Trigonometry is the study of right triangles. So to help us understand what that means and why it’s important, we’ll need some triangles.
Here’s a small right triangle.
Here’s a bigger right triangle.
These two triangles happen to be proportional. Triangle DEF is ten times as big as triangle ABC.
Since these triangles are proportional, that means the angles will be the same. Angle A will equal angle D. Angle B will equal angle E. And angle C will equal angle F.
Another cool fact about these triangles is that since they’re proportional if we look at the ratios of corresponding sides, we get the same answer.
For example, in the small triangle let’s look at the ratio of the shorter leg to the longer leg.
And in the big triangle let’s look at that same ratio. Shorter leg to longer leg.
We get the same answer. Let’s do one more. Now let's do short leg to hypotenuse. First in the small triangle…
And then in the bigger triangle
Same answer again.
Let’s think about this for just a second. We have two triangles of completely different sizes that have the same angles. Even though one is way bigger than the other, the ratios of their sides are the same. This idea, that these ratios only depend on the angles in a triangle, is the basis of trigonometry. It means that if we know some information about the angles in a triangle, we can figure out some information about the sides in a triangle.
So how do we use it?
In trigonometry we give special names to these ratios. And since these ratios depend on the angles in a triangle, we always have to specify an angle when we talk about them.
We say that the sine of an angle is the ratio of the opposite side to the hypotenuse.
We say that the cosine of an angle is the ratio of the adjacent side to the hypotenuse.
And we say that the tangent of an angle is the ratio of the opposite side to the adjacent side.
To help remember this, a lot of people remember “SOH CAH TOA”
Sine S is the opposite O over the hypotenuse H. Cosine C is the adjacent A over the hypotenuse H. Tangent T is the opposite O over the adjacent A.
To get a feel for how these work, we’ll need to look at another triangle.
Let’s say we wanted to find the sine of A. Sine. SOH. S O H. Sine is opposite over hypotenuse. The opposite side is the one across the triangle from A. So that will be 5. The hypotenuse is 13. So sine of A is 5 over 13.
What about the tangent of A? Tangent. TOA. T O A. Tangent is opposite over adjacent. The opposite side, across the triangle, is 5. The adjacent side is the one touching A that’s not the hypotenuse. So that will be 12. So tangent of A is 5 over 12.
We can also talk about trig ratios from angle B. Let’s look at the tangent of B. Tangent was opposite over adjacent. This time, since we’re talking about angle B the opposite side will be across the triangle from B. It’s 12. The adjacent side will be adjacent to B. It’s 5. So the tangent of B is 12 over 5.
Notice how when we used the other angle, the opposite and adjacent sides switched places. This leads to a pretty interesting fact. Let’s look at the cosine of B. Cosine. CAH. C A H. Cosine is adjacent over hypotenuse. The side adjacent to B is 5. The hypotenuse is 13. So, Cosine of B is 5 over 13.
The sine of A and the cosine of B are the same!. This is actually always true in any right triangle. We can sum this up as a rule. If two angles A and B are complementary then the sine of one is equal to the cosine of the other.
This is called the cofunction identity, and it’s the first of many interesting facts there are to discover about trigonometry.