Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 69. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | Applying Trigonometry to a General Triangle The Law of Cosines and the Law of Sines 661 C an acute angle i Figure The coordinates of C are 0 0 the coordinates of A are fe 0 and we ll denote the coordinates of B by x y . Check that regardless of whether we use part i or part ii of Figure we have x cos C - or equivalently x a cos C a y sin C - or equivalently y a sin C. a The distance between points A and B is c. c the distance between A and B yj x - fe 2 y - 0 2 Using the formula for the distance between points c y a cos C fe 2 a sin C 0 2 Squaring both sides gives c2 a cos C fe 2 a sin C 2 c2 a2 cos2 C 2afe cos C fe2 a2 sin2 C c2 a2 cos2 C sin2 C 2afe cos C fe2 c2 a2 fe2 2afe cos C. We have proven the Law of Cosines. When Might We Use the Law of Cosines When we know the lengths of all the sides of a triangle we can use the Law of Cosines to flnd any angle SSS . When we know the lengths of two sides and the angle between them we can use the Law of Cosines to flnd the length of the third side of the triangle SAS . Scenario II An astronomer sights a planet in her telescope. She measures the angle of elevation of her line of sight. She travels 5 miles and repeats the procedure. Can she determine how far away the planet is using the information she has gathered Figure on the following page is not drawn to scale and we are ignoring the curvature of the earth . 662 CHAPTER 20 Trigonometry Circles and Triangles 01 and 02 are the angles of elevation of her line of sight from the two locations. Figure From her measurements of 01 and 02 see Figure she can determine two angles and hence the third as well of the triangle drawn. Knowing the length of the side between them determines the triangle. The Law of Cosines is not useful in this case but the Law of Sines will be. The Law of Sines says sin yy n We can flnd A B and C and we know a so we use sinA sinC This allows us to solve for c. Before proving the Law of Sines we will look at an area .
