In this video, we introduce the sine, cosine and tangent of an acute angle, which are essential ingredients of trigonometry, leading to general definitions of circular functions in the next video. Trigonometry is the study of triangles through a careful analysis of the relationships between lengths and angles. Consider a right-angled triangle with side lengths a, b and c, where c is the hypotenuse, and a and b are the shorter side lengths. Draw the angle between the sides a and c, and give it a name theta. Notice that theta is acute. That is, lies between zero and 90 degrees. We say that the side a of the triangle is adjacent to theta, and side b is opposite theta. Now define the sine, cosine and tangent to the angle theta by the following rules. Sine theta is b on c, the opposite side of the hypotenuse. Note that it's conventional to abbreviate sine by dropping the e. Cos theta is a over c, the adjacent side of the hypotenuse. Note that it's conventional to abbreviate cosine by dropping the ine, and we say cos. Tan theta is b over a, the opposite over the adjacent. Note that it's conventional to abbreviate tangent to just T-A-N, tan. There's a connection to tangent lines which we will explain in the next video. By the theorem of Pythagoras, a squared plus b squared equals c squared. So, taking this equation and dividing through by c squared, we get a squared on c squared, plus b squared on c squared equals one. So that a on c all squared plus b on c all squared equals one. So that in line of our definitions, cos theta all squared plus sine theta all squared equals one. It's conventional to write cos squared theta for cos theta all squared. In this way to try to remove some brackets, which is useful decongest notation when formula become complicated. Similarly, sine squared theta for sine theta all squared. Then, the earlier equation becomes cos squared theta plus sine squared theta equals one. Or equivalently, sine squared theta plus cos squared theta equals one. This is called the circular identity, related in a natural way to circles as we'll see in the next video. If we take the ratio of sine theta with cos theta, something nice happens. Sine theta divided by cos theta is b over c divided by a over c, which equals b over a as the c's cancel, which just becomes the definition of tan theta. Thus, we see that tan theta is just sine theta on cos theta. Let's work out some common trig values. Trig is an abbreviation for trigonometry. Consider first an isosceles triangle where the two shorter sides have equal length. There's no harm scaling and up or down so that its hypotenuse has length one. Using a hypotenuse of one, get you used to interpreting trig values in terms of the unit circle discussed in the next video. Because the angles add up to 180 degrees, and the other two angles are equal, they must both be 45 degrees. Hence theta in our diagram becomes 45 degrees, and b equals a. Thus, one is one squared, which is a squared plus b squared, which is 2a squared. So, a squared is a half. So, a is the square root of a half, which is one over the square root of two, which can you leave just like that. We can also write it as root two over two. Hence the two shorter sides of the triangle are both one over the square root of two. So, sine 45 degrees is opposite over hypotenuse, which is one on root two, divided by one, which is just one over root two. Cos of 45 degrees is adjacent over hypotenuse, which is also one over the square root of two. Tan 45 degrees is opposite over adjacent, which is one over root two over itself, which is just one. Let's next consider a right-angled triangle with hypotenuse one and angle theta equal to 30 degrees. Just from the diagram, the opposite side length you might guess to be half the length of the hypotenuse. So, it should be a half. This turns out in fact to be true. Why is that? To see why it must be a half, you can add the reflection in this triangle in the horizontal. Putting the two triangles together, makes a larger triangle with all angles equal to 60 degrees. This is easy to see because the angle in the middle is got by doubling 30 degrees, and the other 60 degree angles are forced by the fact that angles of a triangle add up to 180 degrees. But then this larger triangle is equilateral. So, all side lengths are equal to one. So, half of the vertical side length, it's just one half, confirming our earlier visual intuition or guess. What then is the adjacent side length a in the original 30 degree triangle? By Pythagoras, one is a squared plus a half all squared, which is a squared plus a quarter. So, a squared is one minus a quarter, which is three-quarters. So, a is equal to the square root of three quarters. That is, a is the square root of three divided by two. So, now we have complete information to read off. Sine 30 degrees, which is a half over one, just a half. Cos of 30 degrees is root three on two over one, which is root three on two. Tan 30 degrees is a half divided by root three on two, which is one on root three. We can use the same triangle to read off trig values for another case, because the angles add up to 180 degrees. So, the remaining angle must be 60 degrees. To make it easier or clearer how to read off the values with our setup, first, just flip over and reorient the triangle. So, the angle 60 degrees is between the hypotenuse and the horizontal side. Now, we can read off sine 60 degrees, opposite over hypotenuse. That's root three on two. Cos 60 degrees, adjacent over hypotenuse, which is just a half. Tan of 60 degrees, opposite over adjacent, which is just root three. Here's the complete table for sine, cos and tan theta for those special angles, theta equal to 45, 30 and 60 degrees respectively. Notice that surd expressions involving the square root of two and square root of three arise quite naturally. In advanced trigonometry, there are more complicated surd expressions for other angles, but we won't need them. Typically, it suffices to use your calculator for most angles. It's an important issue we'll get to later concerning the way angles are measured. For calculus, the most natural angle measure uses radians. We'll discuss that in the next video. We'll finish with an example where we don't even know the value of the angle, but can do some detective work to deduce information about trig values. Suppose we have an acute angle theta, such that tan theta is three. Find sine theta and cos theta. This looks at first sight like there just isn't enough information. We don't even have a diagram of a triangle to guide us. Well, we can quickly remedy that, and a rough sketch suffices. For tan theta to be equal to three means that in a right-angled triangle with angle theta, the ratio of the opposite side length to the adjacent side length is three. So, we draw a triangle with a vertical side length three times as long as the horizontal side length. The diagram is just a guide to finding the solution and doesn't have to be accurate. Label the hypotenuse h say. By Pythagoras, h is the square root of one squared plus three squared, which is the square root of 10. But, now we have enough information to read off sine theta and cos theta. Namely, sine theta, opposite over hypotenuse is three on root 10. Cos theta, adjacent on hypotenuse is one on root 10. We can use a circular identity, sine squared theta plus cos squared theta equal to one as a check. Three on root ten all squared plus one on root 10 all squared, which is nine tenths plus one tenth is one, which checks out. We've made a good start introducing some trigonometry, working towards developing the theory of circular functions in the next video. Today, we first defined the sine, cosine and tangent of an acute angle using ratios of side lengths through an associated right-angled triangle, worked out the details and the special cases where the angle is 45 degrees, 30 degrees and 60 degrees. Also, discussed an example where we could figure out trig values from just one known trig value without actually knowing the value of the angle. Please read the notes accompanying this video, and when you're ready please attempt the exercises. Thank you very much for watching and I look forward to seeing you again soon.
