[MUSIC] In this video, we introduce the radian measure of angles, and extend the trigonometric definitions in the previous video by using points on the unit circle to define the sine, cosine, and tangent of any angle. This leads to the so-called circular functions, y = sin x and y = cos x and their graphs. We start off with the inner circle centered at the origin with radius 1. A typical point, x, y, on the circle, satisfies the equation x squared + y squared = 1. Related to the so called circular identity that we mentioned last time, there's also an angle we call theta between the positive x-axis and the radius that links the point to the origin. How do we measure angles? In the last video, we measured angles in degrees, and even referred to the fact that angles of a triangle add up to 180 degrees. It turns out for calculus that the radian measure of angles is more useful. But first, let's remind ourselves that there are 360 degrees in a single full revolution of the circle. This idea was introduced by the ancient Babylonian mathematicians. It's useful because 360 is a nice number. It's 2 times 180, which is 4 times 90, and so on. And there's so many different divisors that capture lots of useful common angles. We'll keep using degrees, especially for applications, but the radian measure turns out to be the best and most appropriate for calculus. Radians measure arc length on the unit circle. What does that mean? Here's a unit circle again. The point subtending an angle theta with a positive x-axis. The distance we travel along the circle to get to our point is the arc length. And this is defined to be the radian measure of the angle. For example, a full revolution of circle subtends 360 degrees. And the arc length is just the perimeter which is 2 pi times the radius which is just 2 pi since the radius is 1. Hence, 360 degrees corresponds to 2 pi radians. Half of a full revolution subtends 180 degrees and the arc length is just half of the full perimeter, which is pi. Hence 180 degrees corresponds to pi radians. Similarly, 90 degrees corresponds to pi on 2 radians and so on. So what should one degree be in radians? Well we just divide 2 pi by 360 which simplifies to pi on a 180 radians. What should a single radian correspond to in degrees? Well by reciprocating, 1 radian is 180 on pi degrees. And if you type this into a calculator, you get about 57.3 degrees. With radian measure, we drop the word radians and just use numbers. So if an angle is quoted as a number, you know that radians are intended. So 360 degrees becomes a number 2 pi. 180 degrees becomes a number pi and so on. A single degree is represented by the number pi on 180. The number one interpreted as an angle is 180 on pi degrees. Angles then are really just numbers, so it can be positive, negative, or zero. Positive angles are measured by anticlockwise rotation about the unit circle. Negative angles are measured by clockwise rotation. For example, if we move anticlockwise 45 degrees from the starting point, then we've moved pi on 4 radians. If we move clockwise 45 degrees, then we've moved negative pi on 4 radians. Here's some more angles. Note with this last pair, pi and negative pi, we end up at the same point on the unit circle. Angles are said to be equivalent if you end up with the same point on the unit circle. Angles that are large in magnitude can involve multiple revolutions of the unit circle. For example, an angle of 4 pi uses up two revolutions, an angle of 6 pi uses three revolutions and so on. The angles 2 pi, 4 pi, and 6 pi all end up back at the same starting point, so are equivalent. In the last video, we defined sin, cos, and tan of an acute angle in terms of certain ratios right angle triangles. It turns out there's a much more general way of defining these for any angle, exploiting coordinates of points that wind around the unit circle. Here's the unit circle with a point subtending an angle theta, with coordinates x, y say. We now define cos theta to be the x coordinate, and sine theta to be the y coordinate. You move directly to the x axis to get cos theta, the first coordinate and directly across to the y axis to get sin theta, the second coordinate. In this particular diagram the angle, theta, happens to be acute. And there's a right angle triangle formed by the radius, a piece of the x axis and a vertical line segment parallel to the y axis. The horizontal side of this triangle equals the adjacent over the hypotenuse since the hypotenuse is one, giving cosine of theta. And the vertical side length equals the opposite over the hypotenuse, giving sine of theta, again, since the hypotenuse is one. So the coordinates of the point reproduce the definitions of cos theta and sine theta that we gave in the last video, in terms of right angle triangles. But the definitions work anywhere on the circle. Regardless of where you are, you get the cosine of the angle by moving down and possibly out to the horizontal axis. And the sine of the angle by moving directly across to the vertical axis. We define tan of theta always to be sine theta divided by cos theta provided cos theta is not equal to 0. In this diagram, we're in the first quarter of the circle, also called the first quadrant where the coordinates are all positive. Thus, sine theta and cos theta are both positive, and tan theta is also positive. In the second quadrant, the angle theta is now obtuse, and the x and y coordinates have mixed signs. In this case, sine theta is positive. But to get cos theta, you drop down to the negative part of the x axis, so cos theta is negative. In this case, tan theta is the quotient of a positive by a negative number, so it will be negative. In the third quadrant, now both coordinates, cos theta and sin theta are negative. So the tan theta becomes a quotient of two negative numbers, which is positive. In the fourth quadrant, cos theta is positive and sine theta is negative, so tan theta becomes negative. There are some important extreme cases on the boundaries between the quadrants where one of the coordinates is zero and the other is plus or minus one. Corresponding to angle zero, pi on 2 pi and three pi on 2 radians respectively. For example, the point (1,0) equals cos of zero, sine of zero, corresponding to the angle zero. So sine of 0 is 0, and cos of 0 is 1. Hence tan of 0 is just 0 divided by 1, which is 0. The point (0,1) equals cos pi on 2 sin pi on 2 corresponding to an angle of 90 degrees. So sine of pi on 2 is one, and cos of pi on 2 is zero. So if we try to apply the formula for tan pi on 2, we get into trouble because this needs to be one over zero which is undefined. The other points for angles pi and three pi on two are discussed in more detail in the notes. But lets try to understand the geometric significance of the fact that tan of pi on 2 is undefined. Here's the unit circle again. Let's draw a vertical line that just glances the circle where it crosses the x axis at x equals one. This is called a tangent, or tangent line to the unit circle, touching the circle at the x intercept. Now at a point on the unit circle, say making an angle theta with a positive x axis. Then we extend the radius from the origin through this point until we touch the tangent. We can drop a vertical from the point on the circle down to the x axis, creating a right angle triangle with horizontal side length, x equals a say. And vertical side length y equals b say. So a is cos theta, the result of dropping down to the horizontal axis. And b is sine theta, the result of moving across to the vertical axis which form the coordinates of the point. The point where the line extending the radius meets the tangent has a vertical distance h, say, from the x-axis. What is h? Notice that the tangent is parallel to and located one unit from the y-axis. We can work out what h is by locating similar triangles which are highlighted in the diagram. The ratio b on a, coming from the small right angle triangle, is equal to h on one from the larger triangle, but this is just h. So h being b on a is just sine theta over cos theta which remember is tan theta. This relationship with the tangent line explains the terminology involving the word tangent. Thus, tan of an angle theta is just the length of this piece that you get by extending the radius through the point on the circle until you hit the tangent line. If the angle theta happens to be negative, say moving clockwise around the unit circle, but staying in the fourth quadrant, this method also works, but you hit the tangent below the x-axis, and tan theta is then measured as negative. Now we can explain why tan of pi on 2 or 90 degrees should should be undefined. Start off with the unit circle, and the vertical tangent line that touches the circle at the x intercept. Now move the radius anticlockwise, from x equals 1 on the x-axis, along the circle towards y equals 1 on the vertical axis. But extend the radius until it touches the tangent. Make note of the vertical distance from this point on the tangent line to the x-axis. And just keep repeating this as the radius rotates to get closer and closer to y equals 1 on the y-axis. Very quickly, we run out of room on the tangent line in the diagram. The vertical distances from the point on the tangent line to the x-axis, are the tans and the angles and these vertical distances shoot off towards infinity. As theta, the angle made with the positive x-axis gets closer and closer to pi over two, tan theta heads towards infinity. This explains geometrically why tan on pi over 2 should be undefined. Now informing sin theta and cos theta, we're really thinking of theta as an input into a function. But remember, it's traditional to use x as an input. So here's the unit circle again, but think of x now as the arc length subtended by the angle. So x becomes the angle itself. From the unit circle we can read off the sin and cosine angle by moving directly across the vertical axis to get sin of x and straight down to the horizontal axis to get cos of x. Let's just focus on sine x for the time being. Notice as x moves from zero to pi on two covering the first quarter of the unit circle, sine x has moved from zero to one. The blue dot on the circle has carried the pink dot representing the sine of x up the vertical axis. As x continues then to move from pi on one to pi, and sine of x, the pink dot, comes back down from one to zero. As x continues then to move from pi to three pi on two, sine x moves from zero to minus one. Finally, as x moves from three pi on two to two pi, sine x moves from negative one to zero and we're back where we started. As x continues to wind around the circle as many times as you like, sine x will oscillate backwards and forwards between zero, one, zero, minus one, and zero. Now form the function with the rule y=sinx. We can ask, what does this graph look like? Here's part of the graph for x=0 through to x=2pi capturing this oscillation effect that we just looked at a moment ago using a pink dot representing the y values. This curve is called sinusoidal. And the shape continues to repeat as x moves further along the positive x axis, winding through the unit circle as often as you like. This captures movement anticlockwise around the circle. Remember, clockwise movement corresponds to negative angles. So if we allow x to move to the left along the negative half of the real line. And sine x will also oscillate, and the sinusoidal graph repeats ad infinitum to the left, as well as to the right. Thus we get the full sine curve extending along all of the real line, creating a wave that oscillates infinitely often. Let's try to understand now how cosine of x behaves. We'll go around the unit circle again, but remember the cosine of x is located on the horizontal axis represented here by a pink dot again. As the angle x moves from zero to 2 pi, as the blue point makes a full revolutionary circle, the cosine of x moves from 1 to 0 to -1 and back from -1 to 0 to 1. If we continue to wind around the circle, cos x just repeats this pattern. Now form the function with rule y equals cos x. What does its graph look like? Here's the part of the graph x equals zero through x equals 2 pi, capturing this oscillation effect of the y values represented a moment ago by the pink dot. And this pattern is repeated ad infinitum to right and left forming the cosine curve. If we compare it with a sine curve, you'll notice that both have the same shape and only differ by a horizontal translation. So the cosine function is also sinusoidal. The functions y = sine x and y = cos x are said to be circular because a pair of those cos x sine x represents points on the unit circle. There are also so-called hyperbolic functions which form pairs of those that represent points on a hyperbola. But discussion of them goes beyond the scope of this course. The graph of the function y equals tan x is very interesting, and we'll save that up for the next video when we introduce inverse trigonometric functions. Again we've covered a lot of ground in a short space of time. We've introduced radian measures of angles using arc lengths from unit circle and shown how to convert degrees into radians, and back again, introduced rules for forming sine and cosine of any angles using coordinates on the unit circle. And discussed how to form the tangent of an angle, provided the cosine is nonzero and how the tangent of an angle arises geometrically. We defined the circular functions y sin x, and y cos x and illustrated how the graphs arise as sinusoidal curves. Please read and digest the notes which contain more details and when you're ready, please attempt the exercises. Thank you very much for watching, and I look forward to seeing you again soon. [MUSIC]
