Trigonometry: Sine, Cosine and Tangent Ratios

Trigonometry is one of the most useful branches of mathematics, connecting the study of angles and triangles to countless real-world applications, from measuring the height of a building to determining the position of a ship at sea. At the heart of trigonometry lie three fundamental ratios: sine, cosine, and tangent, which describe the relationships between the angles and sides of a right-angled triangle. This note explores these ratios in detail, along with their applications.

The Right-Angled Triangle

Trigonometry begins with the right-angled triangle, a triangle that contains one angle of exactly 90 degrees. The three sides of a right-angled triangle have special names relative to a chosen angle, usually labelled theta (θ). The side opposite the right angle is called the hypotenuse, and it is always the longest side. The side opposite the chosen angle is called the opposite side, and the side next to the chosen angle, which is not the hypotenuse, is called the adjacent side.

The Three Basic Trigonometric Ratios

The three basic trigonometric ratios relate the sides of a right-angled triangle to one of its acute angles. These ratios are commonly remembered using the mnemonic SOHCAHTOA.

• Sine (sin θ) = Opposite ÷ Hypotenuse
• Cosine (cos θ) = Adjacent ÷ Hypotenuse
• Tangent (tan θ) = Opposite ÷ Adjacent

These ratios allow us to find an unknown side or angle of a right-angled triangle as long as we know at least one angle (other than the right angle) and one side, or two sides.

Common Trigonometric Values

Certain angles produce trigonometric values that are used so frequently that it is helpful to memorize them. For example, sin 30° = 0.5, cos 60° = 0.5, tan 45° = 1, sin 90° = 1, and cos 0° = 1. These special values often appear in examination questions and can be derived from special right-angled triangles, such as the 30-60-90 triangle and the 45-45-90 triangle.

The Pythagorean Theorem and Trigonometry

The Pythagorean theorem is closely related to trigonometry and states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides, expressed as a² + b² = c². This relationship is often used alongside trigonometric ratios to solve for missing sides in a right-angled triangle when the necessary angle information is not directly available.

Angles of Elevation and Depression

Two important applications of trigonometry involve the angle of elevation and the angle of depression. The angle of elevation is the angle measured upward from the horizontal to a point above the observer, such as looking up at the top of a tall building. The angle of depression is the angle measured downward from the horizontal to a point below the observer, such as looking down from a cliff to a boat at sea. These concepts are widely used in surveying, navigation, and engineering to calculate heights and distances that would otherwise be difficult to measure directly.

Using Trigonometry to Find Heights and Distances

One of the most practical applications of trigonometry is calculating heights and distances that cannot be measured directly. For example, to find the height of a tree, a surveyor can measure the angle of elevation to the top of the tree from a known distance away, and then use the tangent ratio: height = distance × tan(angle of elevation). This method is widely used in fields such as architecture, construction, aviation, and astronomy.

Bearings and Trigonometry

Bearings are used to describe direction in navigation and are always measured clockwise from the north direction, expressed as a three-digit number, such as 045° or 270°. Trigonometry is often combined with bearings to solve problems involving the distance and direction travelled by ships, aircraft, or hikers, allowing navigators to calculate positions accurately.

The Sine Rule and Cosine Rule

While the basic trigonometric ratios apply specifically to right-angled triangles, the sine rule and cosine rule extend trigonometry to any triangle, whether or not it contains a right angle.

• The sine rule states that a/sin A = b/sin B = c/sin C, and it is useful when we know two angles and one side, or two sides and a non-included angle.
• The cosine rule states that a² = b² + c² − 2bc cos A, and it is useful when we know all three sides, or two sides and the included angle.

Reciprocal Trigonometric Ratios

In addition to sine, cosine, and tangent, there are three reciprocal ratios: cosecant (the reciprocal of sine), secant (the reciprocal of cosine), and cotangent (the reciprocal of tangent). While these are used less frequently at the secondary school level, they become more important in advanced mathematics and physics.

Why Trigonometry Matters

Trigonometry has countless real-world applications beyond the classroom. Engineers use it to design bridges and buildings, pilots and sailors use it for navigation, astronomers use it to calculate distances to stars and planets, and even video game designers use it to create realistic graphics and animations. Learning trigonometry equips students with powerful problem-solving tools that apply far beyond mathematics itself.

Solving Practical Trigonometry Problems Step by Step

When solving a real-world trigonometry problem, it helps to follow a consistent method. First, draw a clear diagram representing the situation, labelling the known angle, known side, and the side or angle you need to find. Second, identify which sides are the opposite, adjacent, and hypotenuse relative to the known angle. Third, choose the correct trigonometric ratio, sine, cosine, or tangent, based on which two sides are involved. Finally, rearrange the formula if necessary and calculate the answer, remembering to include the correct units, such as metres or degrees. Practising this structured approach helps students avoid common mistakes, such as confusing the opposite and adjacent sides or selecting the wrong ratio for a given problem.

Trigonometric Identities

Beyond the basic ratios, trigonometry includes several important identities that describe permanent relationships between trigonometric functions. The most fundamental of these is the Pythagorean identity, sin²θ + cos²θ = 1, which holds true for every angle θ. Another useful relationship is that tangent can be expressed as the ratio of sine to cosine, tan θ = sin θ / cos θ. These identities are extremely useful for simplifying complicated trigonometric expressions and for solving equations involving multiple trigonometric functions at once, and they form the foundation for more advanced topics students will encounter in further mathematics.

Key Terms to Remember

• Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
• Opposite side: The side opposite the angle being considered.
• Adjacent side: The side next to the angle being considered, excluding the hypotenuse.
• Sine, cosine, tangent: The three basic trigonometric ratios relating angles and sides of a right-angled triangle.
• Angle of elevation: The upward angle from the horizontal to a point above the observer.
• Angle of depression: The downward angle from the horizontal to a point below the observer.
• Sine rule and cosine rule: Formulas that extend trigonometry to any triangle, not just right-angled ones.

Summary

Trigonometry studies the relationships between the angles and sides of triangles, particularly right-angled triangles, through the ratios sine, cosine, and tangent, remembered using SOHCAHTOA. These ratios, along with the Pythagorean theorem, allow us to calculate unknown sides and angles. Concepts such as angles of elevation and depression extend trigonometry into practical applications like measuring heights and distances, while the sine rule and cosine rule allow trigonometric methods to be applied to any triangle. From navigation and construction to astronomy and technology, trigonometry remains one of the most widely applied areas of mathematics in the real world.