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Compiled by: Stephen Waweru
Applied trigonometric functions are fundamental in agricultural engineering, especially when dealing with measurements, land surveying, irrigation layout, and structural designs of farm buildings. Understanding angles and their trigonometric properties enables professionals to calculate distances, heights, and slopes accurately, enhancing precision in fieldwork and equipment installation. This chapter focuses on different types of angles and their trigonometric functions, providing worked examples relevant to agricultural engineering tasks in Kenya.
Angles are the foundation of trigonometry and are classified based on their measure in degrees. In agricultural engineering, angles help determine slopes of irrigation canals, roof pitches of farm structures, and alignment of machinery. This section covers acute, obtuse, reflex, and right angles, emphasizing their trigonometric functions and applications.
Acute angles measure between 0° and 90°. Their trigonometric functions, sine, cosine, and tangent, are widely used in agricultural engineering for calculating heights of trees, slopes of furrows, and angles of solar panels. The basic 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 farmer wants to determine the height of a mango tree. Standing 10 m from the tree, the angle of elevation to the top is 30°. Find the height of the tree.
Given: Distance from tree \(= 10\,m\), Angle \(\theta = 30^\circ\)
$$ \tan \theta = \frac{\text{Height}}{\text{Distance}} $$
$$ \tan 30^\circ = \frac{h}{10} $$
$$ 0.577 = \frac{h}{10} $$
$$ h = 0.577 \times 10 $$
Answer: \(h = 5.77\,m\)
Example 2: An irrigation pipe is laid at 15° to the horizontal ground. If the pipe length is 40 m, find the vertical height difference between the pipe ends.
Given: Length \(= 40\,m\), Angle \(\theta = 15^\circ\)
$$ \sin \theta = \frac{\text{Vertical height}}{\text{Length}} $$
$$ \sin 15^\circ = \frac{h}{40} $$
$$ 0.2588 = \frac{h}{40} $$
$$ h = 0.2588 \times 40 $$
Answer: \(h = 10.35\,m\)
Example 3: The angle of depression from the top of a silo to a water tank is 45°. If the silo height is 20 m, find the horizontal distance between the silo and the tank.
Given: Height \(= 20\,m\), Angle \(\theta = 45^\circ\)
$$ \tan \theta = \frac{\text{Height}}{\text{Distance}} $$
$$ \tan 45^\circ = \frac{20}{d} $$
$$ 1 = \frac{20}{d} $$
$$ d = 20 $$
Answer: \(d = 20\,m\)
Example 4: A solar panel is tilted at an angle of 35° from the horizontal. If the length of the panel is 2 m, find the length of its shadow on the ground.
Given: Length \(= 2\,m\), Angle \(\theta = 35^\circ\)
$$ \cos \theta = \frac{\text{Shadow length}}{\text{Length}} $$
$$ \cos 35^\circ = \frac{s}{2} $$
$$ 0.8192 = \frac{s}{2} $$
$$ s = 0.8192 \times 2 $$
Answer: \(s = 1.64\,m\)
Example 5: A ramp for loading farm produce is constructed at an angle of 25°. If the vertical height to be reached is 1.5 m, find the length of the ramp.
Given: Height \(= 1.5\,m\), Angle \(\theta = 25^\circ\)
$$ \sin \theta = \frac{\text{Height}}{\text{Ramp length}} $$
$$ \sin 25^\circ = \frac{1.5}{L} $$
$$ 0.4226 = \frac{1.5}{L} $$
$$ L = \frac{1.5}{0.4226} $$
Answer: \(L = 3.55\,m\)
Obtuse angles measure more than 90° but less than 180°. Although they do not appear directly in right triangles, their trigonometric functions are useful in agricultural engineering when analyzing forces on structures or land plots with wide corner angles. The sine function is positive for obtuse angles, while cosine and tangent are negative.
For an obtuse angle \(\theta\), use the identity:
$$ \sin \theta = \sin (180^\circ - \theta) $$
$$ \cos \theta = -\cos (180^\circ - \theta) $$
$$ \tan \theta = -\tan (180^\circ - \theta) $$
Example 1: Find \(\sin 120^\circ\), \(\cos 120^\circ\), and \(\tan 120^\circ\) for a farm gate opening angle.
Given: \(\theta = 120^\circ\)
$$ \sin 120^\circ = \sin (180^\circ - 120^\circ) = \sin 60^\circ = 0.8660 $$
$$ \cos 120^\circ = -\cos 60^\circ = -0.5 $$
$$ \tan 120^\circ = -\tan 60^\circ = -1.732 $$
Answer: \(\sin 120^\circ = 0.8660\), \(\cos 120^\circ = -0.5\), \(\tan 120^\circ = -1.732\)
Example 2: An agricultural plot has a corner angle of 135°. Calculate the sine, cosine, and tangent values.
Given: \(\theta = 135^\circ\)
$$ \sin 135^\circ = \sin (180^\circ - 135^\circ) = \sin 45^\circ = 0.7071 $$
$$ \cos 135^\circ = -\cos 45^\circ = -0.7071 $$
$$ \tan 135^\circ = -\tan 45^\circ = -1 $$
Answer: \(\sin 135^\circ = 0.7071\), \(\cos 135^\circ = -0.7071\), \(\tan 135^\circ = -1\)
Example 3: Calculate the vertical component of a force of 200 N acting at an angle of 150° to the horizontal on a farm gate.
Given: Force \(F = 200\,N\), Angle \(\theta = 150^\circ\)
$$ F_v = F \sin \theta $$
$$ F_v = 200 \times \sin 150^\circ $$
$$ F_v = 200 \times \sin (180^\circ - 150^\circ) = 200 \times \sin 30^\circ = 200 \times 0.5 $$
Answer: \(F_v = 100\,N\)
Example 4: Find the horizontal component of the same force.
Given: Force \(F = 200\,N\), Angle \(\theta = 150^\circ\)
$$ F_h = F \cos \theta $$
$$ F_h = 200 \times \cos 150^\circ $$
$$ F_h = 200 \times -\cos 30^\circ = 200 \times -0.8660 $$
$$ F_h = -173.2\,N $$
Answer: \(F_h = -173.2\,N\) (Negative indicates direction opposite to reference)
Example 5: A pivot arm rotates through 110°. Find the tangent of this angle.
Given: \(\theta = 110^\circ\)
$$ \tan 110^\circ = -\tan (180^\circ - 110^\circ) = -\tan 70^\circ $$
$$ \tan 70^\circ = 2.747 $$
$$ \tan 110^\circ = -2.747 $$
Answer: \(\tan 110^\circ = -2.747\)
Reflex angles measure between 180° and 360°. These angles are important in agricultural machinery rotation, wind direction analysis, and large-scale land plot angle measurements. The sine function is negative for reflex angles, while cosine and tangent values depend on the specific quadrant.
Use identities:
$$ \sin \theta = -\sin (\theta - 180^\circ) $$
$$ \cos \theta = -\cos (\theta - 180^\circ) $$
$$ \tan \theta = \tan (\theta - 180^\circ) $$
Example 1: Calculate \(\sin 210^\circ\), \(\cos 210^\circ\), and \(\tan 210^\circ\).
Given: \(\theta = 210^\circ\)
$$ \sin 210^\circ = -\sin (210^\circ - 180^\circ) = -\sin 30^\circ = -0.5 $$
$$ \cos 210^\circ = -\cos 30^\circ = -0.8660 $$
$$ \tan 210^\circ = \tan (210^\circ - 180^\circ) = \tan 30^\circ = 0.577 $$
Answer: \(\sin 210^\circ = -0.5\), \(\cos 210^\circ = -0.8660\), \(\tan 210^\circ = 0.577\)
Example 2: A wind vane shows wind direction at 225°. Find sine, cosine, and tangent values.
Given: \(\theta = 225^\circ\)
$$ \sin 225^\circ = -\sin 45^\circ = -0.7071 $$
$$ \cos 225^\circ = -\cos 45^\circ = -0.7071 $$
$$ \tan 225^\circ = \tan 45^\circ = 1 $$
Answer: \(\sin 225^\circ = -0.7071\), \(\cos 225^\circ = -0.7071\), \(\tan 225^\circ = 1\)
Example 3: Calculate the force components of a 150 N force acting at 300° on a farm gate.
Given: Force \(F = 150\,N\), Angle \(\theta = 300^\circ\)
$$ F_v = F \sin 300^\circ = 150 \times -\sin (300^\circ - 180^\circ) = 150 \times -\sin 120^\circ $$
$$ \sin 120^\circ = 0.8660 $$
$$ F_v = 150 \times -0.8660 = -129.9\,N $$
$$ F_h = F \cos 300^\circ = 150 \times -\cos 120^\circ $$
$$ \cos 120^\circ = -0.5 $$
$$ F_h = 150 \times -(-0.5) = 150 \times 0.5 = 75\,N $$
Answer: \(F_v = -129.9\,N\), \(F_h = 75\,N\)
Example 4: Find the tangent of 315°, relevant in pivot irrigation system rotation.
Given: \(\theta = 315^\circ\)
$$ \tan 315^\circ = \tan (315^\circ - 180^\circ) = \tan 135^\circ $$
$$ \tan 135^\circ = -1 $$
Answer: \(\tan 315^\circ = -1\)
Example 5: Determine sine, cosine, and tangent values for 330°.
Given: \(\theta = 330^\circ\)
$$ \sin 330^\circ = -\sin 150^\circ = -0.5 $$
$$ \cos 330^\circ = -\cos 150^\circ = 0.8660 $$
$$ \tan 330^\circ = \tan 150^\circ = -0.577 $$
Answer: \(\sin 330^\circ = -0.5\), \(\cos 330^\circ = 0.8660\), \(\tan 330^\circ = -0.577\)
A right angle measures exactly 90°. It is the basis for defining the trigonometric functions in right-angled triangles, which are pervasive in agricultural engineering for land leveling, roof construction, and machinery alignment. The trigonometric ratios at 90° have special values:
$$ \sin 90^\circ = 1 $$
$$ \cos 90^\circ = 0 $$
$$ \tan 90^\circ = \text{undefined} $$
Example 1: Find \(\sin 90^\circ\), \(\cos 90^\circ\), and \(\tan 90^\circ\).
Given: \(\theta = 90^\circ\)
$$ \sin 90^\circ = 1 $$
$$ \cos 90^\circ = 0 $$
$$ \tan 90^\circ = \text{undefined} $$
Answer: \(\sin 90^\circ = 1\), \(\cos 90^\circ = 0\), \(\tan 90^\circ\) does not exist
Example 2: A ladder is placed vertically against a silo making a 90° angle with the ground. If the ladder length is 6 m, find the height reached.
Given: Ladder length \(= 6\,m\), Angle \(\theta = 90^\circ\)
$$ \sin 90^\circ = \frac{\text{Height}}{6} = 1 $$
$$ \text{Height} = 6 \times 1 = 6\,m $$
Answer: Height = 6 m
Example 3: Calculate the horizontal distance from the base of the ladder to the silo.
Given: Ladder length \(= 6\,m\), Angle \(\theta = 90^\circ\)
$$ \cos 90^\circ = \frac{\text{Distance}}{6} = 0 $$
$$ \text{Distance} = 6 \times 0 = 0\,m $$
Answer: Distance = 0 m (ladder base touches silo base)
Example 4: Find the tangent of 90° for a farm structure analysis.
Given: \(\theta = 90^\circ\)
$$ \tan 90^\circ = \frac{\sin 90^\circ}{\cos 90^\circ} = \frac{1}{0} $$
Undefined due to division by zero.
Answer: \(\tan 90^\circ\) is undefined
Example 5: A ramp with a vertical rise of 3 m forms a right angle with the horizontal. Find the length of the ramp.
Given: Rise \(= 3\,m\), Angle \(\theta = 90^\circ\)
Since the ramp is vertical,
$$ \sin 90^\circ = \frac{3}{L} = 1 $$
$$ L = 3\,m $$
Answer: Ramp length = 3 m
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Create a free accountThis chapter began by exploring different types of angles including acute, obtuse, reflex, and right angles, establishing their definitions and properties. It then examined various triangles such as isosceles, equilateral, right angled, and scalene, focusing on their distinctive characteristics. The core trigonometric ratios were introduced next, covering sine, cosine, tangent, cosecant, secant, and cotangent, with methods to calculate and apply each ratio. The chapter progressed to trigonometric identities, demonstrating proofs and highlighting key Pythagorean identities essential for simplifying expressions. Techniques for solving trigonometric equations were then presented, emphasizing systematic approaches to find angle measures. Finally, the chapter introduced hyperbolic functions including sinh, cosh, cosech, tanh, and sech, explaining their definitions and relevance in applied mathematics. Together, these topics provide a comprehensive foundation for using trigonometry in practical and theoretical contexts.
A farm irrigation pipe is laid at an angle of \(30^\circ\) to the horizontal. Calculate the sine, cosine, and tangent of this angle. (2 marks)
A triangular plot of land has two equal sides measuring 50 m each and an included angle of \(40^\circ\). Calculate the length of the third side using the Law of Cosines. (3 marks)
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