Life and trigonometry are both the same. They both have formulas to be solved, but where which formula is applied; that’s difficult to understand- Anonymous.

This a very impactful trigonometric quote that captures its essence just right. No wonder why kids shy away from studying trigonometric functions or leave it to fate or miracles, maybe. Trigonometry is a super complicated mathematical concept that challenges the children who use it at every stage and makes its application tricky.

Sine and cosine are some of the most important trigonometric functions widely used in many areas of study; hence understanding and studying the concept is highly important. Without the graphical representation of sine and cosine, various real-life processes would only be a sequence of chaotic mathematical equations that would take hours to solve.

So, what are sine and cosine? Imagine a right-angle triangle. The angle opposite to the 90 degrees is the θ. The ratio of the sides of the opposite side of the θ is sine, and the ratio of adjacent sides of θ is cosine.

With this basic understanding of the concept, let’s cover some real-life applications of sine and cosine graphs.

**Real-life application of sine and cosine graphs **

Sine and cosine graphs are of high importance from both science and mathematics perspectives. The following mentioned applications will help in assessing the broad range where sine and cosine are applied. The real-life applications of sine and cosine graphs are-

**1. Developing music of different frequencies **

Trigonometry is applied to various surprising areas, such as the musical industry. Developing and refining music is based on trigonometric functions, such as sine and cosine. The music frequencies vary with each tone, and its graphical representation of sine and cosine is useful in assessing the developed music.

Controlling and adjusting the vibrations, frequencies, pitch, and other musical elements is done through computer software that utilizes various trigonometric functions such as sine and cosine.

Figure showing the sine waves while recording the music

**2. Constructing magnificent architectural pieces **

The elegant curves and arcs of bridges, the domes of large monuments, or even the foundation for the surface of a regular building, sine and cosine of mathematics are widely used in construction. Graphical representation of sine and cosine is useful in making precise measurements of buildings, identifying angles, calculating the height, distance, and width of the buildings, making walls parallel and perpendicular to each other, setting up inclined roofs, and so on.

Trigonometry, especially sine, and cosine, is used to construct every building. Constructing an architectural piece without employing trigonometric functions is challenging and risky.

Image representing how trigonometry can be used to find the distance and height of a building.

**3. Aircraft functioning **

Aircraft designers and pilots are no exception when using trigonometric functions in real life. From designing, maintaining, and constructing to flying aircraft, sine and cosine plays a great role. Graphical representation of sine and cosine helps to ensure that the aircraft’s wings are accurately positioned and in sync with the aircraft’s design and weight.

Additionally, taking off and landing aircraft requires basic trigonometry on the pilot’s part. Landing on any part of the ground could be risky without calculating their speed, distance, and height. Even pilots need to perform trigonometry when in the air to ensure that they are moving in the right direction and are in sync with the wind’s direction.

Trigonometric functions during the landing of an airplane

**4. GPS or Global Positioning System**

GPS, or Global Positioning System, is a high-tech tool used to identify the location and position of objects. GPS can easily track where an object, such as a car, is and in which direction. GPS works on a triangulation system that uses the object’s position concerning the earth to assess the direction and locations.

GPS is in-built to use sine and cosine functions and identify objects’ location concerning the earth’s direction. GPS also helps identify your current location and the directions you are moving in.

Figure showing how trigonometry helps in locating objects with reference to earth

**5. Cartography **

Cartography or map-making is another major area where the application of sine and cosine is highly important. While making the maps, directions, distances, angles, etc., need to be represented accurately. Trigonometric functions not only help in map making but also assist in reading the map and navigating the distance.

Applying the basic principles of sine and cosine functions ensures that the map is accurately developed and all the locations are precisely marked.

**6. Marine biology**

Marine biology is a fascinating field of study, and trigonometry is a highly important aspect of the field. Trigonometry is not directly used in marine biology, but various processes depend on the use of sine and cosine. For instance, how much sunlight a water plant such as algae requires to grow is a question from core biology, but assessing the rate and direction of sunlight using sine and cosine is the medium to find the answer to the question.

Additionally, sine and cosine help to identify the size of various marine species, which is otherwise impossible. Marine biologist uses the graphical representation of sine and cosine using distance and height to identify the size of the different marine creatures. Marine vessels like submarines are designed and constructed using trigonometric functions.

Figure showing the use of trigonometric functions in calculating the depth of the ocean and distances of sea animals

**7. Optics **

Trigonometry functions are applied to various real-life scenarios, and the field of optics is no exception. The mechanism through which light travels, the direction in which light travels, speed, direction, etc., all are calculated and studied using trigonometric functions and other advanced mathematical concepts.

The process of reflection and refraction are also identified and studied using sine and cosine functions. The angle from which light will be reflected or refracted, the direction it will move, and the speed are all studied through the graphical representation of sine and cosine.

Use of sine and cosine in reflection and refraction

**8. Assessing the heights of the tides**

Oceanography is another important field that relies heavily on trigonometric functions, such as sine and cosine. The graphical representation of sine and cosine on water waves or tides helps to assess the length of the tides and if the tides will reach the shore. Sine and cosine also help to assess if tides are near the shore and how much destruction can be anticipated in their wake.

Trigonometric functions are also useful to identify the periodic function of the waves, that is, the possibility of occurring the waves again and the length of the waves if it comes again.

Sine waves showing the height of the waves

**9. Graphics for video games and movies**

Have you ever played or seen someone else playing video games where the main character jumps from a very high surface, gracefully lands on the ground, and is always in the same position? Well, that’s sine and cosine in video games. The same process applies to movies, where one object comes flying and lands on a surface without hurting anyone.

These animations result from applying sine, cosine, and other trigonometric functions. Developers perform trigonometric functions and calculate the objects’ height, distance, and speed and then develop the animations. The landing of objects is not incidental; rather, a predetermined aspect of games and movies carried out through a graphical representation of sine and cosine.

Image representing the use of sine and cosine during a collision in movies or video games

**10. Sinusoidal waves **

The study of sinusoidal waves, which are radio waves used in communication, also requires the application of trigonometric functions. Sinusoidal waves are based on periodic functions and are similar over time. Sinusoidal waves carry messages from one channel to another. Both sine and cosine are types of sinusoidal waves or communication signals, where cosine signals are much advanced by 90 degrees.

The graphical representation of sine and cosine waves helps to identify the behavior of the signals at different times. Other electrical waves and signals also require trigonometric functions like sine and cosine for their analysis.

Sine waves showing the frequency of radio waves over time

**11. Length of the day or weather**

Did you know that not all days are the same in length? Neither all seasons occur for the same duration. Yes, the length of the day and weather varies based on various factors, such as latitudes, movement of other celestial bodies such as the sun and moon, etc. Some days are longer than others in the year, and some are short.

The study of increment and decrement in the length of the days and weather is done through sine, cosine, and other trigonometric functions. The movement of celestial bodies, distance of equator and latitudes, and other factors, combined, determine the length of the days and weather, and sine and cosine functions are a great way to identify them.

**12. Criminology **

Criminology is a branch that deals with crime scenes and criminals. Sine and cosine play a key role in various aspects of criminology. For instance, assessing the reasons and re-creating the accident scenes are done through trigonometric functions such as sine and cosine. Various crime scenes are solved using sine and cosine representation, such as the direction from where an attack was initiated or a bullet was fired.

Trigonometry also helps assess if a certain fall from some building was intended or forceful or from which floor of a building the objects were thrown.

Trigonometric function representing the bullet trajectory

**13. Navigating distances **

Compasses alone are tricky when finding the direction in huge forests or landscapes. Basic knowledge of sine and cosine is highly important for navigation. Navigation becomes easy when trigonometric functions are applied to the position of the sun, horizon, or other celestial bodies along with the distance represented by a map or compass.

Finding shore to the sea, identifying the straight paths in huge landscapes, and finding directions, such as the northeast, southwest, etc., all require a basic graphical representation of sine and cosine.

Navigating the path to shore using horizon and sun position

**14. Archaeology **

The history of human civilization is equal parts mystery and fascination. The remarkable buildings, precise tools and weapons, underground tunnels, water systems, emergency exits, routes, etc., all are beautifully designed and represent the advanced knowledge of early humans. Although, they also represent the great use of trigonometric functions used to design and construct whole civilizations.

Archaeologists use graphical representations of sine and cosine to find the whole areas of underground civilizations, and the evidence of smart use of trigonometry is also evident in those found civilizations.

Representation of trigonometry in early monuments

**Wrapping up **

Teaching trigonometry to young minds can sometimes be very challenging. Simply introducing the concept can leave them confused and uninterested in the topic. Hence, teachers can use some fun and engaging trigonometric function activities, games, real-life examples, and applications to make the concepts easier and more interesting. Even some trigonometric jokes can help children understand the trigonometric concepts they previously struggled with.

Real-life applications of trigonometric functions such as sine and cosine can help children better relate to and understand the concepts thoroughly. Active support from educators and parents can further strengthen their understanding of sine and cosine functions.

I am Sehjal Goel, a psychology student, and a writer. I am currently pursuing my Masters’s from Banaras Hindu University, Varanasi. Child psychology has always fascinated me and I have a deep interest in learning about disabilities in children and spreading awareness regarding the same. My other areas of interest are neuropsychology and cognitive psychology. Connect me on Linkedin