Trigonometry - In the real world

Why do we learn Trig? You'll be amazed how much Trig is used in everyday life. Including computer game design, engineering and robotics.

Egyptian Papyrus Did you know the Egyptians were using Trig almost 5,000 years ago -On this image of the Rhind papyrus, you can see early calculations.

Engineering and Construction

• Building Structures: Trigonometry helps engineers calculate the correct angles and dimensions for structures like bridges, pyramids, skyscrapers, and tunnels. It’s used to ensure stability and balance.

• Roof Design: The slope of a roof is calculated using trigonometric functions to determine the correct angle for drainage.

• Surveying: Land surveyors use trigonometry to measure distances and angles between points on the earth’s surface.

How can I measure the height of the Burj Khalifa? Trig will help with that. See lesson 4 in our trigonometry series

Navigation

• Marine and Air Navigation: Pilots and sailors use trigonometry to determine their position using GPS, stars, and radio signals. The angles and distances between various waypoints are calculated using trigonometric functions.

• Satellite Systems: Trigonometry is essential for calculating the orbits of satellites and determining where they will be at a given time.

Physics

• Wave Motion: Trigonometric functions (sine and cosine) describe wave patterns, such as sound waves, light waves, and electrical waves.

• Projectile Motion: The trajectory of objects (like a thrown ball or a rocket) is modeled using trigonometric equations.

• Forces and Angles: Trigonometry is used in physics to break down forces into components, such as in the analysis of inclined planes or tension in cables.

  
  

Imagine being in a submarine during world war 2. You need to fire a torpedo to sink a ship. What angle do you shoot it at? Calculators were not invented yet!!

Astronomy

• Distance to Stars and Planets: Astronomers use trigonometric parallax to measure the distance to nearby stars. By observing a star from two different positions in Earth's orbit, they use trigonometry to calculate its distance.

• Planetary Orbits: The orbits of planets and celestial bodies are calculated using principles of trigonometry to predict future positions.

  
  

Geology

• Measuring Slopes and Heights: Geologists use trigonometry to calculate the height of mountains, the slope of geological formations, and to measure distances in areas that are difficult to access.

• Seismology: In studying earthquakes, trigonometry helps determine the epicenter of seismic waves.

Architecture

• Designing Complex Structures: Architects use trigonometry to design buildings, especially in irregular shapes, domes, and curves. They rely on it to calculate load-bearing angles and optimise space and materials.

• Interior Design: Trigonometric calculations help in determining the right angles for natural light, optimal placement of furniture, and the geometry of rooms.

Computer Graphics and Animation

• Rendering 3D Models: Trigonometry is used to rotate, scale, and move objects in 3D space in video games and movies. It is crucial in simulating realistic perspectives and shadows.

• Image Processing: Trigonometric functions are applied to algorithms for manipulating and filtering digital images.

Music Theory

• Sound Waves: The frequency and pitch of musical notes are represented by sine and cosine waves, which are fundamental trigonometric functions. Trigonometry helps in understanding harmonics and resonance.

Medical Imaging

• CT Scans and MRIs: Trigonometry is involved in creating cross-sectional images of the body in medical imaging technologies. It helps in reconstructing 3D images from multiple 2D slices.

• Radiation Therapy: In cancer treatments, trigonometric calculations are used to ensure radiation is targeted accurately at tumors without damaging surrounding tissue.

Robotics and Machine Learning

• Motion and Path Planning: Robots use trigonometry to calculate trajectories and navigate through spaces. Algorithms that drive autonomous robots often rely on trigonometric principles to determine angles and distances.

• Computer Vision: Robots and AI systems use trigonometry to process visual information and interpret objects' positions, angles, and distances.

Trigonometry - Moments in History

Below you will find some examples of how useful Trig has been in history.

Egyptian pyramid building - Using Trig for Construction

Khufu’s pyramid is perhaps the most colossal single building ever erected on the planet. Its sides rise at an angle of 51°52′ and are accurately oriented to the four cardinal points of the compass. The Great Pyramid’s core is made of yellowish limestone blocks, the outer casing (now almost completely gone) and the inner passages are of finer light-coloured limestone. The interior burial chamber is built of huge blocks of granite. Approximately 2.3 million blocks of stone were cut, transported, and assembled to create the 5.75-million-ton structure, which is a masterpiece of technical skill and engineering ability.

Stella in Cairo looking for right angled triangles

The ancient Egyptians built pyramids primarily as tombs for their pharaohs and important figures. They believed in an afterlife where the pharaohs would continue to rule, so the pyramids served as eternal resting places. The structures were filled with treasures, food, and other items necessary for the afterlife. Pyramids symbolised the rays of the sun, and their shape was meant to help the soul of the pharaoh ascend to the heavens.

• Great Pyramid's Precision: The Great Pyramid of Giza was originally 146.6 meters tall and is aligned almost perfectly with the cardinal points—north, south, east, and west.

• Oldest of the Seven Wonders: The Great Pyramid is the oldest of the Seven Wonders of the Ancient World and the only one still largely intact.

• Construction: Built around 4,500 years ago, it is believed to have taken 20 years and around 2.3 million stone blocks, each weighing 2.5-15 tons.

• Not Built by Slaves: Contrary to popular belief, it’s now thought that skilled labourers, not slaves, built the pyramids. Workers were housed in nearby villages and likely well-compensated.

• Chambers and Mysteries: Inside the Great Pyramid are multiple chambers, including the King’s and Queen’s chambers. Many mysteries still surround its construction techniques and purpose.

Construction

The Egyptians used a mix of observation, practical knowledge, and early mathematical concepts to build the pyramids. Their understanding of geometry and measurement was advanced. They likely used simple tools like ropes and plumb bobs for alignment and leveling. The seked, an ancient unit of slope, helped them calculate the angles of the pyramid sides. By carefully measuring the base and using astronomical alignments (such as aligning with the cardinal points), they achieved precise results.

Pyramids of Giza, trigonometry is everywhere, Learn trig fast

• Ramp Theories: Several theories explain how the massive stones were lifted, with the most popular suggesting long straight or spiral ramps built around the pyramid.

• Limestone Casing: The pyramids were originally covered in smooth white limestone casing stones, giving them a shiny, polished appearance.

• Labor Organization: Workers were organized into highly skilled teams, with evidence of specialized roles like quarrying, transporting, and stone setting.

• Transportation of Stones: Stones were likely transported using sledges over wetted sand, reducing friction.

https://www.electrummagazine.com/2021/05/trigonometry-and-math-egyptian-style/

The sinking of the Luscitania - Using trig for firing torpedos

The sinking of the Lusitania on May 7, 1915, was a major event during World War I. A British ocean liner, it was torpedoed by a German U-boat submarine off the coast of Ireland, leading to the deaths of 1,198 people, including 128 Americans.

Trig is used to navigate and sink ships

Germany had warned that ships entering British waters could be targeted due to the ongoing war. Despite these warnings, Lusitania continued its voyage from New York to Liverpool. The attack caused a significant diplomatic rift between Germany and neutral countries like the United States, contributing to shifting public opinion toward involvement in the war. The sinking is considered one of the key factors leading to the U.S. joining World War I in 1917

Difficult to calculate. Needs trig

The RMS Lusitania was located about 11 miles off the southern coast of Ireland near the Old Head of Kinsale when it was struck. The German U-boat, U-20, commanded by Kapitänleutnant Walther Schwieger, was patrolling the area off the southern Irish coast. Schwieger spotted the Lusitania and fired a torpedo at the ship, which struck on the starboard side, leading to its rapid sinking​

U-20 surfaced again at 12:45 as visibility was now excellent. At 13:20, something was sighted and Schwieger was summoned to the conning tower: at first it appeared to be several ships because of the number of funnels and masts, but this resolved into one large steamer appearing over the horizon. At 13:25, the submarine submerged to periscope depth of 11 metres and set a course to intercept the liner at her maximum submerged speed of 9 knots. When the ships had closed to 2 nautical miles (3.7 km) Lusitania turned away, Schwieger feared he had lost his target, but she turned again, this time onto a near ideal course to bring her into position for an attack. At 14:10, with the target at 700 m range he ordered one gyroscopic torpedo to be fired, set to run at a depth of three metres

https://en.wikipedia.org/wiki/Sinking_of_the_RMS_Lusitania

Trigonometry - Sums with Stella

Learn trig fast and demystifying Trigonometry – the word itself might sound intimidating, conjuring up memories of complex angles and equations. This article will help unravel the mysteries of trigonometry and guide you through the exciting world of triangles and angles.

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	Video Lesson 1 Introduction and learning how to name sides of a right angled triangle.
	Video Lesson 2 Introducing SOHCAHTOA. Understanding when to use each ratio. Finding the unknown length of a side
	Video Lesson 3 How to find an unknown angle using trigonometry.
	Video Lesson 4 Finding an angle of elevation or depression.
	Video Lesson 5 Bearings

Real World Scenarios Blog post

Moments Trig has been used in History Blog post

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Introduction

Trigonometry is the branch of mathematics that deals with the relationship between the angles and sides of triangles. It has applications in various fields, from engineering and physics to astronomy and even art. By mastering the basics of trigonometry, you can open up a world of possibilities and enhance your problem-solving skills. Learn trigonometry fast.

We use trig when pythagoras theory is not possible.

To find a side we need at least one side and an angle.

To find an angle we need 2 sides

Getting Started: The Trigonometric Ratios (Sin/Cos/ Tan)

To calculate the angles and lengths, you need to first determine which formula or ratio to use. Each one has a different purpose.

Sin, cos, and tan are just ways to describe how big parts of a triangle are compared with each other. They are all about right triangles (triangles with a 90° angle).

These ratios define the relationship between the angles and sides of a right angled triangle. Let's break them down:

• Sine (sin): The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

• Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

• Tangent (tan): The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the adjacent side.

Practical Application: Solving Triangles

Now that you understand the basic trigonometric ratios, let's put them to use in solving triangles. Given an angle and a side length in a right triangle, you can use trigonometry to find missing side lengths or angles. This process, known as trigonometric solving, is a fundamental skill in trigonometry.

Step 1: Identify the sides of a triangle

Step 2: Choose the correct formula (trig function)

Step 3: Solve the equation

Lesson 1: Identify and label the sides of a triangle

Every angle will have a matching ratio and every ratio will have a matching angle

Step 1: A square will indicate the right angle of the triangle.
Step 2: Label the side opposite the right angle (the square) "Hypotenuse"
Step 3: Find the angle you need to calculate and draw it (sometimes this is given a greek symbol such as
Step 4: Label the side opposite to the angle the "Opposite" side
Step 5: Label the remaining side the "Adjacent" side

Lesson 2: Finding which formula to use

Calculate the unknown sides of a right angled triangle

Key Concepts to Remember

• SOH-CAH-TOA : A mnemonic device to remember the trigonometric ratios – Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

  
  

Pronounced: "So-Ka-Toe-Ah" - remember this!!!

• Unit Circle : An essential tool in trigonometry that helps visualise angles, values of trigonometric functions, and trigonometric identities.

Lesson 3: Finding unknown angles of a right angled triangle

Lesson 4: Angles of Elevation and Depression

​Angles of elevation and depression are concepts in trigonometry that describe the angles formed between a horizontal line and an observer's line of sight to an object. The angle of elevation occurs when an observer looks upward at an object above the horizontal line; it's the angle between the horizontal and the line of sight. Conversely, the angle of depression is formed when an observer looks downward at an object below the horizontal line; it's the angle between the horizontal and the downward line of sight.

These angles are commonly used in real-world applications, such as determining the height of a building or the distance of a ship from a lighthouse. By applying trigonometric ratios—sine, cosine, and tangent—one can calculate unknown distances or heights when at least one side length and an angle are known

Lesson 5: Bearings

​In trigonometry, bearings are used to describe the direction of one point relative to another, measured clockwise from the north direction. This system is essential in navigation, surveying, and various real-world applications. Bearings are typically represented as three-figure angles; for example, an angle of 45 degrees is written as 045°. Understanding and calculating bearings allow for precise navigation and positioning, which are crucial in fields such as aviation, marine travel, and land surveying. 

Mastering Trigonometry

To excel in trigonometry, practice is key. Work through problems, delve into real-world applications, and explore the beauty of trigonometry in nature and architecture. Here are some tips to enhance your trigonometry skills:

• Practice : Regular practice with trigonometry problems will help reinforce your understanding of concepts and improve your problem-solving abilities.

• Engage with Resources : Dive into online tutorials, textbooks, and interactive tools to deepen your knowledge of trigonometry.

• Real-World Applications : Explore how trigonometry is used in fields like engineering, architecture, and physics to appreciate its practical significance.

Conclusion

Trigonometry, often feared for its complexity, is a fascinating branch of mathematics with diverse applications in various fields. By understanding the basic concepts of trigonometry, you can unlock a world of possibilities and enrich your mathematical skills. Embrace the challenge, practice diligently, and soon you will find yourself navigating the world of angles and triangles with confidence and ease. Let sums with Stella help you learn trigonometry fast. We call it easy trig!!

Remember to keep exploring, learning, and challenging yourself – trigonometry is just the beginning of a mathematical adventure waiting to unfold.