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If u understand the answer to this the whole unit circle becomes really easy no more memorizing at all!! Using the unit circle diagram, draw a line "tangent" to the unit circle where the hypotenuse contacts the unit circle. So sure, this is a right triangle, so the angle is pretty large. Let be a point on the terminal side of the road. And why don't we define sine of theta to be equal to the y-coordinate where the terminal side of the angle intersects the unit circle? It tells us that the cosine of an angle is equal to the length of the adjacent side over the hypotenuse. I need a clear explanation... How can anyone extend it to the other quadrants?

Let Be A Point On The Terminal Side Of The Road

So our x is 0, and our y is negative 1. Want to join the conversation? The distance of this line segment from its tangent point on the unit circle to the x-axis is the tangent (TAN). The ray on the x-axis is called the initial side and the other ray is called the terminal side.

I can make the angle even larger and still have a right triangle. Instead of defining cosine as if I have a right triangle, and saying, OK, it's the adjacent over the hypotenuse. Even larger-- but I can never get quite to 90 degrees. See my previous answer to Vamsavardan Vemuru(1 vote). Let -7 4 be a point on the terminal side of. 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. When you graph the tangent function place the angle value on the x-axis and the value of the tangent on the y-axis. The angle line, COT line, and CSC line also forms a similar triangle. I hate to ask this, but why are we concerned about the height of b? The sign of that value equals the direction positive or negative along the y-axis you need to travel from the origin to that y-axis intercept. This is similar to the equation x^2+y^2=1, which is the graph of a circle with a radius of 1 centered around the origin. No question, just feedback.

The advantage of the unit circle is that the ratio is trivial since the hypotenuse is always one, so it vanishes when you make ratios using the sine or cosine. It would be x and y, but he uses the letters a and b in the example because a and b are the letters we use in the Pythagorean Theorem. Let be a point on the terminal side of . find the exact values of and. We can always make it part of a right triangle. You are left with something that looks a little like the right half of an upright parabola. This height is equal to b. The section Unit Circle showed the placement of degrees and radians in the coordinate plane.

Let -7 4 Be A Point On The Terminal Side Of

So essentially, for any angle, this point is going to define cosine of theta and sine of theta. The distance from the origin to where that tangent line intercepts the y-axis is the cosecant (CSC). It's equal to the x-coordinate of where this terminal side of the angle intersected the unit circle. Now, can we in some way use this to extend soh cah toa? For example, If the line intersects the negative side of the x-axis and the positive side of the y-axis, you would multiply the length of the tangent line by (-1) for the x-axis and (+1) for the y-axis.

Affix the appropriate sign based on the quadrant in which θ lies. So to make it part of a right triangle, let me drop an altitude right over here. The angle shown at the right is referred to as a Quadrant II angle since its terminal side lies in Quadrant II. Well, this is going to be the x-coordinate of this point of intersection. So Algebra II is assuming that you use prior knowledge from Geometry and expand on it into other areas which also prepares you for Pre-Calculus and/or Calculus. To determine the sign (+ or -) of the tangent and cotangent, multiply the length of the tangent by the signs of the x and y axis intercepts of that "tangent" line you drew. What happens when you exceed a full rotation (360º)? Extend this tangent line to the x-axis. Well, we've gone 1 above the origin, but we haven't moved to the left or the right. Standard Position: An angle is in standard position if its vertex is located at the origin and one ray is on the positive x-axis. If θ is an angle in standard position, then the reference angle for θ is the acute angle θ' formed by the terminal side of θ and the horizontal axis.

Other sets by this creator. Anthropology Exam 2. You will find that the TAN and COT are positive in the first and third quadrants and negative in the second and fourth quadrants. Give yourself plenty of room on the y-axis as the tangent value rises quickly as it nears 90 degrees and jumps to large negative numbers just on the other side of 90 degrees. Say you are standing at the end of a building's shadow and you want to know the height of the building. So this theta is part of this right triangle. Why is it called the unit circle? It doesn't matter which letters you use so long as the equation of the circle is still in the form. In the next few videos, I'll show some examples where we use the unit circle definition to start evaluating some trig ratios. This seems extremely complex to be the very first lesson for the Trigonometry unit. Now, what is the length of this blue side right over here? So what's this going to be? In this second triangle the tangent leg is similar to the sin leg the angle leg is similar to the cosine leg and the secant leg (the hypotenuse of this triangle) is similar to the angle leg of the first triangle.

Let Be A Point On The Terminal Side Of . Find The Exact Values Of And

So it's going to be equal to a over-- what's the length of the hypotenuse? At negative 45 degrees the tangent is -1 and as the angle nears negative 90 degrees the tangent becomes an astronomically large negative value. Now, with that out of the way, I'm going to draw an angle. Does pi sometimes equal 180 degree. You could view this as the opposite side to the angle. The problem with Algebra II is that it assumes that you have already taken Geometry which is where all the introduction of trig functions already occurred.

And so what I want to do is I want to make this theta part of a right triangle. Anthropology Final Exam Flashcards. So you can kind of view it as the starting side, the initial side of an angle. 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). It the most important question about the whole topic to understand at all! Well, this hypotenuse is just a radius of a unit circle. Terms in this set (12).

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. Cosine and secant positive. If you were to drop this down, this is the point x is equal to a. What would this coordinate be up here? Key questions to consider: Where is the Initial Side always located? So this is a positive angle theta. The y value where it intersects is b. Well, to think about that, we just need our soh cah toa definition. He keeps using terms that have never been defined prior to this, if you're progressing linearly through the math lessons, and doesn't take the time to even briefly define the terms. You can also see that 1/COS = SEC/1 and 1^2 + TAN^2 = SEC^2.

And so you can imagine a negative angle would move in a clockwise direction. We've moved 1 to the left. What is the terminal side of an angle?