Welcome back! In our last lesson, you did a great job mastering the Pythagorean theorem to find missing sides in right-angled triangles. We established that the longest side, opposite the right angle, is always called the hypotenuse.
Today, we're going to build directly on that knowledge. We'll learn how to label the other two sides. Instead of just calling them "legs," we'll give them specific names: opposite and adjacent. This skill is the absolute key to unlocking the main part of trigonometry, and by the end of this lesson, you'll be able to correctly label any right-angled triangle, setting you up perfectly for your test.
A New Point of View
In the last lesson, the only angle that mattered was the 90-degree right angle. For trigonometry, we need to focus on one of the other two angles in the triangle. This angle is our "point of view" and determines how we label the sides. It's often marked with a symbol, like the Greek letter theta ().
There are three labels we use:
- Hypotenuse (H): You already know this one!
- Opposite (O): The side directly opposite the marked angle ().
- Adjacent (A): The side next to the marked angle () that isn't the hypotenuse.
The following resource gives a very clear, visual guide to these three labels.
Introduction to trigonometry for right-angled triangles
This BBC Bitesize article provides a simple, step-by-step method for labelling the sides of a right-angled triangle. It's the foundation for everything we'll do next.
Please read the first part of the article, from the beginning down to the end of the first example. Pay close attention to the three steps shown in the images for labelling the hypotenuse, opposite, and adjacent sides. Read from the start of the main text.
A Simple 3-Step Process
As the article shows, there's a reliable order to follow every single time, which makes this much easier.
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Find the Hypotenuse (H) first. This is your anchor. It's always the longest side, and it's always opposite the square symbol (the 90° angle). This side's name never changes, no matter which other angle we look at.
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Find the Opposite (O) second. Locate the marked angle (). The side that is directly across from it, not touching it at all, is the opposite side.
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Find the Adjacent (A) last. The word "adjacent" just means "next to". This is the side that is next to the marked angle, but is not the hypotenuse. It's the side that's left over after you've identified the hypotenuse and the opposite.
Let's look at a diagram together and apply this process.
Introduction to trigonometry for right-angled triangles
In this triangle, the marked angle is \theta. The side opposite the right angle is the hypotenuse (h). The side across from \theta is the opposite (o). The side next to \theta that isn't the hypotenuse is the adjacent (a).
The Labels Depend on the Angle
This is the most important concept in today's lesson. The "hypotenuse" is always the hypotenuse. But the labels "opposite" and "adjacent" swap if we change our point of view to the other angle.
The next resource from Khan Academy has some excellent diagrams that show this very clearly.
Hypotenuse, opposite, and adjacent (article) | Khan Academy
This article shows side-by-side comparisons of the same triangle labelled from two different angles. This will help you see exactly how the 'opposite' and 'adjacent' sides can change.
Start at the beginning and read the definitions for hypotenuse, opposite, and adjacent. Then, carefully compare the two diagrams under the headings "from the perspective of ∠A" and "from ∠B". Notice which label stays the same and which two swap places.
Let's break this down with our own example. Imagine this triangle:
Let's say angle C is the 90° right angle. Side AB is the hypotenuse.
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If we focus on angle A:
- The side opposite angle A is BC.
- The side adjacent to angle A is AC.
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Now, let's switch our focus to angle B:
- The side opposite angle B is now AC.
- The side adjacent to angle B is now BC.
Do you see how side BC was "opposite" from angle A's perspective, but "adjacent" from angle B's perspective? This is why it's so important to first identify which angle a question is asking you about.
Conclusion
Excellent work today. This is a short but very significant lesson. Being able to correctly label the sides of a right-angled triangle from a specific angle is a non-negotiable skill for trigonometry, and you've now covered it thoroughly.
Here are the key takeaways:
- There are three labels for the sides of a right-angled triangle in trigonometry: Hypotenuse, Opposite, and Adjacent.
- The Hypotenuse is always the longest side, opposite the right angle. Its label never changes.
- The Opposite and Adjacent sides are named in relation to a marked angle ().
- The Opposite side is across from the marked angle.
- The Adjacent side is next to the marked angle (and is not the hypotenuse).
In our next lesson, we will finally answer the question of why we need these labels. We'll introduce the three famous trigonometric ratios—sine, cosine, and tangent—and see how to use these labels to choose the correct one to solve a problem.