Trigonometry, defined as the branch of mathematics that deals with the relationships between the angles and sides of triangles, especially right-angled triangles, is a key topic in the GCSE Maths syllabus. It helps to explain the distances and angles in the world around us. From the ancient architects who built the pyramids to engineers designing skyscrapers, trigonometry is one of the most indispensable tool in maths.

The following are commonly called trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). These ratios let us understand the properties of right-angled triangles, enabling advancements in science, engineering, and even navigation.

## Basic Trigonometric Ratios

The study of trigonometry dates back to ancient civilizations, which used the principles of trigonometry for astronomy and land surveying. The word 'trigonometry' comes from the Greek 'trigonon' (triangle) and 'metron' (measure). Early trigonometry was focussed on the computation of the lengths and angles of triangles. For a deeper look into the history of trigonometry and its development through the ages, look at the History of Trigonometry by MacTutor.

### Sine, Cosine and Tangent

Trigonometric ratios are the ratios of the sides of a triangle in relation to one of its acute angles (less than a 90 degrees angle).

There are six trigonometric ratios. The three common trigonometric ratios we focus on are listed below:

**Sine (sin)**- defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.**Cosine (cos)**- the ratio of the length of the adjacent side to the hypotenuse.**Tangent (tan)**- defined as the ratio of the opposite side to the adjacent side.

These ratios are fundamental to the theoretical study of trigonometry and its practical applications. Whether it's in navigation, architecture, or even in the creation of video games, understanding these ratios is essential.

### Soh-cah-toa

We can remember this by using the trigonometric mnemonic SOH-CAH-TOA. This stands for Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent. It's a catchy way to ensure you never forget how to calculate each ratio.

This mnemonic is not just a memory aid but a representation of the profound relationships that exist within right-angled triangles. It shows that complex relationships in trigonometry can be understood in terms of simple ratios of the sides of the triangle.

With just three ratios, trigonometric angles and distances open up to you. From calculating the distance to a star, to working out the size of an area of land, these ratios are crucial.

## Applications of Trigonometric Ratios

The practical uses of trigonometry are numerous and varied. From the fields of engineering and physics to the realms of space exploration or computer programming, trigonometry plays a pivotal role in various scientific advancements.

Imagine standing at the base of The Shard, looking up towards its peak and wondering about how angle is defined of elevation from where you stand to the top of the building. Or picture yourself trying to calculate the height of a tree without climbing it. These are examples of real-world problems that trigonometric ratios can help you solve.

**Navigation and Geography**

Trigonometry plays a crucial role in navigation, helping sailors and pilots find their positions and chart their courses. Trigonometric ratios are used to calculate distances across water or land by navigators, ensuring safe and efficient travel. For example, by measuring the angle between the horizon and the sun or a star, navigators can use trigonometric ratios to calculate their latitude.

**Architecture and Engineering**

Astronomers use trigonometry to calculate the distances to stars and galaxies. The parallax method, which involves observing an astronomical object from two different points on Earth's orbit around the sun, uses trigonometric ratios to measure vast distances in space. This helps us understand the size and scale of the universe.

**Astronomy**

Astronomers use trigonometry to calculate the distances to stars and galaxies. The parallax method, which involves observing an astronomical object from two different points on Earth's orbit around the sun, uses trigonometric ratios to measure vast distances in space. This helps us understand the size and scale of the universe.

**Art and Music**

Even in art and music, trigonometric ratios find their application. Artists use perspective drawing to create depth in their paintings, requiring an understanding of angles and proportions. In music, the theory behind sound waves and harmonics involves trigonometry, allowing for the creation of different tones and sounds.

### 1

You're designing a ramp for wheelchair access to a building. The height of the entrance is 1.5 meters off the ground and the ramp must have a maximum incline of 5 degrees for safety. What is the minimum length of the ramp?

To further explore some of the exciting ways trigonometry is applied in real-world scenarios, such as space travel and satellite technology, consider the article on real-world applications of trigonometry by Byjus.

## The formula for Trigonometric Ratios

A fundamental skill in trigonometry, and indeed in GCSE Maths, is the ability to calculate trigonometric ratios. These calculations enable us to solve many problems, from the heights of mountains to the angles of sunlight.

**Step-by-step guide:**

**Identify the sides in relation to the angle you're interested in**, excluding the 90-degree angle. The terms you'll use are:

**Opposite**: The side opposite the angle.**Adjacent**: The side adjacent to the angle, forming a right angle with the hypotenuse.**Hypotenuse**: The longest side, lying opposite the right angle, abbreviated as 'hyp'.

**Select the appropriate ratio**based on the sides and angle involved. Remember the mnemonic SOH-CAH-TOA:

**Sine (sin)**involves the opposite side and the hypotenuse.**Cosine (cos)**involves the adjacent side and the hypotenuse.**Tangent (tan)**involves the opposite side and the adjacent side.

**Substitute the known values into the chosen formula**. If you're solving for a side length, rearrange the formula to isolate the unknown side.

**Calculate using a calculator**for the ratio or the angle, depending on what's known. Ensure your calculator is in the correct mode (degrees for most GCSE problems).

**Example in action**

Imagine you're at a park and there's a tall statue. You want to find out how tall it is without measuring directly. Standing 50 meters away (adjacent side), you measure the angle of elevation to the top of the statue to be 30 degrees.

To find the height of the statue (opposite side), we can use the tangent ratio, since we have the angle and the adjacent side:

tan(30°) = Height of the statue / 50

By rearranging the formula to solve for the height and plugging in the values, we find the height of the statue. This process demonstrates the tangible application of trigonometric ratios in solving real-world problems.

## Trigonometric tables and identities

Mastering trigonometry involves understanding trigonometric tables and identities. These tools simplify calculations and provide a deeper insight into the functions of trig.

**Trigonometric ratios tables**

Trigonometric tables list the values of sine, cosine, and tangent functions for specific angles. These tables were indispensable for solving trigonometric problems before calculators existed. Today, they serve as a quick reference and a means to understand the symmetry and patterns within trigonometric functions.

**Understanding the table**: Trigonometric tables usually cover standard angles such as 0°, 30°, 45°, 60°, and 90°. For example, the sine of 30° is always 0.5, regardless of the triangle's size. This consistency provides a foundation for solving problems and understanding the behaviour of trigonometric functions across different quadrants.

**Application**: When calculating the dimensions of a structure or the angle of elevation to a star, knowing the exact trigonometric values can simplify calculations significantly. For instance, if you know you need to find the cosine of 45°, a quick reference to the table tells you the value is 1/the square root of 2 or approximately 0.707.

## Trigonometric identities

Trigonometric identities are equations that express relationships between trigonometric functions. These identities include Pythagorus and are not merely curiosities; they are powerful tools for simplifying expressions, solving equations and proving theorems.

**Key identities**: Among the most crucial identities are the Pythagorean identities, which relate the squares of the sine and cosine of an angle to 1. For example, sin

^{2}+ cos

^{2}= 1. This identity is fundamental in transitioning between sine and cosine functions and in solving trigonometric equations.

**Angle sum and difference identities**: These identities provide formulas for the sine, cosine and tangent of the sum or difference of two angles. They are particularly useful in solving problems involving angles not found in standard trigonometric tables.

**Double angle and half angle identities**: These allow the calculation of trigonometric values for twice an angle or half an angle, facilitating the solving of more complex problems.

## Practice questions

Through practical exercises, we can apply trigonometric ratios, tables and identities to solve problems, enhancing our skills and confidence. Below is an exercise followed by a quiz question to test your understanding of trigonometric ratios and identities.

**Exercise: Calculating heights and distances**

Imagine you are trying to determine the height of a building without direct measurement. You measure a shadow cast by the building and find it to be 20 meters long. You also measure the angle of elevation from the tip of the shadow to the top of the building, which is 45°.

Using trigonometric ratios, calculate the height of the building.

**Steps**:

- Sketch the scenario as a right-angled triangle, identifying the angle, opposite side (building height) and adjacent side (shadow length).
- Apply the tangent ratio, since you have the angle and the adjacent side and you're solving for the opposite side: tan(45°) = Height of the building / 20
- Solve for the height. Given tan(45°)=1, it implies that the height of the building is equal to the length of the shadow, hence 20 meters.

### 2

Which of the following is a correct representation of the Pythagorean identity?

## Overcoming challenges with Trigonometric Ratios

Understanding how to approach and solve trigonometric problems can make this area of mathematics much more accessible. Here, we explore some of the common challenges students face with trigonometric ratios and offer strategies to overcome them.

**Memorising Trigonometric Ratios**

One of the first hurdles is memorising the sine, cosine and tangent ratios, along with their reciprocal functions, cosecant, secant and cotangent.

Rather than rote memorisation, understand the concept of SOH-CAH-TOA as it relates to the sides of a right-angled triangle. Visual aids and mnemonic devices can be incredibly helpful. Additionally, regularly practising with these ratios in different problems can help cement your understanding.

**Applying Trigonometric Identities**

Trigonometric identities, especially the Pythagorean identity, are crucial but can be tricky to apply correctly.

Start with the basics and gradually move to more complex identities. Practice is key here. Use visual representations where possible, such as the unit circle, to grasp how these identities work. Also, try to understand the derivations of these identities, as it can provide deeper insights into their applications.

**Solving Real-World Problems**

Applying trigonometry to solve real-world problems can be challenging due to the abstract nature of the concepts.

Break down the problem into smaller parts and sketch a diagram whenever possible. Identifying which trigonometric ratio to use and correctly setting up the equation are crucial steps. Practice with a variety of problems, from simple height and distance calculations to more complex applications in physics and engineering.

**Switching Between Degrees and Radians**

Another common challenge is the need to switch between degrees and radians, depending on the context of the problem.

Familiarise yourself with the conversion between degrees and radians (180 degrees = pi radians). Practice converting angles in both directions and using them in various trigonometric functions. Over time, this will become second nature.

## Final thoughts on Trigonometric Ratios

Trigonometry is more than topic a subject in GCSE Maths; it's a tool that enhances our understanding of the world around us. We've learned about trigonometric ratios, including their practical applications in fields as diverse as astronomy, geography, architecture and even digital animation, trigonometry plays a pivotal role. By mastering trigonometric ratios and their applications, you're both preparing for your exams and opening doors to future careers.

Remember that there are plenty of maths resources to support your learning. Online platforms, tutoring services and educational apps offer endless opportunities for learning and engagement. For those looking to deepen their understanding or require additional support, GCSE Maths tutors are available to guide you through the intricacies of trigonometry and beyond.

*This post was updated on 05 Jul, 2024.*