Trigonometry is one of the core topics in GCSE Maths. Trigonometric ratios - trig ratios for short - are therefore crucial as they help to traverse this field, providing a bridge between the angles and sides of a right-angled triangle. This article will explain the six basic ratios: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec) and cosecant (cosec). We will include quiz questions to test your understanding and real-world examples so you can see how this concept is applied.

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## The definition of basic Trigonometric Ratios

Trigonometric ratios show the relationship between the angles and sides of the triangle. These ratios are crucial in maths, allowing us to calculate and understand geometric properties and angles within a triangle. For GCSE Maths, these ratios allow you to solve complex trigonometric problems.

The main six trigonometric ratios are sine, cosine, tangent, cosecant, secant, and cotangent.

**Sine (sin)**: The ratio of the length of the opposite side to the hypotenuse. For example, in a right-angled triangle, if the length of the side opposite the angle is 6 units and the hypotenuse is 10 units, then the sine of the angle is calculated as*sin = opposite / hypotenuse = 6 / 10 = 0.6*.**Cosine (cos)**: The ratio of the length of the adjacent side to the hypotenuse. For example, if the length of the adjacent side to the angle is 8 units and the hypotenuse is 10 units, then the cosine of the angle is*cos = adjacent / hypotenuse = 8 / 10 = 0.8*.**Tangent (tan)**: The ratio of the length of the opposite side to the adjacent side. With the opposite side being 6 units and the adjacent side being 8 units, the tangent of the angle is*tan = opposite / adjacent = 68 = 0.75*.**Cotangent (cot)**: The reciprocal of tangent, or the ratio of the length of the adjacent side to the opposite side. Using the tangent example, the cotangent of the angle, being the reciprocal of tangent, is*cot = 1 / tan= 1 / 0.75 = 8 / 6 = 1.33*.**Secant (sec)**: The reciprocal of cosine, or the ratio of the hypotenuse to the adjacent side. From the cosine example, the secant of the angle, being the reciprocal of cosine, is*sec = 1 / cos = 1 / 0.8 = 10 / 8 = 1.25*.**Cosecant (cosec)**: The reciprocal of sine, or the ratio of the hypotenuse to the opposite side. Based on the sine example, the cosecant of the angle, being the reciprocal of sine, is*cosec = 1 / sin = 1 / 0.6 = 10 / 6 = 1.67*.

### 1

Which trigonometric ratio is the ratio of the length of the opposite side to the length of the hypotenuse?

## Breaking down each Ratio

Now that the basic ratios have been introduced, we will go into more detail with each. This will include explaining their definitions and applications.

### Sine (sin)

The sine of an angle in a right-angled triangle is defined by the formula

*sin = opposite side / hypotenuse*. This relationship enables the calculation of an angle's sine given the lengths of the triangle's sides. Sine is crucial in real-world applications such as calculating the height of inaccessible objects or determining the distance across water bodies when direct measurement isn't possible. For example, by measuring the angle of elevation from a particular point to the top of a tower and knowing the distance to the tower, you can calculate the tower's height.### Cosine (cos)

The formula cosine is defined as is

*cos = adjacent side / hypotenuse*. This ratio calculates how much of the hypotenuse is "covered" by the adjacent side when viewed from the angle of interest. Cosine values can solve problems involving distances, such as finding the straight-line distance between two points on a coordinate plane. In astronomy, physics and other fields that require precision, cosine helps calculate the position of stars and the dynamics of moving bodies, using the adjacent side's proportion to the hypotenuse for essential calculations.### Tangent (tan)

Tangent combines the ratios of sine and cosine, represented by the formula

*tan = sin / cos = opposite side / adjacent side*. This ratio is particularly useful in applications involving slopes e.g. determining the gradient of a hill or the angle of elevation to an airplane in the sky from a ground observer. The tangent function's ability to relate the opposite and adjacent sides makes it useful when the relationship between these two sides is more relevant than their relationship to the hypotenuse.### Cotangent (cot)

Cotangent is the reciprocal of tangent. The formula is

*cot = 1 / tan = adjacent side / opposite side*. This ratio compares the adjacent side's length directly to the opposite side's length. It's less commonly used in basic trigonometry. Instead, it is used in advanced mathematics and physics, where the method can solve equations and analyse systems. This is particularly the case when the cotangent's specific relationship abbreviates calculations.### Secant (sec)

Secant is the reciprocal of cosine. It is defined as

*sec = 1 / cos = hypotenuse / adjacent side*. This function is especially useful when you're trying to find the length of the hypotenuse relative to the adjacent side. It isn't as common in basic trigonometry. However, secant plays a crucial role in certain areas of calculus and geometry, including in the derivation and integration of trigonometric functions and identities.### Cosecant (cosec)

Cosecant is the reciprocal of sine and uses the formula

*cosec = 1 / sin = hypotenuse / opposite side*. This ratio highlights the relationship between the hypotenuse and the side opposite the angle. It gives us an alternate approach to understanding angle-side relationships in a triangle. Cosecant is crucial in various mathematical and engineering fields, helping to abbreviate and solve complex trigonometric equations and identities.### 2

Which trigonometric ratio is defined as the reciprocal of sine?

## The significance of Standard Angles

In the study of trigonometry, specific angles are considered more significant than others due to their frequent appearance and the simplicity of their trigonometric ratios. These are known as standard angles and they include 0°, 30°, 45°, 60° and 90 degrees. Understanding the trigonometric ratios for these angles is crucial, as they serve as the foundation for solving a wide array of trigonometric problems.

**0° and 90° Angles**: At these two angles, the trigonometric functions demonstrate unique values. For instance, the sine of 0° is 0, reflecting the absence of an opposite side, while the sine of 90° is 1, indicating the opposite side is equal to the hypotenuse.

**30°, 45° and 60° Angles**: These acute angles are particularly noteworthy due to their appearance in equilateral and isosceles right-angled triangles, making their ratios easy to memorise and apply. For example, the sine of 30° is 1/2, a simple ratio that stems from the geometry of an equilateral triangle divided in half.

The values for the trigonometric functions at these angles have been tabulated and widely used for quick reference. Knowing these values not only aids in the swift calculation of trigonometric problems but also in understanding the graphical representation of sin, cos and tan functions.

### 3

What is the value of tan 45°?

## Solve real-world problems with Trigonometric Ratios

Trig ratios are used to deal with for various applications in science, engineering, architecture and beyond. This section will gives examples of their usage in the real world.

**Height and distance calculations**: One of the most common uses of trigonometric ratios is to calculate heights and distances that can't be measured directly. For example, we can use the tangent ratio to find the height of a building or a mountain from a certain distance, given the angle of elevation.

**Navigation and surveying**: Trigonometric ratios are important in navigation and surveying. The sine and cosine functions are used to calculate distances across the Earth's surface, aiding in map making and the planning of various construction projects.

**Architecture and construction**: Architects and construction engineers often use trig ratios to design buildings and structures, ensuring stability and aesthetic appeal. Using the trigonometric ratios allows for the calculation of angles and dimensions necessary for the materialisation of complex designs.

**Physics and engineering**: In physics, trigonometric ratios help to understand wave properties, oscillations and forces. Engineers use these ratios to design mechanical systems, electrical circuits and to analyse stress and strain on materials.

A useful place to learn more about trigonometry is the trigonometric ratio course run by Khan Academy.

### 4

Which field commonly uses trigonometric ratios for designing and planning structures?

## Final thoughts on trig ratios

Trig ratios are powerful tools that have practical applications in a range of fields, from the precision required in engineering and architecture to the calculations necessary for navigation and physics. They bridge the gap between theoretical mathematics and real-world problem-solving.

Understanding the ratios sine, cosine, tangent and their reciprocals opens up a world of possibilities for solving complex problems with elegance and efficiency. To further your understanding, it's a good idea to seek out additional resources, such as past papers and practice questions. If you need further support, you can search TeachTutti's list of Maths GCSE tutors to revise this topic.

## Frequently asked questions

Trigonometric ratios are mathematical relationships that compare the lengths of sides of a right triangle to its angles. The six primary trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec) and cosecant (cosec).

Trigonometric ratios are used in various fields, including engineering, architecture, navigation and physics. They help calculate distances, heights, angles and other dimensions that can't be directly measured.

Yes, the basic definitions of trigonometric ratios apply specifically to right-angled triangles. However, trigonometric functions can be applied to any angle, not just those formed by right-angled triangles.

Trigonometric ratios refer specifically to the relationships between the sides of right-angled triangles and their angles. Trigonometric functions extend these concepts to all angles, providing a way to work with trigonometry in a broader mathematical context, including analysis and calculus.

A common mnemonic for remembering the primary trigonometric ratios is SOH-CAH-TOA or SohCahToa, where SOH stands for Sine = Opposite/Hypotenuse, CAH for Cosine = Adjacent/Hypotenuse and TOA for Tangent = Opposite/Adjacent.

Trigonometric ratios are fundamental to understanding relationships between angles and distances, crucial for solving problems in geometry, physics and various engineering fields. Learning about them at the GCSE level provides a solid foundation for further studies and practical applications in science and mathematics.

Yes, trigonometric ratios and the functions derived from them are integral to calculus, especially in studying derivatives and integrals involving trigonometric functions, which are pivotal in many areas of mathematics and applied sciences.

## Glossary

- Angle - A figure formed by two rays, called the sides of the angle, sharing a common endpoint called the vertex.
- Adjacent Side - In a right-angled triangle, the side that is next to the angle in question and forms one side of the right angle.
- Cosine - A trigonometric ratio is defined as the length of the adjacent side divided by the length of the hypotenuse in a right-angled triangle.
- Cosecant - The reciprocal of the sine function, defined as the length of the hypotenuse divided by the length of the opposite side in a right-angled triangle.
- Cotangent - The reciprocal of the tangent function, defined as the length of the adjacent side divided by the length of the opposite side in a right-angled triangle.
- Hypotenuse - The longest side of a right-angled triangle, opposite the right angle.
- Opposite Side - In a right-angled triangle, the side opposite the angle in question. It does not form the right angle with the hypotenuse.
- Radian - A unit of angle measure used in mathematics to express angles, defined such that a full circle is 2 radians.
- Secant - The reciprocal of the cosine function, defined as the length of the hypotenuse divided by the length of the adjacent side in a right-angled triangle.
- Sine - A trigonometric ratio is defined as the length of the opposite side divided by the length of the hypotenuse in a right-angled triangle.
- Tangent - A trigonometric ratio is defined as the length of the opposite side divided by the length of the adjacent side in a right-angled triangle.
- Trigonometric Functions - Functions that relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are sine, cosine and tangent, along with their reciprocals: cosecant, secant and cotangent.
- Trigonometric Ratios - Ratios of the lengths of sides of a right-angled triangle in relation to its angles. The six common trigonometric ratios are sine, cosine, tangent, cosecant, secant and cotangent.

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