Trigonometry is the branch of math that studies triangles, with a particular focus on the relationships between angles and the lengths of corresponding sides.
Interestingly enough, the trigonometric functions that define those relationships are also closely tied to circles.
Needless to say, this makes trig one of the hardest topics in math for students to grasp intuitively.
Part of that is the way it’s taught. Students are taught the “unit circle” and its relationship to trigonometry, but many fail to make the leap on how crucial circles are for trig functions.
With static graphs and equations, it’s possible to get a handle on the rules of what various functions do and mean. However, it’s still hard to get an intuitive sense of the relationship between the circle and the trigonometric functions and the triangles.
For starters, here’s what you should really think when you see the number π:
Many people are confused about what radians are. Well, there’s a GIF for that:
Next, think about the relationship between sine, cosine and the circle.
Here’s an illustration of the fundamental relationship between the three.
Notice how the crank moves in a circle, and the bars — which correspond to sine and cosine — move up and down and side to side in a wave-like formation:
Here’s a more traditional demonstration of sine and cosine. You make your way around the circle (black). As you do so, the values of Y translates to sine (red line) and the values of X translate to cosine (blue line):
Now, let’s start linking this relationship between the functions and circles to triangles:
The triangle relationship is crucial to the definition of the tangent() function. The intersection of the triangle’s hypotenuse line with the vertical line along the right side of the circle defines the function.
Here’s another way of looking at it, without the triangle: