Hello wonderful mathematics people. I'm Anna Cox from Kella Community College. Trigonometric definitions. We're going to look at a circle with the center at 00 and a radius R. The equation of the circle is X ^2 + y ^2 equal R-squared, where X&Y are the sides or the legs of a triangle and R is the hypotenuse or the side opposite the right angle. We're going to look at unit circle where unit circle is the center still at the origin 00 but its radius is 1. So CAH toa is one way that some people remember the definitions for sine, cosine and tangent. Sine being opposite over hypotenuse, so the sine opposite over hypotenuse, CAH cosine adjacent over hypotenuse to a tangent opposite over adjacent. So when we look at this, the sine is the opposite leg of the angle over the hypotenuse of the triangle, cosine is the adjacent leg of the angle divided by the hypotenuse of the triangle, and tangent is the opposite leg divided by the adjacent leg. We also have the reciprocals of these. So cosecant Theta is 1 divided by sine Theta or r / y Secant Theta is 1 divided by cosine Theta, which is r / X. Cotangent Theta is 1 divided by tangent Theta X / y. We also frequently think about tangent Theta, which a moment ago we said was y / X. We could think of Y as sine over cosine because if we have y / r / X / r, those Rs would cancel out and we'd get y / X. Hence cotangent can be thought of as cosine over sine and a unit circle X ^2 + y ^2 equal 1. Now remember, cosine Theta is just the adjacent side of the right triangle over the hypotenuse. So in a unit circle it's X / 1 or plain old X. Sine Theta is the opposite side of the right triangle over the hypotenuse, or y / 1, or just Y. And tangent Theta is y / X. So we want to look at the quadrant angles. If we think about this location here, its X value is one because we want one to the right, its Y value is 0. Now if we wanted the tangent, the tangent would be 0 / 1 or 0. So in these ordered pairs, actually ordered triplets, we're going to write it cosine, sine, tangent. And the reason is the X is our cosine, the Y is our sine, and the tangent is y / X for the unit circle. So here we went left to right zero, but we went up one and 1 / 0 is undefined. We're going to use AU there. Over here we went left one up zero and 0 / -1 is 0 and here we went zero -1 negative 1 / 0 is undefined if we think about these angles. This went 90° which is also Pi have radians, this one is 180° which is equal to π radians, this one 270° or three π / 2 radians, and finally 360° or 2π radians. This location is also thought of as 0° 0 radiance and a 4545 right triangle. We know that the two sides of the triangle are the same, so in our unit circle we have other angles besides the quadrant angles. We have 45. This is our central angle. If it's a unit circle, we know the radius is 1, so we know that the two legs of the triangles, let's call them a, have to be the same length. So a ^2 + a ^2 has to equal 1 ^2 by the Pythagorean theorem. If we solve for a, we get 2A squared equaling 1A squared equal 1/2, A equal positive negative sqrt 1 half. And then if we want to rationalize that, we would multiply by sqrt 2 / sqrt 2 and that would give us a equaling positive negative sqrt 2 / 2. Now, because we're actually referring to a distance, we're going to talk about just the positive root 2 / 2. So it means that X went root 2 / 2 to the right or left and root 2 / 2 up and down. So if we think about our cosine sine, remember our X is like our cosine in a unit circle, our Y is like our sine in a unit circle, and our tangent is y / X so cosine sine tangent. So what happens here is we have root 2 / 2 for our X value or our cosine root 2 / 2 for our Y value and one for our tangent. Now we have other angles in this unit circle that have 4545 right triangles. This in here would be a 45° triangle. It's just a reflection. Well, if we look at this angle here, that angle is 180, the straight line -45 or 135° angle. This is 135°, which is also three Pi 4th radiance. And the way this location works is instead of going right root 2 / 2, we've now gone left. So our X value would be negative root 2 / 2. For our cosine, we still went up, so root 2 / 2 for our Y and then a positive divided by a negative is a negative. So -1 we have another one down here using the same concept. Here's 45°. Well, now we want this entire angle going this way. So we have our straight line of 180 plus the 45 now. So this new angle is 225° or five Pi force. This time we went left and we went down. So it's negative root 2 / 2 for our cosine, negative root 2 / 2 for our sine, and +1 for our tangent. And the last angle would be in quadrant 4. This time it's all all the way around, and the easiest way to do this one is to think about it's the full circle minus the 45° minus this piece in here. So 360° -, 45° would give us 315°, which is 7 Pi fours. This time we've gone to the right, so our cosine is positive, root 2 / 2 we've gone down, so our sine is negative, root 2 / 2, and a negative divided by a positive is a negative. So the tangent is -1. Our unit circle also has some other special angles, our 306090 right triangle. If we thought about putting in a sixty degree 60° equilateral triangle, this angle would be 60 and this one's 60 and this one's 60, and we know that the radius of the unit circle had to be one. If we then drop an altitude, what does that do to that top angle? It splits it in half. So now we have a 30° and a 60° and a 90° right triangle. Well, if it splits the angle in half, it also splits the side in half. So let's call this 1/2 and 1/2. Now could we actually find the height? Well, this height, let's call it H using the Pythagorean theorem, we know that 1/2 ^2 plus H ^2 has to equal 1 ^2. 1/2 ^2 is one 4th H ^2, 1. Subtract we'd get H ^2 equaling 3/4 because one is 4, fourths -1 fourth. If we square root, we'd get positive negative sqrt 3 / 2 because sqrt 4 is 2. So this point up here is going to be a 60° angle. 60° angle is π third's radians, and the cosine would be 1/2. The sine. How high did we go would be root 3 / 2 and the tangent is sine divided by cosine. So those twos are going to cancel and we'd get root 3. Now, just like in the 4545 triangles, we have lots of angles on this triangle that will be 60°. So here's another 60° triangle. How do we find this angle? Well, it's a straight line 180 minus the 60° in here. So it's going to be 120° or 2π thirds radians. How far left did we go? Well we have the reflection so we went left half, so it's going to be a negative 1/2. For the cosine we still went up, so we have root 3 / 2 and then sine divided by cosine would give us negative root 3 for the tangent. Another triangle in this unit circle with a 60° would be right here. Well, how do we find what this angle is? It's that full line plus another 60. So 180° + 60 would give us 240°, which is going to be 4 Pi thirds. This time we went and left, so our cosine is negative 1/2. We went down, so our sine is negative root 3 / 2 and our tangents a negative divided by a negative or root 3. And our last triangle in this unit circle with a 60° is right here. Well, how do we find this angle? It's the full circle minus that 60°. So now we have a three 100° which would also be five Pi thirds. We went to the right this time, so we're going to have 1/2. We went down negative. Root 3 / 2 is our sine and our tangent. A negative divided by a positive is a negative. An important angle on our unit circle is also a 30°. And if we thought about using an equilateral triangle having another 30° coming down this direction connecting those endpoints, we would then have radius here of one, radius here of one, IE it's a equilateral triangle which we add the 30 and 30, which means that this here had to be a 60 and that's a right triangle. This base then is going to be split evenly, so that base is 1/2. Now we need to find this length here. Let's call it B for the base of a triangle. So b ^2 + 1/2 ^2 would have to equal 1 ^2 because it's a right triangle by Pythagorean theorem. So b ^2 would equal 1 - 1/4 or b ^2 = 3/4. B equal positive negative sqrt 3 / 2. So when we look at this location, this is π six which is also 30° and we went to the right B value of root 3 / 2. We went up 1/2 and the tangent is just sine divided by cosine, which would be one over root 3, but we always rationalize, so it would be root 3 / 3. One over root 3 rationalizes by multiplying the top and the bottom each by root 3, so we'd get root 3 / 3. Now we have some other triangles on this unit circle that we get by reflection. If this degree here is 30°, how do we find from the initial ray over? Well, it'd be the straight line of 180 -, 30°, so we'd have 150°, which is also five Pi 6 radians. We went to the left this time, so it's negative root 3 / 2. For our cosine, we went up so it's positive 1/2 and then the sine divided by cosine would give us a negative root 3 / 3. We also have a triangle in the third quadrant. If it's 30° here in the third quadrant, how do we get this new angle? Well, it's a straight line of 180 plus our 30, so it's 210°, which is also seven Pi 6 radians. We went to the left this time, so it's negative root 3 / 2. For our cosine, we went down, so it's negative 1/2 for our sine and then tangent. The sine divided by cosine negative divided by a negative is a positive. So root 3 / 3 over here. How do we get this angle? Well it's a full circle, IE 360 minus our 30°, so it's 330° which is 11 Pi 6 radians. This time we went right so we went root 3 / 2 but we went down so we have -1 half and then the tangent is just sine divided by cosine negative divided by a positive is a negative. Thank you and have a wonderful day. This is Anna Cox.