So to make it part of a right triangle, let me drop an altitude right over here. Created by Sal Khan. So our x is 0, and our y is negative 1. In the concept of trigononmetric functions, a point on the unit circle is defined as (cos0, sin0)[note - 0 is theta i. e angle from positive x-axis] as a substitute for (x, y). Let be a point on the terminal side of the doc. What happens when you exceed a full rotation (360º)? And let me make it clear that this is a 90-degree angle. Or this whole length between the origin and that is of length a.
Some people can visualize what happens to the tangent as the angle increases in value. So this theta is part of this right triangle. A²+b² = c²and they're the letters we commonly use for the sides of triangles in general. Do these ratios hold good only for unit circle? Let -8 3 be a point on the terminal side of. At negative 45 degrees the tangent is -1 and as the angle nears negative 90 degrees the tangent becomes an astronomically large negative value. The y value where it intersects is b. At the angle of 0 degrees the value of the tangent is 0. Why is it called the unit circle? A positive angle is measured counter-clockwise from that and a negative angle is measured clockwise. Now, exact same logic-- what is the length of this base going to be?
And especially the case, what happens when I go beyond 90 degrees. And what is its graph? We've moved 1 to the left. So how does tangent relate to unit circles? The angle shown at the right is referred to as a Quadrant II angle since its terminal side lies in Quadrant II. Well, we've gone 1 above the origin, but we haven't moved to the left or the right. This line is at right angles to the hypotenuse at the unit circle and touches the unit circle only at that point (the tangent point). Well, the opposite side here has length b. This seems extremely complex to be the very first lesson for the Trigonometry unit. Because soh cah toa has a problem. The second bonus – the right triangle within the unit circle formed by the cosine leg, sine leg, and angle leg (value of 1) is similar to a second triangle formed by the angle leg (value of 1), the tangent leg, and the secant leg. Why don't I just say, for any angle, I can draw it in the unit circle using this convention that I just set up? And the cah part is what helps us with cosine. Let be a point on the terminal side of . find the exact values of and. The angle line, COT line, and CSC line also forms a similar triangle.
Instead of defining cosine as if I have a right triangle, and saying, OK, it's the adjacent over the hypotenuse. You could use the tangent trig function (tan35 degrees = b/40ft). So the first question I have to ask you is, what is the length of the hypotenuse of this right triangle that I have just constructed? It doesn't matter which letters you use so long as the equation of the circle is still in the form. 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 so what would be a reasonable definition for tangent of theta? Using the unit circle diagram, draw a line "tangent" to the unit circle where the hypotenuse contacts the unit circle. I think the unit circle is a great way to show the tangent. So our sine of theta is equal to b. So what would this coordinate be right over there, right where it intersects along the x-axis?
Well, to think about that, we just need our soh cah toa definition. And this is just the convention I'm going to use, and it's also the convention that is typically used. When you graph the tangent function place the angle value on the x-axis and the value of the tangent on the y-axis. It starts to break down. Even larger-- but I can never get quite to 90 degrees. At2:34, shouldn't the point on the circle be (x, y) and not (a, b)? I saw it in a jee paper(3 votes).