Welcome to the circle. Circles symbolize a lot of things, and they’re a basic shape for building life. To understand circles, we need to understand triangles. First we’ll review the XY coordinate system, and I tried to make it fun. Well, it was fun, for me. I love math, drawing, music … { music by Louise Goffin: Sometimes a circle feels like a direction } Thanks for the perfect song, Louise. We draw a circle and then figure it out Circles have angles, { music by Louise Goffin: This noise in my head. Some things are easier done than said } and we’ll find them Understanding the Unit Circle helped me tremendously. I hope it can help you too. Hi, this is crystal. Here’s my point, or maybe, your point. it is the Origin, the place you start. To give it some reference, this is an x-axis going through it horizontally, and a y-axis going through it vertically. Imagine that you are that point. Let’s make you a little bigger. Here is where you stand. Now, imagine a grid around you. It goes from -5 to +5 on the x-axis, which is horizontal, and from -5 to +5 on the y-axis, which is vertical. The address of a point is called a coordinate. A coordinate is a pair of values where the first number shows horizontal distance to the left or the right, along the x-axis, and the second value shows the vertical distance down or up on the y-axis. A coordinate is a set of 2 numbers, separated by a comma, and written in parentheses. You are at (0,0) From now on, I’ll just say the coordinate values without saying parentheses. aah, warm sunshine When you take one step to the right, not up or down, just one step horizontally, your coordinate is (1,0) Numbers to the right of the origin are positive. One more step to the right, more positively, the coordinate is (2,0) Numbers to the left of the origin are negative. If you walk 3 steps back, from where you are, 2-3, you are left of the origin at coordinate (-1,0) So far, you are only moving horizontally along the x-axis. Whew, its getting cold. aah, steps to the sun! When you walk left and right, you move along the x-axis. When you go up steps, you can also climb. Then you are also changing your y-coordinate. After climbing the steps, you’re closer to the sun, and that makes you happy. Not only did you move along the x-axis, but you also moved up the y-axis. You went over 3 along X, from -1 to 2, and up 3 on Y, from 0 to 3. The coordinate is now (2,3) On the y-axis, numbers up from the origin are positive, and numbers down are negative. { music by Louise Goffin: Up and down, still lookin’ for perfection } Nice arabesque, Leigh Ann, but it can be dangerous doing arabesques on the top of steps Here comes Mr Wind and he blows you off your step As you bounce down, your y-coordinate decreases Oh no, its raining too! You’re slipping and sliding and spinning. As you move around system, your XY coordinates change. Now you bounce up, over, and down, landing into the soft, peaceful water. The rain stops and here comes the sun again A lovely rainbow! Oh, my lucky stars, what a big pot of gold. The XY coordinate system can be divided into 4 quadrants. The first quadrant is NE of the origin, where x and y are both positive In the second quadrant, NW of the origin x is negative and y is positive In the third quadrant, SW of the origin x and y are both negative. In the fourth quadrant, SE of the origin, x is positive and y is negative Back at the origin, your point, you see a another point, and draw a line of sight to that point. If you continue to look around as you draw points from your line of sight, you get a circle. A circle is a round shape where all points on the circle are the same distance from the center. You can think of a circle as a perimeter that encloses the area inside it. The perimeter distance around a circle is called a circumference. When you flatten the circumference into into a straight line, you can more easily measure it, and see it is a little over 6 units long, where this circle is 1 unit wide and 1 unit tall. The line from the from the center to the edge is called a radius. Therefore, a radius is a line from the center of a circle to a point on the edge of a circle. If we were to extend the radius to another point on the circle, to go all the way across, we get a diameter. A diameter is a line that goes from one point on a circle to another and passes through the center. The diameter is represented by a variable called d. It’s a variable because it can vary. Now lets label our grid with numbers. If the diameter is 1, then we can see that the circumference is a bit more than 3. The ratio of the circumference of a circle to its diameter is called Pi. This is true for every circle, no matter how big or small it is. Pi is irrational, which means it can’t be expressed as whole number or a fraction. Pi turns out to be about 3.14 The digits go on and on with no pattern. Pi with five decimal places is 3.14159 and Pi with 15 decimal places is 3.141592653589793 Even though there isn’t a repeating pattern for pi in decimals, there is a pattern to calculate it. Pi is 3 + 4/(2*3*4) – 4/(4*5*6) + 4/(6*7*8) – 4/(8*9*10) + … – … plus, minus, da dada. Its always 4 over … and the denominator will then just repeat the pattern. We can approximate Pi with fractions. 22/7 yields 3.14 but the digits after those 2 decimal places are different. 355/113 is a closer fraction for approximating pi, accurate to 6 decimal places, 3.141592 Many have calculated pi out to lots and lots and lots of decimals places, and it just never repeats. Why is this ratio called Pi? Pi is the greek letter for ‘p’. Perimeter starts with P. The area inside a circle can be calculated using pi r^2 { music by Louise Goffin: Sometimes a circle feels like a direction } Here we are … back to a point. { music by Louise Goffin: … like a direction } The distance around the circle is called ? … the circumference. It is abbreviated as C and is always equal to pi times the diameter. The diameter of a circle is the radius times … 2 The radius of a circle is half the diameter; and the diameter is twice the radius. In our equation for circumference, we can substitute 2r for d. r means radius and d means diameter. C is? the circumference. 2 is constant because its value doesn’t change. 2 is always 2 (and pi is always pi) no matter what anything else in the equation is. r is a variable because the radius changes, or varies, depending on how big the circle is. By convention, constants are displayed first, so our equation becomes C=2 pi r The length of the perimeter, the circumference, is equal to 2 times pi times the radius of the circle. How many degrees are in a complete circle? A circle is 360° It is also 2pi radians. Radian is the SI unit for measuring angles. If we divide 360° by 2pi, we find out that a radian is about 57.3° Since a circle ends where we start, one can say that a circle never ends. This point is 0 … its also 2 pi, and 4 pi, and 6 pi, and 8 pi, and pi pi on How many radians are in half a circle? What’s 360° divided by 2? It’s 180°. What’s 2 pi divided by 2? It’s pi. By convention, angles are measured counter-clockwise from the positive x-axis. Half of a half is a fourth, and a fourth of a circle is half of pi radians, so it’s pi/2 radians, and it’s half of 180°, so it’s 90° Three quarters of a circle is 1-1/2 times pi, which is generally expressed as 3 pi/2 radians. This is also 270° The original radius we drew is actually at an angle of half of 90° — does it look like that? 45°, or … how many radians is it? Whats a fourth of pi? pi/4. It’s pi/4 radians Halway between pi/2 and pi is 3/4 of pi This is also 135° Here, x is negative while y is still positive. To put the pi fractions all in terms of fourths, I’ll label the angles we’ve already covered for pi to be over 4. 1/2 pi is 2/4 pi. 1 pi … what’s 1? … how many fourths is 1? 1 pi is 4/4 pi. 3/2 pi is 6/4 pi and 2 pi is 8/4 pi. In the third quadrant, or SW quadrant, halfway between 4 pi/4, and 6 pi/4, is 5pi/4 … or how many degrees? What’s halfway between 180° and 270? 225° Here, both x and y are negative. What’s halfway between 6 pi/4 and 8 pi/4? 7 pi/4. This is the middle of the fourth quadrant, and also 315° We’ve gone all the way around a circle. Hopefully you’re getting comfortable with pi radians instead of degrees Degrees, totally unrelated, are also used to measure temperature. Its Fall and the trees are turning beautiful colors Its 60°F outside and you might also hear the pitter pat of rain. Lets focus on our radius at pi/4, or 45° If we draw a vertical line from the point on the circle down to the x-axis, the height of the line is the y-coordinate. That tells us how far up on the y-axis we are. If we draw a line on the x-axis to the place under the point on the circle, the width of this line is the x-coordinate. What we end up with is a triangle, specifically a right triangle. A triangle is called a “right” triangle when one of the angles is 90° since a right angle is — how many degrees? A right angle is 90°. The little square where the right angle is indicates that angle is 90°. Pythagoras was an ancient Greek mathematician. He discovered that a^2+b^2=c^2 for right triangles. This equation is known as the Pythagorean theorem. a and b are the short sides of the triangle. c is the long side, which is also called the hypotenuse. Changing the variable names to our example, a and b, the short sides, are x and y c, the hypotenuse, is our radius, r. For a circle, we can write the Pythagorean theorem as x^2 + y^2=r^2 45°, or pi/4, is a special place because the x and y coordinates are equal, so it’s a good place to use the Pythagorean theorem to determine the actual values of x and y. Whenever you’re solving an equation, it’s a good idea to list the “knowns” That means you should write down what you know. At this point, x=y Since this is a unit circle, the radius is 1 so r=1 1×1 , or 1 squared, is still 1, so this makes things a lot easier. Let’s make some substitutions. x^2 + y^2=1 when you’re trying to solve an equation with more than one variable, and you know how the variables relate to each other, get rid the other variables by writing them in terms of the one you will solve for. Since x=y, then y=x. Therefore, x^2 + x^2=1 We can simplify this to be 2x^2=1 If we divide each side by 2, we get x^2=1/2 If we take the square root of each side, we get x=SquareRoot(1/2) Square roots are hard to calculate. Lets change 1/2 from a fraction. 1/2 is also 0.5, so x is the square root of 0.5 Let’s estimate it. 7×7 is 49 0.7×0.7 is 0.49 So we know that the square root of 0.5, is a little more than the square root of 0.49 x is actually 0.707 since y=x, then y is also 0.707 Remember, we’re on a unit circle, so the biggest number we’re going to have on the circle is 1 And the smallest number will be -1. With me so far? Great, you’re doing great. Now that we’ve calculated what x and y are, when the angle is 45°, or pi/4 using the Pythagorean theorem, lets put our grid back. This time, distances are labeled, so you can see that x and y indeed look correct. Trigonometry is the branch of mathematics that studies relationships between side lengths and angles of triangles In Greek, ‘tri’ means 3 and ‘trigonon’ means 3 angles. The Greek word ‘metron’ means measure. A closed shape using straight lines with 3 angles is a triangle. Our pi/4 and 45° angle is represented by a little arc across the angle. This is commonly labeled with the Greek letter ‘theta’. The sine of an angle is defined to be the length of the opposite side divided by the length of hypotenuse, or long side. For our triangle, this is the y-coordinate divided by the radius, or simply, y/r The cosine of an angle is defined to be the length of adjacent side divided by the length of hypotenuse. For our triangle, this is x/r Another basic trignometric function is Tangent. This is the change in y over the change in x. In our example, this is y/x, which is also Sine divided by Cosine. Lets make a table to write down what we’ve calculated, or can observe. We’ll list angle, sine, cosine, and tangent. This is where visualizing the unit circle is super helpful. Lets start with 0° Look at our point on the circle. At 0°, what is the y-coordinate? It is 0. That’s the sine What’s the x-coordinate? It’s 1. That’s the cosine The tangent is y/x. 0/1 is 0 At 45°, or pi/4, what’s the y-coordinate? We figured out that this is 0.707, which is the sine. What is the x-coordinate? Here, x and y are equal, so x is also 0.707 That’s the cosine. The tangent is y/x and since they’re both the same, the tangent at 45° is 1 The slope of a line is the change in y over the change in x, so the tangent is also the slope. When our arc is a fourth of the circle, the angle is pi/2, or 90° The sine is the y-coordinate, which is 1 The cosine is the x-coordinate, which is 0 The tangent of 90° would be 1/0, but math doesn’t let you divide by 0, so the tangent of 90° is undefined. At 3pi/4, or 135°, which is 90° + 45°, x is negative and y is still positive. The sine of 135° is the same as the sine of 45°, 0.707 In the second quadrant, x is negative. Its amplitude, or absolute value, is the same as x in the first quadrant, so the cosine is -0.707 The tangent is sine divided by cosine. Since the amplitude of both of the values is the same, but one of them is negative, then the tangent is -1. When we are halfway around the circle, we are on the x-axis, so the y-coordinate is 0. This is the sine. The x-coordinate, or cosine, is -1. Tangent is y/x. 0 over any number (except 0 since it’s not defined) is 0, so the tangent is 0. In the third quadrant, at 5 pi/4, which is 180° + 45°, both x and y are negative. The sine is -0.707, and so is the cosine. Any number divided by the same number is 1 So the tangent here is also 1, just as what we got for 45°. A positive tangent means the direction of a line is from SW to NE. If we extend the radius from the 3rd quadrant, we can see the slope is the same as in the first quadrant. Three quarters of the way around the circle, at 3 pi/2 radians, we can see that the value of y is -1. That’s the sine. X is 0, which is the cosine. The tangent is -1/0. Again, math can’t divide by 0, so the tangent is undefined. In the fourth quadrant, at 7 pi/4, the value of y is negative, and the value of x is positive. The angle in degrees is 270° + 45°, which is also 360°-45°. The sine is -0.707, same as the third quadrant, and the cosine is 0.707, same as the first quadrant. The tangent is -1, which is the same tangent at 3 pi/4. A negative tangent means the direction of a line is from NW to SE. 360°, or 2 pi, takes us back to 0. The sine is 0, the cosine is 1, and the tangent is 0. A tangent of 0 means that there is no change in y as x changes because the line is horizontal. Y is always the same. There are charts to lookup sine, cosine, and tangent for any angle. We calculated them so you can understand where they come from. We didn’t talk about tangent much, except to calculate it. In case you didn’t pick it up, the slope of our radius represents the tangent. What if you know x and y but not the angle? You can use an inverse trigonometric function called ArcTangent to determine the angle. This might be abbreviated as arctan, atan, or, simply, atn. What if you know x and r but not the angle, what could you use to get the angle? ArcCosine How about if you know y and r? You can use ArcSine to get the angle. If you’ve followed all of this, fantastic! That means you’ll forever understand angles and trignometric functions. Its not rocket science. When you can visualize it though, it’s really simple. What good does it do to know this? How can you apply this knowledge? Why might this be useful? Lets say you want to know how tall a tree is? Do you have to climb to the top and drop a line? No, there is a much easier way. We can use the fact that triangles with the same angles have the same ratios between their side lengths. Ask a friend to stand between you and the tree with a pole. Then move until your line of sight to the top of the tree is also at the top of the pole. Mark your spot! x1 is the horizontal distance to the pole x2 is the horizontal distance to the tree y is the height to your eyeballs, from the ground y1 is the vertical height between your eye height and the pole top, so subtract your eyeball height from the pole height. y2, what we want to figure out, is the height of the tree – the height to your eyeballs The ratio between y2 and x2 is the same as the ratio between y1 and x1. Therefore, y2=y1/x1 times x2 Once you calculate y2, add y, the height to your eyes, and you will know the height of the tree. You can use trigonometry to figure out the height of buildings and mountains; to visualize sound waves and adjust parameters like pitch, volume, attack, and decay; how to make darts and figure out dimensions of fabric for sewing; and to map terrains and mark boundaries in surveying. Trigonometry is used in electronics and machining parts, building bridges, roads, houses, and other construction, sciences such as architecture, aviation, engineering, and physics. Trigonometry is also used in video games. And you can use Trigonometry to figure out where a bullet came from and the location of cars, where they were before they collided in an accident. In pool, if you try to play it well, you can also use trigonometry to make that shot. Whether you think about it or not, Trigonometry plays a part in a lot of what you do every day. We discussed points, lines, and the XY coordinate system. We drew a circle and showed how the circumference of a circle is 2pi * r We learned about trignometric functions like sine, cosine, and tanglent. We got comfortable with radians and degrees. What prompted me to make this video was to provide background to teach how an analog clock application I made for Access works. By knowing an angle and a radius you can calculate XY coordinates of any point. As I was creating this tutorial, I saw a bigger purpose. I hope this helps you as much as its helped me. The graphics for this video were drawn with … ready for it … { drum roll } Microsoft Access! Access is a database management application, not a graphics program, but if you can imagine it, and simplify it, Access can do it. Thanks to Louise Goffin for her great circle song. { music by Louise Goffin: Sometimes a circle feels like a direction } Louise grew up with musical parents, who taught her about music and encouraged her to develop her talent. And she did! Its no surprise that her music is billiant and wonderful. Thanks, Michelle Johnson, also a great musician and performer, for putting me in touch with Louise. I’d also like to thank my mother for teaching me the Unit Circle at a young age. It’s taken many years to realize that what I understand now is largely because she taught me so much about math. Thanks, mom Are you a teacher? Do you want to show this to your class? If you can’t get to YouTube during class, you can download this lesson free. Email me and I’ll send you test questions too! If you’re looking for a math tutor, or someone to help you with an Access application, I’d love to hear from you. Visit my website, msAccessGurus.com, and send me a message. Connect to me, let’s build it together. Thanks for joining me. Through sharing, we will all get better.