I think everyone’s mathematical journey would have been so much easier if they showed you the sine, cosine, and tangent unit circle (R=1) animation in precalc
Below is sine and cosine graph in relation to circle (R undefined)
Below is tangent graph in relation to circle (R=1)
below is all six trigonometric functions on unit circle (R=1)
i have to be honest this makes it worse
it’s very nice to look at but i think i actually understand less
if i understood those at any point, i would’ve lost that knowledge after this
For anyone struggling to understand, the problem is that you are thinking about it like numbers and angles and math, when what you need to do is go back to preschool and think about it like a kid playing with shapes.
So, lets use our imagination to play with shapes! Pretend you have a ring or a wheel. Something that is a circle that you can play with. Now, you take a marker and put a dot on the outside of the ring. Now roll the circle along the ground. You see how the dot is moving around the circle?
We get curious about how the dot moves around the circle as it rolls along. Round and round it goes. Sometimes it is closer to the ground. Sometimes it is farther away. Depending on how far we turn the circle it can be in a bunch of different places. But it is always the same distance from the center of the circle. Isn’t that weird and cool?
What if we had words that we could use to talk about the circle and the dot and the ground? Well, it turns out we do! The words are all about where the dot is as we turn the circle round and round, depending on how far we turn the circle.
So go look at that first gif, but only the circle part, and ignore all the math stuff. Just the circle and the dot. See how the circles rolls and the dot moves? The dot always starts at the very right and the circle always rolls to the left, so the dot goes around counter clockwise. How far we have turned the circle is called the Angle. But because math nerds are lazy, we don’t want to write so many letters! one letter! So we just use one letter, θ, but it still means angle. We measure how far the circle is turned using degrees. There are 360 degrees in a full turn of the circle.
Now, you see how no matter how much we turn the circle, the dot is always just as far from the center of the circle? We call that the Radius. But again, math nerds are lazy, so we just use r. It could be a foot, or a centimeter, or 564 miles, but it is always called r.
Now, lets come up with more words!
One of those words is sine, pronounced like ‘sign’. But again, lazy math nerds don’t like using lots of letters, so we usually just spell it sin. Sine means “How far away is the dot from the center of the circle, but only up or down?” If you go look at our rotating circle in the first gif, you can see that the dot goes up and down as the circle turns. If we have not turned it at all, the dot is zero distance up or down away from the center of the circle. If we turn it 90 degrees, it is r distance up from the center. If you look at the circle turning, you can see how it goes up until we turn 90 degrees, then it goes down until we turn 270 degrees, then it goes back to the center. So, if we want to know how far away the dot is from the center, but only vertical, we could ask:
“How many Radius’ is the dot from the center of the circle, but only up or down, if we have turned the circle 90 degrees?” The answer is one r. But that is a lot of words, and we are very lazy! So instead we write sin(90). That is easier.
sin(0) = 0 sin(90) = 1 sin(270) = -1 etc. Use a calculator to get the exact number, it is easier that way.
Cosine is another word we use to talk about the dot and circle, pronounced “co-sign”. It is a lot like sine, except instead of up or down it is right and left. The lazy version of cosine is cos. Again, look at the circle. It turns and the dot moves left and right.
cos(0) = 1 cos(90) = 0 cos(180) = -1 etc.
The last one I am going to talk about is tangent. The lazy version of tangent is tan. Look at the second gif. See how there is a line going through the center of the circle so it also goes through the dot? Pretend that line is a laser shooting out of the circle in both directions. Unless the laser is pointing straight up and down it will hit the wall somewhere. Tangent means “If we are rolling the circle up a wall, where does the laser hit the wall?”
tan(0) = 0 tan(90) = it doesn’t! (undefined) tan (45) = 1
In the third gif, we show a different, more mathy way to think about tangent. It is useful because computers and calculators think using math. tan is the same as sin divided by cos. The lazy math version is:
tan(θ) = sin(θ)/cos(θ)
The basic trigonometric functions are hard if you think of them just as numbers, but they are actually simple questions about shapes. Using the answer to those simple questions, we can figure out all sorts of really useful things. But remember, it is all just playing with shapes.
