We organize our trigonometric measurements by comparing the sides of a right triangle, creating three trigonometric ratios: sine, cosine, and tangent.
Looking at ∠A, we can see clearly which side is adjacent, which is opposite, and which is the hypotenuse. Using these terms, we can define our three basic trigonometric ratios.
sin(A) = opposite / hypotenuse = a / h
cos(A) = adjacent / hypotenuse = b / h
tan(A) = opposite / adjacent = a / b
An easy way to remember which ratio is which is to just remember what someone who stubbed their foot would do and say. They'd soak their toe and say, "Ahhh."
Soak a toe, ahhh. SOHCAHTOA. Each letter stands for something:
Sine: Opposite over Hypotenuse
Cosine: Adjacent over Hypotenuse
Tangent: Opposite over Adjacent
It's important to know that these ratios depend on angles, not side lengths. That means it doesn't matter if a triangle is the size of Godzilla or the new iPod nano. If its angles are the same, it'll have the same trigonometric ratios.
We could stare at these fractions all day, but the only surefire way to learn trigonometry inside and out is to practice. Are you ready for this?
We'll start simple and find the sine, cosine, and tangent for ∠A of this triangle.
That's all there is to it, putting one side over the other. For ∠B, though, these ratios would be different.
Since these ratios never change for similar angles, we can use them to find the side lengths of similar triangles. That's kind of the whole point of trigonometry.