Unit circle - Wikipedia
What is the difference between the trigonometric ratio tangent and the tangent to the curve? What is the relationship with a tangent line and tangent as a trigonometric ratio? To answer the question, consider TT, the tangent line of our curve at a given point. The "Unit Circle" is a circle with a radius of 1. Being so simple, it is a great way to learn and talk about lengths and angles. The center is put on a graph where the. Trigonometry is a branch of mathematics that studies relationships involving lengths and Sumerian astronomers studied angle measure, using a division of circles into degrees. They, and later the Babylonians, studied the ratios of the .
And scholars might study haversine, exsecant and gamsinlike biologists who find a link between your tibia and clavicle.
And because triangles show up in circles… …and circles appear in cycles, our triangle terminology helps describe repeating patterns! The Dome Instead of staring at triangles by themselves, like a caveman frozen in ice, imagine them in a scenario, hunting that mammoth. The angle you point at determines: Want the screen the furthest away?
Point straight across, 0 degrees. The height and distance move in opposite directions: Trig Values Are Percentages Nobody ever told me in my years of schooling: How do we compute the percentage? Cosine becomes negative when your angle points backwards. Every circle is really the unit circle, scaled up or down to a different size.
So work out the connections on the unit circle and apply the results to your particular scenario. This should make sense: The Wall One day your neighbor puts up a wall right next to your dome.
But can we make the best of a bad situation? What if we hang our movie screen on the wall? You point at an angle x and figure out: It starts at 0, and goes infinitely high. You can keep pointing higher and higher on the wall, to get an infinitely large screen! Tangent is just a bigger version of sine!
How about secant, the ladder distance? Secant starts at 1 ladder on the floor to the wall and grows from there Secant is always longer than tangent. The leaning ladder used to put up the screen must be longer than the screen itself, right? But secant is always a smidge longer. Remember, the values are percentages. The Ceiling Amazingly enough, your neighbor now decides to build a ceiling on top of your dome, far into the horizon. Oh, the naked-man-on-my-wall incident… Well, time to build a ramp to the ceiling, and have a little chit chat.
You pick an angle to build and work out: Our intuitive facts are similar: If you pick an angle of 0, your ramp is flat infinite and never reachers the ceiling. Cosecant is the full distance from you to the ceiling. The triangles have similar facts: If we increase the angle, we reach the ceiling before the wall: So the hypotenuse has length 1. Now, what is the length of this blue side right over here? You could view this as the opposite side to the angle. Well, this height is the exact same thing as the y-coordinate of this point of intersection.
So this height right over here is going to be equal to b. The y-coordinate right over here is b.
The unit circle definition of sine, cosine, and tangent | Khan Academy
This height is equal to b. Now, exact same logic-- what is the length of this base going to be? The base just of the right triangle? Well, this is going to be the x-coordinate of this point of intersection. If you were to drop this down, this is the point x is equal to a. Or this whole length between the origin and that is of length a. Now that we have set that up, what is the cosine-- let me use the same green-- what is the cosine of my angle going to be in terms of a's and b's and any other numbers that might show up?
Well, to think about that, we just need our soh cah toa definition.
That's the only one we have now. We are actually in the process of extending it-- soh cah toa definition of trig functions. And the cah part is what helps us with cosine. It tells us that the cosine of an angle is equal to the length of the adjacent side over the hypotenuse.
So what's this going to be? The length of the adjacent side-- for this angle, the adjacent side has length a. So it's going to be equal to a over-- what's the length of the hypotenuse? Well, that's just 1. So the cosine of theta is just equal to a. Let me write this down again. It's equal to the x-coordinate of where this terminal side of the angle intersected the unit circle. Now let's think about the sine of theta. And I'm going to do it in-- let me see-- I'll do it in orange.
So what's the sine of theta going to be? Well, we just have to look at the soh part of our soh cah toa definition. It tells us that sine is opposite over hypotenuse. Well, the opposite side here has length b. And the hypotenuse has length 1. So our sine of theta is equal to b. So an interesting thing-- this coordinate, this point where our terminal side of our angle intersected the unit circle, that point a, b-- we could also view this as a is the same thing as cosine of theta.
And b is the same thing as sine of theta. We just used our soh cah toa definition. Now, can we in some way use this to extend soh cah toa? Because soh cah toa has a problem.
It works out fine if our angle is greater than 0 degrees, if we're dealing with degrees, and if it's less than 90 degrees.
We can always make it part of a right triangle. But soh cah toa starts to break down as our angle is either 0 or maybe even becomes negative, or as our angle is 90 degrees or more. You can't have a right triangle with two degree angles in it. It starts to break down. Let me make this clear. So sure, this is a right triangle, so the angle is pretty large. I can make the angle even larger and still have a right triangle.
Even larger-- but I can never get quite to 90 degrees. At 90 degrees, it's not clear that I have a right triangle any more. It all seems to break down. And especially the case, what happens when I go beyond 90 degrees. So let's see if we can use what we said up here.
Let's set up a new definition of our trig functions which is really an extension of soh cah toa and is consistent with soh cah toa. Instead of defining cosine as if I have a right triangle, and saying, OK, it's the adjacent over the hypotenuse.