Trigonometry is all about the relationships between the angles and side lengths of triangles. To start, we need a clear way to talk about the sides of a right-angled triangle.

First, there's the hypotenuse. This is always the longest side, and it's the one directly across from the right angle. The other two sides are called legs. But their names change depending on which angle we're focusing on.

Let's pick one of the acute angles (one of the angles that isn't the 90° one) and call it $θ$ (theta).

• The Opposite side is the leg directly across from our angle $θ$.
• The Adjacent side is the leg next to our angle $θ$ that is not the hypotenuse.

If you switch to the other acute angle, the opposite and adjacent sides also switch. The hypotenuse always stays the same.

For any given angle $θ$ in a right triangle, the ratio of its side lengths is always the same, no matter how big or small the triangle is. Mathematicians gave these constant ratios special names: sine, cosine, and tangent.

It can be tough to keep these straight. A popular mnemonic for remembering them is "SOH CAH TOA".

Let's work with a classic 3-4-5 right triangle. The side lengths are 3, 4, and 5. The hypotenuse must be the longest side, which is 5.

Let's find the trigonometric ratios for angle $α$.

• Opposite side to $α$: 3
• Adjacent side to $α$: 4
• Hypotenuse: 5

Using SOH CAH TOA, we get:

Now, let's look at angle $β$. From its perspective, the opposite and adjacent sides are different.

• Opposite side to $β$: 4
• Adjacent side to $β$: 3
• Hypotenuse: 5 (still 5!)

This gives us a new set of ratios for $β$:

These ratios form the foundation of trigonometry. By knowing just one angle and one side length in a right triangle, you can use these relationships to find all the other sides and angles.