Continuing our discussion of analytical geometry and trigonometry, now I want to look at trigonometry. And firstly, we define the simple function sine, cosine, etc. from a right triangle where R is the hypotenuse, X the adjacent, and Y opposite relative to the angle theta. We have the Law of Sines, which is A over sin A is equal to B over sin B is equal to C over sin C. In other words, the length of this side A divided by the sin of the angle which is opposite to us. Is equal to the length of b divided by the sin of its opposite angle, etcetera. And another convenient law or equation is the law of cosines, which can be written in terms of the three different angles as shown here. A squared is equal to b squared plus c squared minus 2bc cosin a. And similar equations for the other two angles. The sum of the interior angles in a triangle is equal to 180 degrees, and more general, for polygons of arbitrary number of sides, the sum of the interior angles Is equal to the number of sides or the number of angles, minus two, multiplied by 180 degrees. And for regular polygons, such as sketched here where all the angles are equal, and all the lengths of the sides are equal, the magnitude of the interior angles is number of sides minus two, times 180 divided by the number of sides or angles, N. Let’s do an example on that. And the question is an observer surveys a building and observes that the vertical to its top is 40 degrees. He then walks 50 meters farther away and observes the vertical angle to be now 30 degrees. The height of the building is most nearly which of these alternatives? So, here's a sketch. So we have the building and here is the first observation point which are labelled A and he observes at the angle to the top of the building 40 degrees. Then he walks 50 metres away so this distance is 50 metres To the point b, where he observes that the angle is 30 degrees, and the question is, how high is the building? In other words, the height CD in those triangles. So the steps involved here, first we want to find this angle, theta. By looking at the triangle ADC, the right angle triangle. So the sum of the angles in that triangle is 180 degrees, so 40 degrees plus 90 degrees plus theta is 180. Therefore, theta is 50 degrees. Next we want to find the Angle phi by observing the triangle BDC. And again, the sum of the angles in that triangle is equal to 180 degrees, therefore 30, plus the right angle, plus phi plus theta. Is 180. And we already know theta, so therefore the angle phi is ten degrees. Next we'll use the law of sines on the triangle ABCABC to find the length of the inclined side AC. Is equal to 144 meters, and the final step is to look at the triangle ADC to find the height of the building. And there we see that dc is equal to ac sin 40 degrees. In other words, 144 sin 40. Which is 92.6 m, so the answer is C. There are a great number of trigonometric identities given in the handbook, and I've listed some of them here. And it's not clear which or any of these will be used for, but a few of them might be for example. Sin square theta plus cosine square theta is equal to one and all of the others of course you can look up in the handbook as needed for any particular problem. Next, the handbook discovers, discusses mensuration Or measurement of areas of various shapes for example parabola, ellipse, circular segment, et cetera. And an example is given over here the the area of a parabola of half a parabola of height h and width b is given by this formula A is equal to 2bh over 3, where a is the area and p is the parameter. And it also gives volumes if this is a three-dimensional object. So this is the area of the parabola. Above the curve there Above the concave curve and above a convex curve, the corresponding equation is A is equal to Bh over 3, and similar equations are given for all of the shapes which are listed over on the left-hand side, here. An example on that. We have a circular segment with a length of one meter and a radius of two meters. Its area is most nearly which of these alternatives? So, here's the definition sketch, which is similar to the one that's contained in the handbook. We have a radius, r A height d, and a perimeter or length of the segment s, and the interior angle is phi. So the relevant equations given for a segment like this in the handbook are phi. The interior angle is s over r or 2 cosign arc r minus d. Over R. So, in this case, we have phi, the interior angle, is equal to S which is given as one meter, divided by the radius, in other words to half, is equal to 0.5. And remember, of course, that the units here are radiants when you compute it in this way. The next equation given is the area in terms of the radius and the angle, r squared phi minus sine phi over 2. So, in this case, and again, you have to be careful. Remember that phi is given, generally speaking, in radians in this equation, so that is equal to 2 squared Radius is 2, phi is 0.5 minus sin phi. And here I've converted the angle phi to degrees. You don't need to. You can leave it in radians if you like, but I've converted there to degrees divided by 2, and the answer is 0.0396 square meters. So the answer is D. And this concludes our discussion of mensuration of areas.
