Trigonometry is one of the most important topics you will encounter in Secondary 3 Elementary Mathematics (E Math). It deals with the relationships between angles and lengths in right-angled triangles, forming the foundation for many more complex questions that appear in your O-Level examinations. From calculating heights and distances to solving real-world geometry problems, trigonometry is a key tool in your mathematical toolkit. Mastering it early not only helps you score but also builds the confidence you need for related topics. Whether you are just starting out or brushing up before exams, understanding trigonometry is a must.
Trigonometric Ratios of Acute and Obtuse Angles
In trigonometry, we use three main ratios to describe the relationship between the sides and angles of a right-angled triangle: sine (sin), cosine (cos), and tangent (tan). For acute angles (angles between 0° and 90°), these ratios are defined based on the position of the angle in the triangle:
- sin θ = opposite / hypotenuse
- cos θ = adjacent / hypotenuse
- tan θ = opposite / adjacent
These formulas help us find unknown side lengths or angles in right-angled triangles. But trigonometry does not stop at acute angles!
We can also find the trigonometric values for obtuse angles (angles between 90° and 180°) by using the concept of the unit circle and the idea that trigonometry ratios extend beyond just triangle geometry. Understanding how these ratios behave in different quadrants is crucial for solving a wide range of O-Level questions—including those involving angles greater than 90°.
Understanding the Sine Rule: Solving Non-Right-Angled Triangles
The Sine Rule is used when dealing with non-right-angled triangles, especially when you are given two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). It allows you to find unknown sides or angles using the formula:
Here, a, b, and c are the sides of the triangle, and A, B, and C are the angles opposite those sides.
Cracking the Cosine Rule: Finding Unknowns in Any Triangle
The Cosine Rule is another essential tool for solving non-right-angled triangles, especially when you are given two sides and the included angle (SAS) or all three sides (SSS) and need to find a missing side or angle. The formula is:
This rule is like an extended version of Pythagoras’ Theorem that works even when there is no right angle. It helps you calculate the third side of a triangle or find an angle when all sides are known. The Cosine Rule is often tested in O-Level exams, particularly in questions involving bearings, navigation, and geometry problems. Knowing when and how to apply it gives you a big advantage in securing those harder marks.
Finding the Area of a Triangle Using Trigonometry
When you do not have the height of a triangle, you can still find its area using trigonometry—specifically when you know two sides and the included angle (SAS). The formula is:
Here, a and b are two sides of the triangle, and C is the angle between them. This formula works for any triangle, not just right-angled ones, and is especially useful in O-Level questions where height is not given.
Trigonometry is more than just formulas—it is a powerful tool that helps you solve real-world problems involving angles and distances. By understanding key concepts like trigonometric ratios, the Sine Rule, the Cosine Rule, and the area formula for triangles, you will be well-prepared to tackle both school exams and O-Level questions with confidence. The earlier you master these skills, the easier it becomes to apply them in more advanced topics in Sec 4 and beyond.
