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Apply trigonometric functions play a vital role in diverse professional fields across Kenya, from healthcare facility layout planning to agricultural land measurement and business logistics. Understanding how to calculate angles and lengths using trigonometric functions enables precise decision-making and problem-solving in everyday tasks. This chapter introduces the application of trigonometric functions to different types of angles, supporting students from all sectors to develop practical mathematical skills essential for their professions.
Angles are fundamental in trigonometry as they define the relationship between the sides of triangles. In professional contexts such as surveying land for a county government office, arranging hospital equipment, or planning hotel room layouts, knowing how to work with angles helps solve real-world problems. This section covers acute, obtuse, reflex, and right angles with their corresponding trigonometric functions.
Acute angles measure less than 90 degrees and are common in many practical situations, such as determining the slope of a wheelchair ramp at a hospital or the angle of sunlight for solar panels in a university. Trigonometric functions for acute angles include sine, cosine, and tangent, which relate the angle to ratios of sides in a right triangle.
The primary formulas for an acute angle \( \theta \) 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 county hospital needs to install a ramp with a slope angle of \( 30^\circ \). If the ramp length (hypotenuse) is 5 m, find the vertical height (opposite side).
Given: \( \theta = 30^\circ \), Hypotenuse \( = 5 \, m \)
$$ \sin 30^\circ = \frac{\text{Opposite}}{5} $$
$$ \text{Opposite} = 5 \times \sin 30^\circ $$
$$ \text{Opposite} = 5 \times 0.5 $$
$$ \text{Opposite} = 2.5 \, m $$
Answer: 2.5 m
Example 2: A retail store places a security camera at a height of 4 m on a wall. The camera is angled downwards at \( 45^\circ \). Find the horizontal distance from the wall to the point the camera covers.
Given: \( \theta = 45^\circ \), Opposite side \( = 4 \, m \)
$$ \tan 45^\circ = \frac{4}{\text{Adjacent}} $$
$$ \text{Adjacent} = \frac{4}{\tan 45^\circ} $$
$$ \text{Adjacent} = \frac{4}{1} $$
$$ \text{Adjacent} = 4 \, m $$
Answer: 4 m
Example 3: A cooperative plans to build a triangular storage shed with a base of 6 m and an angle of \( 60^\circ \) between the base and one side. Find the length of the side adjacent to the base.
Given: Base \( = 6 \, m \), \( \theta = 60^\circ \)
$$ \cos 60^\circ = \frac{6}{\text{Hypotenuse}} $$
$$ \text{Hypotenuse} = \frac{6}{\cos 60^\circ} $$
$$ \text{Hypotenuse} = \frac{6}{0.5} $$
$$ \text{Hypotenuse} = 12 \, m $$
Answer: 12 m
Example 4: A farm water pipe is laid at an angle of \( 25^\circ \) to the horizontal. If the horizontal length is 15 m, find the length of the pipe.
Given: \( \theta = 25^\circ \), Adjacent side \( = 15 \, m \)
$$ \cos 25^\circ = \frac{15}{\text{Hypotenuse}} $$
$$ \text{Hypotenuse} = \frac{15}{\cos 25^\circ} $$
$$ \text{Hypotenuse} = \frac{15}{0.9063} $$
$$ \text{Hypotenuse} \approx 16.56 \, m $$
Answer: 16.56 m
Example 5: A solar panel at a university is tilted at \( 40^\circ \). The height from the base to the top edge is 3 m. Find the length of the panel along the slope.
Given: \( \theta = 40^\circ \), Opposite side \( = 3 \, m \)
$$ \sin 40^\circ = \frac{3}{\text{Hypotenuse}} $$
$$ \text{Hypotenuse} = \frac{3}{\sin 40^\circ} $$
$$ \text{Hypotenuse} = \frac{3}{0.6428} $$
$$ \text{Hypotenuse} \approx 4.67 \, m $$
Answer: 4.67 m
A hotel stairway has a rise of 2.5 m and an angle of elevation of \( 35^\circ \). Find the length of the stairway. (5 marks)
A KRA office places a signboard at 4 m height angled down \( 60^\circ \). Find the horizontal distance covered by the signboard’s shadow. (5 marks)
A cooperative farm has a triangular plot with one angle \( 50^\circ \) and adjacent side 8 m. Find the length of the hypotenuse. (5 marks)
Obtuse angles measure between 90 degrees and 180 degrees and appear in situations like large roof pitches in hotels or the angle between roads in county government planning. Trigonometric functions for obtuse angles use the sine function’s property that \( \sin \theta = \sin (180^\circ - \theta) \), while cosine and tangent may be negative.
The key formula for an obtuse angle \( \theta \) is:
$$ \sin \theta = \sin (180^\circ - \theta) $$
Cosine and tangent are calculated directly but their signs depend on the quadrant.
Example 1: A county office plans a roof with an obtuse angle of \( 120^\circ \). Find \( \sin 120^\circ \).
Given: \( \theta = 120^\circ \)
$$ \sin 120^\circ = \sin (180^\circ - 120^\circ) $$
$$ \sin 120^\circ = \sin 60^\circ $$
$$ \sin 120^\circ = 0.8660 $$
Answer: 0.8660
Example 2: A retail store’s roof pitch forms an angle of \( 135^\circ \). Calculate \( \cos 135^\circ \).
Given: \( \theta = 135^\circ \)
$$ \cos 135^\circ = -\cos (180^\circ - 135^\circ) $$
$$ \cos 135^\circ = -\cos 45^\circ $$
$$ \cos 135^\circ = -0.7071 $$
Answer: -0.7071
Example 3: A hotel conference room wall forms an obtuse angle of \( 150^\circ \) with the adjacent wall. Find \( \tan 150^\circ \).
Given: \( \theta = 150^\circ \)
$$ \tan 150^\circ = -\tan (180^\circ - 150^\circ) $$
$$ \tan 150^\circ = -\tan 30^\circ $$
$$ \tan 150^\circ = -0.5774 $$
Answer: -0.5774
Example 4: A farm plot corner angle is \( 110^\circ \). Find the sine and cosine of this angle.
Given: \( \theta = 110^\circ \)
$$ \sin 110^\circ = \sin (180^\circ - 110^\circ) = \sin 70^\circ = 0.9397 $$
$$ \cos 110^\circ = -\cos (180^\circ - 110^\circ) = -\cos 70^\circ = -0.3420 $$
Answer: \(\sin 110^\circ = 0.9397\), \(\cos 110^\circ = -0.3420\)
Example 5: A university auditorium has an obtuse angle of \( 140^\circ \) between two walls. Calculate \( \tan 140^\circ \).
Given: \( \theta = 140^\circ \)
$$ \tan 140^\circ = -\tan (180^\circ - 140^\circ) = -\tan 40^\circ = -0.8391 $$
Answer: -0.8391
Calculate \( \sin 135^\circ \), \( \cos 135^\circ \), and \( \tan 135^\circ \). (6 marks)
A retail shop has a corner angle of \( 145^\circ \). Find the sine and cosine values. (5 marks)
A county road junction angle is \( 120^\circ \). Find the tangent of the angle. (5 marks)
Reflex angles measure between 180 degrees and 360 degrees, often encountered in navigation, architecture, and land surveying. Trigonometric functions for reflex angles depend on the angle’s reference to 360 degrees and the quadrant in which they lie. The sine function is negative for reflex angles, while cosine and tangent vary.
Key formula:
$$ \sin \theta = - \sin (360^\circ - \theta) $$
Example 1: Calculate \( \sin 210^\circ \).
Given: \( \theta = 210^\circ \)
$$ \sin 210^\circ = -\sin (360^\circ - 210^\circ) = -\sin 150^\circ $$
$$ \sin 150^\circ = 0.5 $$
$$ \sin 210^\circ = -0.5 $$
Answer: -0.5
Example 2: Calculate \( \cos 225^\circ \).
Given: \( \theta = 225^\circ \)
$$ \cos 225^\circ = -\cos (360^\circ - 225^\circ) = -\cos 135^\circ $$
$$ \cos 135^\circ = -0.7071 $$
$$ \cos 225^\circ = -(-0.7071) = 0.7071 $$
Answer: 0.7071
Example 3: Calculate \( \tan 300^\circ \).
Given: \( \theta = 300^\circ \)
$$ \tan 300^\circ = \tan (360^\circ - 300^\circ) = \tan 60^\circ = 1.732 $$
Answer: 1.732
Example 4: Calculate \( \sin 270^\circ \), \( \cos 270^\circ \), and \( \tan 270^\circ \).
Given: \( \theta = 270^\circ \)
$$ \sin 270^\circ = -1 $$
$$ \cos 270^\circ = 0 $$
$$ \tan 270^\circ = \text{undefined} $$
Answer: \( \sin 270^\circ = -1 \), \( \cos 270^\circ = 0 \), \( \tan 270^\circ \) undefined
Example 5: Calculate \( \tan 315^\circ \).
Given: \( \theta = 315^\circ \)
$$ \tan 315^\circ = -\tan (360^\circ - 315^\circ) = -\tan 45^\circ = -1 $$
Answer: -1
Find \( \sin 240^\circ \), \( \cos 240^\circ \), and \( \tan 240^\circ \). (6 marks)
Calculate \( \sin 330^\circ \) and \( \cos 330^\circ \). (5 marks)
A hotel roof angle is \( 300^\circ \). Find the tangent value. (4 marks)
Right angles measure exactly 90 degrees and form the foundation of trigonometric calculations in right-angled triangles. Many practical problems involve right angles, such as measuring heights of buildings, depths of water tanks, or distances in farms. The Pythagorean theorem complements trigonometric functions in these cases.
Key formulas for right triangle with angle \( \theta \):
$$ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} $$
$$ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} $$
$$ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} $$
The Pythagorean theorem:
$$ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 $$
Example 1: A county government building has a triangular window with a right angle. The adjacent side is 3 m and the opposite side is 4 m. Find the hypotenuse.
Given: Adjacent = 3 m, Opposite = 4 m
$$ \text{Hypotenuse}^2 = 3^2 + 4^2 $$
$$ \text{Hypotenuse}^2 = 9 + 16 $$
$$ \text{Hypotenuse}^2 = 25 $$
$$ \text{Hypotenuse} = \sqrt{25} = 5 \, m $$
Answer: 5 m
Example 2: A hospital stair has a height of 1.5 m and a slope length of 5 m. Find the angle of elevation.
Given: Opposite = 1.5 m, Hypotenuse = 5 m
$$ \sin \theta = \frac{1.5}{5} = 0.3 $$
$$ \theta = \sin^{-1} 0.3 = 17.46^\circ $$
Answer: 17.46 degrees
Example 3: A security camera is mounted 6 m high on a wall and angled at \( 60^\circ \) to the wall. Find the horizontal distance it covers.
Given: Height (Opposite) = 6 m, Angle \( \theta = 60^\circ \)
$$ \tan 60^\circ = \frac{6}{\text{Adjacent}} $$
$$ \text{Adjacent} = \frac{6}{\tan 60^\circ} = \frac{6}{1.732} = 3.46 \, m $$
Answer: 3.46 m
Example 4: A farm irrigation pipe is 10 m long and makes a right angle with the ground. The pipe is raised to a height of 8 m. Find the horizontal distance from the base.
Given: Hypotenuse = 10 m, Opposite = 8 m
$$ \text{Adjacent} = \sqrt{10^2 - 8^2} = \sqrt{100 - 64} = \sqrt{36} = 6 \, m $$
Answer: 6 m
Example 5: A cooperative plans to build a ladder leaning against a wall at \( 75^\circ \). The ladder length is 4 m. Find the height reached on the wall.
Given: Hypotenuse = 4 m, Angle \( \theta = 75^\circ \)
$$ \sin 75^\circ = \frac{\text{Opposite}}{4} $$
$$ \text{Opposite} = 4 \times \sin 75^\circ = 4 \times 0.9659 = 3.86 \, m $$
Answer: 3.86 m
A county office flagpole is 7 m high. Find the length of the rope if it forms a right angle with the ground and the base is 3 m from the pole. (5 marks)
A hotel stair rises 2 m with a slope length of 6 m. Find the angle of elevation. (5 marks)
A water tank pipe is at \( 50^\circ \) angle with the ground. If the pipe length is 8 m, find the height reached. (5 marks)
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Create a free accountThis chapter explored the application of trigonometric functions beginning with the classification of angles into acute, obtuse, reflex, and right angles, highlighting their properties in various geometric contexts. It then examined different types of triangles including isosceles, equilateral, right angled, and scalene, establishing the foundation for trigonometric analysis. The chapter introduced the six fundamental trigonometric ratios: sine, cosine, tangent, cosecant, secant, and cotangent, explaining their definitions and relationships within right-angled triangles. Trigonometric identities were then discussed, focusing on their proofs and the essential Pythagorean identities that underpin many trigonometric calculations. Techniques for solving trigonometric equations were presented, enabling the determination of unknown angles or sides in applied problems. Finally, the chapter covered hyperbolic functions such as sinh, cosh, cosech, tanh, and sech, expanding the scope of trigonometric functions to include these important analogues with applications in engineering and physics. Overall, the content provided a comprehensive understanding of trigonometric principles and their practical use in solving mathematical problems.
At a county referral hospital, a staff member measures the angle of elevation to the top of a water tank as \(35^\circ\). Calculate the sine, cosine, and tangent of this angle. (4 marks)
A farmer notices that the shadow of a tree forms an angle of \(120^\circ\) with the ground. Identify the type of angle and calculate its sine value. (4 marks)
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