Log in Vamsavardan Vemuru 12 years agoPosted 12 years ago. Direct link to Vamsavardan Vemuru's post “Do these ratios hold good...” Do these ratios hold good only for unit circle? What if we were to take a circles of different radii? • (194 votes) Matthew Daly 12 years agoPosted 12 years ago. Direct link to Matthew Daly's post “The ratio works for any c...” The ratio works for any circle. 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. (345 votes) Hemanth 11 years agoPosted 11 years ago. Direct link to Hemanth's post “What is the terminal side...” What is the terminal side of an angle? • (120 votes) apattnaik1998 10 years agoPosted 10 years ago. Direct link to apattnaik1998's post “straight line that has be...” straight line that has been rotated around a point on another line to form an angle measured in a clockwise or counterclockwise direction (30 votes) Rory 11 years agoPosted 11 years ago. Direct link to Rory's post “So how does tangent relat...” So how does tangent relate to unit circles? And what is its graph? • (21 votes) William Hunter 11 years agoPosted 11 years ago. Direct link to William Hunter's post “I think the unit circle i...” I think the unit circle is a great way to show the tangent. While you are there you can also show the secant, cotangent and cosecant. I do not understand why Sal does not cover this. (147 votes) Jason 11 years agoPosted 11 years ago. Direct link to Jason's post “I hate to ask this, but w...” I hate to ask this, but why are we concerned about the height of b? What is a real life situation in which this is useful? Graphing sine waves? • (18 votes) Aaron Sandlin 11 years agoPosted 11 years ago. Direct link to Aaron Sandlin's post “Say you are standing at t...” Say you are standing at the end of a building's shadow and you want to know the height of the building. you only know the length (40ft) of its shadow and the angle (say 35 degrees) from you to its roof. you could use the tangent trig function (tan35 degrees = b/40ft) (32 votes) Mari 4 years agoPosted 4 years ago. Direct link to Mari's post “This seems extremely comp...” This seems extremely complex to be the very first lesson for the Trigonometry unit. 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. No question, just feedback. • (18 votes) David Severin 4 years agoPosted 4 years ago. Direct link to David Severin's post “The problem with Algebra ...” 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. 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. So if you need to brush up on trig functions, use the search box and look it up or go to the Geometry class and find trig functions. (24 votes) Ram kumar 11 years agoPosted 11 years ago. Direct link to Ram kumar's post “In the concept of trigono...” 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). This is true only for first quadrant. how can anyone extend it to the other quadrants? i need a clear explanation... I think trigonometric functions has no reality( it is just an assumption trying to provide definition for periodic functions mathematically) in it unlike trigonometric ratios which defines relation of angle(between 0and 90) and the two sides of right triangle( it has reality as when one side is kept constant, the ratio of other two sides varies with the corresponding angle).... i think mathematics is concerned study of reality and not assumptions.... how can you say sin 135*, cos135*...(trigonometric ratio of obtuse angle) because trigonometric ratios are defined only between 0* and 90* beyond which there is no right triangle... i hope my doubt is understood..... if there is any real mathematician I need proper explanation for trigonometric function extending beyond acute angle. • (15 votes) Noble Mushtak 11 years agoPosted 11 years ago. Direct link to Noble Mushtak's post “[cos(θ)]^2+[sin(θ)]^2=1 w...” [cos(θ)]^2+[sin(θ)]^2=1 where θ has the same definition of 0 above. (8 votes) Scarecrow786 9 years agoPosted 9 years ago. Direct link to Scarecrow786's post “At 2:34, shouldn't the po...” At • (7 votes) Kyler Kathan 9 years agoPosted 9 years ago. Direct link to Kyler Kathan's post “It would be x and y, but ...” 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 (20 votes) chasephuayoung88 a year agoPosted a year ago. Direct link to chasephuayoung88's post “You should at least expla...” You should at least explain what sin and cosine are. This is intro to trigonometry • (8 votes) ks 10 months agoPosted 10 months ago. Direct link to ks's post “sin is opposite/hypotnuse...” sin is opposite/hypotnuse. cosine is adjacent/hypotnuse (5 votes) Bryan Bao 9 months agoPosted 9 months ago. Direct link to Bryan Bao's post “I'm a little confused. Ca...” I'm a little confused. Can someone help explain how to use a unit circle to help solve sine, cosine, and tangent? • (6 votes) Aristotle 9 months agoPosted 9 months ago. Direct link to Aristotle's post “Determine angle(theta) yo...” Determine angle(theta) you are finding the cosine, sine, or tangent of. Using SOHCAHTOA in relation to theta, the angle in the math equation you are searching for, then use SOHCAHTOA to search for the proper opposite, adjacent, or hypotenuse. SOHCAHTOA changes based on the location of the angle you are searching for. For example, if the angle was top right corner, instead of at the right angle, SOHCAHTOA measures and inputs would be different. Fast forward to based on this what sides correspond to the adjacent side, the opposite side, and the hypotenuse side BASED ON the location of theta you are solving for? now what is the sine, cosine, and tangent? (SOHCAHTOA to solve for the missing sides) NOW here is the final part to understand. The length of the "cosine" side will now act as the "x-coordinate", the length of the "sine" side will act as the "y-coordinate", and the hypotenuse will act and be measured by the y/x coordinate value. (the side length is like measuring an x coordinate, counting from 0 to 8 for example on the x-axis. the side value now equals 8 and the "cosine" coordinate will also be equal to 8; represented as (8,0) SOHCAHTOA will help you connect the dots, if that makes sense? the dots change in location to the angle you are solving for. When I say change location, this is because of either a clockwise rotation or counterclockwise rotation. this means that the "terminal side" will change location based on the math question's rotation and must constantly be reevaluated to ensure you are solving correctly. if you're still confused, let this information soak for a day and revisit this information. Rewatch the video explanation a few times too. Google "terminal angle/side trigonometry" and look at the graphs in images. Practice the practice problems and then revisit this information and do this a few times. If still confused, ask and I'll explain more. (9 votes) em 10 months agoPosted 10 months ago. Direct link to em's post “At minute 3:00 they talk ...” At minute • (4 votes) Kim Seidel 10 months agoPosted 10 months ago. Direct link to Kim Seidel's post “The unit circle by defini...” The unit circle by definition has a radius of 1, so the hypotenuse will always be 1. (9 votes)Want to join the conversation?
Using the unit circle diagram, draw a line “tangent” to the unit circle where the hypotenuse contacts the unit circle. 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). Extend this tangent line to the x-axis. The distance of this line segment from its tangent point on the unit circle to the x-axis is the tangent (TAN). If you extend the tangent line to the y-axis, the distance of the line segment from the tangent point to the y-axis is the cotangent (COT). 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. 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. You will find that the TAN and COT are positive in the first and third quadrants and negative in the second and fourth quadrants.
As a bonus, the distance from the origin (point (0,0)) to where that tangent line intercepts the x-axis is the secant (SEC). The sign of that value equals the direction, positive or negative, along the x-axis you need to travel from the origin to that x-axis intercept. The distance from the origin to where that tangent line intercepts the y-axis is the cosecant (CSC). 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.
Some people can visualize what happens to the tangent as the angle increases in value. When the angle is close to zero the tangent line is near vertical and the distance from the tangent point to the x-axis is very short. As the angle nears 90 degrees the tangent line becomes nearly horizontal and the distance from the tangent point to the x-axis becomes remarkably long. You can, with a little practice, “see” what happens to the tangent, cotangent, secant and cosecant values as the angle changes.
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. 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. When you compare the sine leg over the cosine leg of the first triangle with the similar sides of the other triangle, you will find that is equal to the tangent leg over the angle leg. Therefore, SIN/COS = TAN/1 . You can also see that 1/COS = SEC/1 and 1^2 + TAN^2 = SEC^2. A bunch of those almost impossible to remember identities become easier to remember when the TAN and SEC become legs of a triangle and not just some ratio of other functions. The angle line, COT line, and CSC line also forms a similar triangle.
-------------------------------------
When you graph the tangent function place the angle value on the x-axis and the value of the tangent on the y-axis. At the angle of 0 degrees the value of the tangent is 0. At 45 degrees the value is 1 and as the angle nears 90 degrees the tangent gets astronomically large. You are left with something that looks a little like the right half of an upright parabola. At negative 45 degrees the tangent is -1 and as the angle nears negative 90 degrees the tangent becomes an astronomically large negative value. This portion looks a little like the left half of an upside down parabola. This pattern repeats itself every 180 degrees. Do yourself a favor and plot it out manually at least once using points at every 10 degrees for 360 degrees. It may not be fun, but it will help lock it in your mind. 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.
40ft * tan35 = b
28ft = b
Now you can use the Pythagorean theorem to find the hypotenuse if you need it.
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. This is how the unit circle is graphed, which you seem to understand well.
Based on this definition, people have found the THEORETICAL value of trigonometric ratios for obtuse, straight, and reflex angles. This value of the trigonometric ratios for these angles no longer represent a ratio, but rather a value that fits a pattern for the actual ratios. I hope this helped!
Proof of [cos(θ)]^2+[sin(θ)]^2=1:
https://www.khanacademy.org/cs/cos2sin21/6138467016769536
2:34a²+b² = c² and they're the letters we commonly use for the sides of triangles in general. It doesn't matter which letters you use so long as the equation of the circle is still in the form a²+b² = 1. 8:00
Terminal side is like saying, based on the location of theta(angle) in this "unit circle equation"....... 3:00
Video transcript
What I have attempted todraw here is a unit circle. And the fact I'mcalling it a unit circle means it has a radius of 1. So this length fromthe center-- and I centered it at the origin--this length, from the center to any point on thecircle, is of length 1. So what would this coordinatebe right over there, right where it intersectsalong the x-axis? Well, x would be1, y would be 0. What would thiscoordinate be up here? Well, we've gone 1above the origin, but we haven't moved tothe left or the right. So our x value is 0. Our y value is 1. What about back here? Well, here our x value is -1. We've moved 1 to the left. And we haven't moved up ordown, so our y value is 0. And what about down here? Well, we've gone a unitdown, or 1 below the origin. But we haven't movedin the xy direction. So our x is 0, andour y is negative 1. Now, with that out of the way,I'm going to draw an angle. And the way I'm goingto draw this angle-- I'm going to define aconvention for positive angles. I'm going to say apositive angle-- well, the initial sideof the angle we're always going to do alongthe positive x-axis. So you can kind of viewit as the starting side, the initial side of an angle. And then to draw a positiveangle, the terminal side, we're going to move in acounterclockwise direction. So positive angle meanswe're going counterclockwise. And this is just theconvention I'm going to use, and it's also the conventionthat is typically used. And so you can imaginea negative angle would move in aclockwise direction. So let me draw a positive angle. So a positive angle mightlook something like this. This is the initial side. And then from that, I go ina counterclockwise direction until I measure out the angle. And then this isthe terminal side. So this is apositive angle theta. And what I want to do isthink about this point of intersectionbetween the terminal side of this angleand my unit circle. And let's just say it hasthe coordinates a comma b. The x value whereit intersects is a. The y value whereit intersects is b. And the whole pointof what I'm doing here is I'm going to see howthis unit circle might be able to help us extend ourtraditional definitions of trig functions. And so what I wantto do is I want to make this theta partof a right triangle. So to make it partof a right triangle, let me drop an altituderight over here. And let me make it clear thatthis is a 90-degree angle. So this theta is partof this right triangle. So let's see whatwe can figure out about the sides ofthis right triangle. So the first questionI have to ask you is, what is thelength of the hypotenuse of this right triangle thatI have just constructed? Well, this hypotenuse is justa radius of a unit circle. The unit circlehas a radius of 1. So the hypotenuse has length 1. Now, what is the length ofthis blue side right over here? You could view this as theopposite side to the angle. Well, this height isthe exact same thing as the y-coordinate ofthis point of intersection. So this height right over hereis going to be equal to b. The y-coordinateright over here is b. This height is equal to b. Now, exact same logic--what is the length of this base going to be? The base just ofthe right triangle? Well, this is goingto be the x-coordinate of this point of intersection. If you were to dropthis down, this is the point x is equal to a. Or this whole length between theorigin and that is of length a. Now that we haveset that up, what is the cosine-- let meuse the same green-- what is the cosine of my angle goingto be in terms of a's and b's and any other numbersthat might show up? Well, to thinkabout that, we just need our soh cah toa definition. That's the only one we have now. We are actually in the processof extending it-- soh cah toa definition of trig functions. And the cah part is whathelps us with cosine. It tells us that thecosine of an angle is equal to the lengthof the adjacent side over the hypotenuse. So what's this going to be? The length of theadjacent side-- for this angle, theadjacent side has length a. So it's going to beequal to a over-- what's the length of the hypotenuse? Well, that's just 1. So the cosine of thetais just equal to a. Let me write this down again. So the cosine of thetais just equal to a. It's equal to the x-coordinateof where this terminal side of the angleintersected the unit circle. Now let's think aboutthe sine of theta. And I'm going to do it in-- letme see-- I'll do it in orange. So what's the sineof theta going to be? Well, we just have to look atthe soh part of our soh cah toa definition. It tells us that sine isopposite over hypotenuse. Well, the oppositeside here has length b. And the hypotenuse has length 1. So our sine oftheta is equal to b. So an interestingthing-- this coordinate, this point where ourterminal side of our angle intersected theunit circle, that point a, b-- we couldalso view this as a is the same thingas cosine of theta. And b is the samething as sine of theta. Well, that's interesting. We just used our sohcah toa definition. Now, can we in some way usethis to extend soh cah toa? Because soh cahtoa has a problem. It works out fine if our angleis greater than 0 degrees, if we're dealing withdegrees, and if it's less than 90 degrees. We can always make itpart of a right triangle. But soh cah toastarts to break down as our angle is either 0 ormaybe even becomes negative, or as our angle is90 degrees or more. You can't have a right trianglewith two 90-degree angles in it. It starts to break down. Let me make this clear. So sure, this isa right triangle, so the angle is pretty large. I can make the angle evenlarger and still have a right triangle. Even larger-- but I can neverget quite to 90 degrees. At 90 degrees, it'snot clear that I have a right triangle any more. It all seems to break down. And especially thecase, what happens when I go beyond 90 degrees. So let's see if we canuse what we said up here. Let's set up a new definitionof our trig functions which is really anextension of soh cah toa and is consistentwith soh cah toa. Instead of defining cosine asif I have a right triangle, and saying, OK, it's theadjacent over the hypotenuse. Sine is the oppositeover the hypotenuse. Tangent is oppositeover adjacent. Why don't I justsay, for any angle, I can draw it in the unit circleusing this convention that I just set up? And let's just say thatthe cosine of our angle is equal to the x-coordinatewhere we intersect, where the terminalside of our angle intersects the unit circle. And why don't wedefine sine of theta to be equal to they-coordinate where the terminal side of the angleintersects the unit circle? So essentially, forany angle, this point is going to define cosineof theta and sine of theta. And so what would be areasonable definition for tangent of theta? Well, tangent of theta--even with soh cah toa-- could be definedas sine of theta over cosine of theta,which in this case is just going to be they-coordinate where we intersect the unit circle overthe x-coordinate. In the next few videos,I'll show some examples where we use the unitcircle definition to start evaluating some trig ratios.
