lengths and positions in welding fixtures and joints.
The reference angle \( \alpha \) for an obtuse angle \( \theta \) is:
$$\alpha = 180^\circ - \theta$$
The trigonometric functions for \( \theta \) relate to those of \( \alpha \) as:
$$\sin \theta = \sin \alpha$$
$$\cos \theta = -\cos \alpha$$
$$\tan \theta = -\tan \alpha$$
Example 1: A welding fixture forms an angle of 120°. Find \(\sin 120^\circ\), \(\cos 120^\circ\), and \(\tan 120^\circ\).
Given: \( \theta = 120^\circ \)
Calculate reference angle:
$$\alpha = 180^\circ - 120^\circ = 60^\circ$$
Calculate sine:
$$\sin 120^\circ = \sin 60^\circ = 0.866$$
Calculate cosine:
$$\cos 120^\circ = -\cos 60^\circ = -0.5$$
Calculate tangent:
$$\tan 120^\circ = -\tan 60^\circ = -\sqrt{3} = -1.732$$
Answer:
\(\sin 120^\circ = 0.866\), \(\cos 120^\circ = -0.5\), \(\tan 120^\circ = -1.732\)
Example 2: A weld joint angle is 135°. Calculate the length of the adjacent side if the hypotenuse is 14 cm.
Given: \( \theta = 135^\circ \), hypotenuse \( h = 14 \) cm
Reference angle:
$$\alpha = 180^\circ - 135^\circ = 45^\circ$$
Use cosine relation:
$$\cos 135^\circ = -\cos 45^\circ = -\frac{\sqrt{2}}{2} = -0.707$$
Calculate adjacent:
$$\cos \theta = \frac{\text{Adjacent}}{h}$$
$$-0.707 = \frac{\text{Adjacent}}{14}$$
$$\text{Adjacent} = -0.707 \times 14 = -9.9 \text{ cm}$$
Answer: -9.9 cm (direction indicates opposite side of reference)
Example 3: A welding jig forms an obtuse angle of 110°. The opposite side length is 8 cm. Find the adjacent side length.
Given: \( \theta = 110^\circ \), opposite \( o = 8 \) cm
Reference angle:
$$\alpha = 180^\circ - 110^\circ = 70^\circ$$
Calculate tangent:
$$\tan 110^\circ = -\tan 70^\circ = -2.747$$
Use tangent definition:
$$\tan \theta = \frac{o}{a}$$
$$-2.747 = \frac{8}{a}$$
$$a = \frac{8}{-2.747} = -2.91 \text{ cm}$$
Answer: -2.91 cm
Example 4: Calculate the sine of 150° and use it to find the height of a weld bevel if the hypotenuse is 16 cm.
Given: \( \theta = 150^\circ \), hypotenuse \( h = 16 \) cm
Reference angle:
$$\alpha = 180^\circ - 150^\circ = 30^\circ$$
Calculate sine:
$$\sin 150^\circ = \sin 30^\circ = 0.5$$
Calculate height:
$$\sin \theta = \frac{\text{Height}}{h}$$
$$0.5 = \frac{\text{Height}}{16}$$
$$\text{Height} = 0.5 \times 16 = 8 \text{ cm}$$
Answer: 8 cm
Example 5: A welding plate forms an angle of 140°. If the adjacent side length is 7 cm, find the hypotenuse.
Given: \( \theta = 140^\circ \), adjacent \( a = 7 \) cm
Reference angle:
$$\alpha = 180^\circ - 140^\circ = 40^\circ$$
Calculate cosine:
$$\cos 140^\circ = -\cos 40^\circ = -0.766$$
Use cosine definition:
$$\cos \theta = \frac{a}{h}$$
$$-0.766 = \frac{7}{h}$$
$$h = \frac{7}{-0.766} = -9.14 \text{ cm}$$
Answer: -9.14 cm
Reflex angles exceed 180° but are less than 360°. In welding, reflex angles rarely appear directly but understanding them helps in layout and fabrication of complex joints where angles wrap around corners or components.
The reference angle \( \beta \) is:
$$\beta = 360^\circ - \theta$$
Trigonometric functions for reflex angles relate to those of \( \beta \) as:
$$\sin \theta = -\sin \beta$$
$$\cos \theta = \cos \beta$$
$$\tan \theta = -\tan \beta$$
Example 1: Calculate \(\sin 210^\circ\), \(\cos 210^\circ\), and \(\tan 210^\circ\).
Given: \( \theta = 210^\circ \)
Calculate reference angle:
$$\beta = 360^\circ - 210^\circ = 150^\circ$$
Calculate sine:
$$\sin 210^\circ = -\sin 150^\circ = -0.5$$
Calculate cosine:
$$\cos 210^\circ = \cos 150^\circ = -0.866$$
Calculate tangent:
$$\tan 210^\circ = -\tan 150^\circ = -(-0.577) = 0.577$$
Answer:
\(\sin 210^\circ = -0.5\), \(\cos 210^\circ = -0.866\), \(\tan 210^\circ = 0.577\)
Example 2: A welding plate forms a reflex angle of 225°. Calculate the length of the adjacent side if the hypotenuse is 20 cm.
Given: \( \theta = 225^\circ \), hypotenuse \( h = 20 \) cm
Reference angle:
$$\beta = 360^\circ - 225^\circ = 135^\circ$$
Calculate cosine:
$$\cos 225^\circ = \cos 135^\circ = -0.707$$
Calculate adjacent:
$$\cos \theta = \frac{a}{h}$$
$$-0.707 = \frac{a}{20}$$
$$a = -0.707 \times 20 = -14.14 \text{ cm}$$
Answer: -14.14 cm
Example 3: Find the sine of 300° and calculate the height of a weld bevel with a hypotenuse of 18 cm.
Given: \( \theta = 300^\circ \), hypotenuse \( h = 18 \) cm
Reference angle:
$$\beta = 360^\circ - 300^\circ = 60^\circ$$
Calculate sine:
$$\sin 300^\circ = -\sin 60^\circ = -0.866$$
Calculate height:
$$\sin \theta = \frac{\text{Height}}{h}$$
$$-0.866 = \frac{\text{Height}}{18}$$
$$\text{Height} = -0.866 \times 18 = -15.59 \text{ cm}$$
Answer: -15.59 cm
Example 4: A joint angle is 315°. If the adjacent side length is 10 cm, find the hypotenuse.
Given: \( \theta = 315^\circ \), adjacent \( a = 10 \) cm
Reference angle:
$$\beta = 360^\circ - 315^\circ = 45^\circ$$
Calculate cosine:
$$\cos 315^\circ = \cos 45^\circ = 0.707$$
Calculate hypotenuse:
$$\cos \theta = \frac{a}{h}$$
$$0.707 = \frac{10}{h}$$
$$h = \frac{10}{0.707} = 14.14 \text{ cm}$$
Answer: 14.14 cm
Example 5: Calculate \(\tan 330^\circ\) and find the adjacent side if the opposite side is 6 cm.
Given: \( \theta = 330^\circ \), opposite \( o = 6 \) cm
Reference angle:
$$\beta = 360^\circ - 330^\circ = 30^\circ$$
Calculate tangent:
$$\tan 330^\circ = -\tan 30^\circ = -0.577$$
Calculate adjacent:
$$\tan \theta = \frac{o}{a}$$
$$-0.577 = \frac{6}{a}$$
$$a = \frac{6}{-0.577} = -10.4 \text{ cm}$$
Answer: -10.4 cm
For a right triangle with angle \( \theta \):
$$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$
$$\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$
$$\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$$
Special values for \( \theta = 90^\circ \):
$$\sin 90^\circ = 1$$
$$\cos 90^\circ = 0$$
$$\tan 90^\circ \text{ is undefined}$$
Example 1: A welding plate forms a right angle at one corner. If the opposite side is 9 cm and the adjacent side is 12 cm, find the hypotenuse.
Given: \( \theta = 90^\circ \), opposite \( o = 9 \) cm, adjacent \( a = 12 \) cm
Use Pythagoras theorem:
$$h = \sqrt{o^2 + a^2}$$
$$h = \sqrt{9^2 + 12^2}$$
$$h = \sqrt{81 + 144}$$
$$h = \sqrt{225} = 15 \text{ cm}$$
Answer: 15 cm
Example 2: Calculate \(\sin 90^\circ\), \(\cos 90^\circ\), and \(\tan 90^\circ\).
Given: \( \theta = 90^\circ \)
$$\sin 90^\circ = 1$$
$$\cos 90^\circ = 0$$
$$\tan 90^\circ \text{ is undefined}$$
Answer:
\(\sin 90^\circ = 1\), \(\cos 90^\circ = 0\), \(\tan 90^\circ\) undefined
Example 3: A welding joint forms a right angle. If the hypotenuse is 13 cm and the adjacent side is 5 cm, find the opposite side.
Given: \( \theta = 90^\circ \), hypotenuse \( h = 13 \) cm, adjacent \( a = 5 \) cm
Use Pythagoras theorem:
$$o = \sqrt{h^2 - a^2}$$
$$o = \sqrt{13^2 - 5^2}$$
$$o = \sqrt{169 - 25}$$
$$o = \sqrt{144} = 12 \text{ cm}$$
Answer: 12 cm
Example 4: Find the angle \( \theta \) if the opposite side is 8 cm and the adjacent side is 15 cm in a right triangle.
Given: \( o = 8 \) cm, \( a = 15 \) cm
Use tangent:
$$\tan \theta = \frac{o}{a} = \frac{8}{15} = 0.533$$
Calculate angle:
$$\theta = \tan^{-1}(0.533) = 28.07^\circ$$
Answer: 28.07°
Example 5: A fillet weld forms a right angle between two plates. If the hypotenuse is 10 cm and the opposite side is 6 cm, find the adjacent side.
Given: \( h = 10 \) cm, \( o = 6 \) cm
Use Pythagoras theorem:
$$a = \sqrt{h^2 - o^2}$$
$$a = \sqrt{10^2 - 6^2}$$
$$a = \sqrt{100 - 36}$$
$$a = \sqrt{64} = 8 \text{ cm}$$
Answer: 8 cm
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Create a free accountThis chapter began by exploring various types of angles including acute, obtuse, reflex, and right angles, establishing their definitions and properties relevant to trigonometry. It then examined different triangles such as isosceles, equilateral, right-angled, and scalene, highlighting their characteristics and significance in trigonometric applications. The focus shifted to trigonometric ratios, detailing sine, cosine, tangent, cosecant, secant, and cotangent functions and how they relate to angle measurements in triangles. Key trigonometric identities were introduced and proved, with particular emphasis on the Pythagorean identities that form the foundation for solving trigonometric problems. The chapter also covered methods to solve trigonometric equations, equipping learners with techniques to find unknown angles or side lengths. Finally, hyperbolic functions including sinh, cosh, cosech, tanh, and sech were presented, expanding the scope of trigonometric functions to hyperbolic contexts and their practical applications. Throughout, the chapter integrated theory with progressive calculations to ensure mastery of trigonometric concepts.
A welding joint forms an acute angle of \(30^\circ\). Calculate the sine, cosine, and tangent of this angle. (2 marks)
A steel plate is cut to form an obtuse angle of \(120^\circ\). Find the sine and cosine of this angle. (2 marks)
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