Trigonometry: 1. Trigonometry

Only a few minutes to exercise your fears and phobias of trigonometry. All architecture students are always freaked out over trigonometry. It is so simple. Anyway, let's see if we can get started. Measuring angles. Counterclockwise is positive. Clockwise is negative. So here's a plus 30 degrees. It's a little less than 45. Here's a minus 30. Minus, plus. Very good. So let's talk about the sine, cosine, and tangent of an angle. And these are ratios in a triangle. Very good. So let's think of the sine of an angle when it's measured with the positive x-axis. Let's think of the sine of an angle as its rise, the cosine of an angle as its run, and the tangent of an angle as its rise over run or slope. Very good. So, in that vein, let me say this is 0, this is plus 90, and 180, and 270 degrees. So, if we consider the angle 0 degrees, we should recognize that its rise is 0, but its run is the largest it can be. And if this circle has a radius of 1, then the largest value of a run is 1 unit. So, the cosine of 0 degrees is its run, which is 1 unit. Its rise is 0, and its run is 1. Versus 90 degrees. With 90 degrees, the run is 0. It's all vertical. The rise is 1, so must be the sine of 90 is 1. The cosine or its run, its horizontal, is 0. Excellent. At 45, your rise equals your run. And the tangent of 45, which is the ratio of rise over run, if those are equal, then the tangent of 45 degrees is 1. Excellent. For 270 degrees, which is down here, 270 degrees, the rise is minus 1, the run is 0. So, sine and cosine are fractions, and they vary between minus 1 and plus 1. Tangent, on the other hand, can be a number larger than 1 or less than minus 1. Okay, so let's think of this for a minute. if your angle is less than 45 degrees, then your run is larger than your rise. The run is larger than the rise, which means the cosine is greater than the sine for angles less than 45. For angles greater than 45, the rise is greater than the run. And forget the plus minus for a minute. the rise is greater than the run, therefore the sine is greater than the cosine, at least in absolute value. Both are less than, sorry, both are between minus 1 and 1. Very good. So let's look at another way of expressing sine, cosine, and tangent. Given, for example, the slope of a roof is 5 and 12, the rise is 5, the run is 12, Well, then the hypotenuse is 5 square plus 12 square square root, which is 13. So the sine of this angle is the side opposite divided by the hypotenuse, or 5 in 13. The cosine is the side adjacent divided by the hypotenuse. And the tangent is the side opposite divided by the side adjacent or the rise over the run. So the tangent of this angle is 5 over 12. So whatever these fractions are, those are the sines and cosines of whatever this angle is. I have no idea what the angle is, but I assure you its sine is 5 over 13. Its cosine is 12 over 13, and its tangent is opposite over adjacent, or 5 over 12. That's it. I don't know what the angle is, but I know all its properties. And if you insist on knowing what the angle is, we can go in there, and there is a key above the tangent key that says second tangent, or inverse tangent, or arc tangent, just above on your calculator, above the tangent key. So you can say, give me an angle. whose tangent is 5 over 12. Make sure you're in degrees. And this is the tangent inverse of, how much is that? 0.4167. So there is an angle out there. I don't know it, but there is an angle out there whose tangent is 0.4167. And when you do tangent inverse, you're asking for the angle. And so the angle comes back as 22.62 degrees. Happy days. Guess what? The sine of 22.62 is this much. The cosine of 22.62 degrees is this much. And the tangent of this angle is 5 over 12. I didn't need to know what the angle is to know its properties when I'm given a rise-run slope. Very good. So, when I have an angle, for example, of 30 degrees, it's less than 45. I should recognize that the run is larger than the rise. So I will go to a calculator and come back with the properties of 30 degrees. And I will find that the sine of 30 degrees is 0.5. The cosine of 30 degrees is 0.866. And now I just look at my diagram and I see here's my 45 degree. If the angle is less than 45, then the run is bigger than the rise. the run is bigger than the rise, the cosine is larger than the sine. So I can break down this force based on sine and cosine, and look at my diagram. I see that the larger of the two is the horizontal, the smaller of the two is vertical. The shallower the angle, let me see if I can fix this a little bit. The shallower the angle, the larger the run, the smaller the rise. That's what happens. Until you reach 44 degrees, at 44 degrees, your rise and run are almost equal. At 46 degrees, your rise and run are almost equal, but your rise is slightly larger, until you go taller. And then, once your angle is larger, your rise is much larger than your run. At 90 degrees, it's all rise and no run. And the slope of 90 degrees, or the tangent, is infinity. Okay. Excellent. So, once you pass 45 degrees, your rise is larger than your run, which means those angles will have a sine greater than a cosine. And therefore, the tangent, which is sine over cosine, which is a larger number over a smaller number, will be more than one. Versus for angles less than 45, the rise will be smaller than the run. The tangent will be a fraction. Excellent. When an angle is given with the vertical, we cannot think anymore of rise-run slope because it's given with the vertical. We can rotate everything 90 degrees and then go from there. But whatever happens for this angle, which is 120 degrees or 30 degrees with the vertical, I just recognize one thing. I will go to a calculator and get the sine and cosine of 30 degrees. These are fractions, and for 30 degrees, it's 0.5 and 0.866. Now, I'm not going to confuse myself with sine, cosine. Whatever happens, this is the smaller of the two, and this one is the larger of the two. Just visually, it has a lot more rise than run. Must be, in this case, this is the sine of 120 degrees, because it's very steep, versus the cosine of 120 degrees must be the smaller of the two. Excellent. So, let's summarize all this in this diagram here. Thinking of sine and cosine as rise and run. This one here has 0 degrees, has a run, has a rise of 0, has a run of 1. Therefore, its tangent is 0 over 1 or 0. This one has a rise of 1 and a run of 0. And a tangent is 1 over 0. 1 over 0, you cannot divide by 0, so it's undefined, it's infinity. Very good. So up to 45 degrees, the rise is larger than the run. The sine is larger, sorry, the sine is smaller than the cosine. The cosine is larger than the sine. Both are positive though, because we're in this first quadrant. The rise and the run are positive. Versus in quadrant number two, this is quadrant number one. In quadrant number two, your run is negative. Your rise is positive. So the sine here will be positive, the cosine will be negative. And if you're more than 45 degrees, the sine is greater than the cosine. If you are less than 45 degrees, then the sine is less than the cosine. Looking at quadrant 3, it looks like we're to the left and down in this quadrant. Down and to the left. So it looks like both will be negative here. the sine and the cosine will be negative, and if it's steeper than 45, which means in this area, then the sine, or the rise, be it negative, will still be greater than the cosine, versus if it's between 0 and minus 45, or between 0 and 225, which is 180 plus 45, then your run is larger than your rise, and the sine is smaller than the cosine. Finally, in quadrant number 4, the run is positive, the rise is negative. So the rise is negative, the run is positive. I want you to think of run as cosine and rise as sine. Okay, and again, negative for the sine, positive for the cosine. If we're in this area, then the rise is larger than the run, but if we are more than 45 or in this quadrant, then the run is larger than the rise. The cosine is larger than the sine. Excellent. And one more thing. The sine square of any angle plus the cosine square of any angle will always equal to 1. And the reason that is so is because in any right-angled triangle, a square plus b square equals c square. And if c is 1, we said a trigonometric circle has a radius of 1. So in that case, a square plus b square equals c square, or the rise square plus the run square is equal to the slope square or the hypotenuse square. Very good. So the sine squared plus the cosine squared equal 1 means both are fractions. That's enough. Good. Plenty of trigonometry. We don't need more than that.