Introduction to Trigonometric Ratios
Trigonometry is one of the most powerful tools in mathematics — it allows us to calculate distances and angles that we can't easily measure directly. Imagine you're standing at the base of a tall building and you want to know its height. You can't climb it with a tape measure, but if you know the angle you're looking up at and how far away you're standing, trigonometry gives you the answer.
Trigonometric ratios describe the relationships between the sides and angles of a right-angle triangle. These ratios are constant for any given angle, no matter how large or small the triangle is. This is what makes them so useful — once you know an angle, you know the ratio of the sides.
A branch of mathematics that studies the relationships between the sides and angles of triangles. The word comes from the Greek trigonon (triangle) and metron (measure).
Think of a ramp. Whether the ramp is 2 metres long or 20 metres long, if it rises at the same angle, the shape of the triangle it forms is the same — only the size changes. Trigonometric ratios capture the shape, not the size.
Labelling the Sides of a Right-Angle Triangle
Before we can use trigonometric ratios, we need to know how to label the three sides of a right-angle triangle relative to a chosen angle (which we often call θ, pronounced "theta").
The longest side of a right-angle triangle, always opposite the right angle (90°). It is never adjacent to the angle θ (unless θ is the right angle itself, which we don't use in trig ratios).
The side that is directly across from the angle θ. It does not touch the angle θ at all.
The side that is next to the angle θ (and is not the hypotenuse). It forms one of the arms of the angle θ along with the hypotenuse.
The key idea: which side is "opposite" and which is "adjacent" changes depending on which angle you choose as θ. The hypotenuse, however, is always the same — it's always opposite the 90° angle.
A very common mistake is mislabelling the sides. Always identify the hypotenuse first (longest side, opposite the right angle), then determine opposite and adjacent relative to your chosen angle θ. If you pick the other acute angle, the opposite and adjacent sides swap!
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