tailieunhanh - Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 67

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 67. 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 | Triangles We Know and Love and the Information They Give Us 641 EXAMPLE SOLUTION You re interested in knowing the height of a very tall tree. You position yourself so that your line of sight to the top of the tree makes a 60 angle with the horizontal. You measure the distance from where you stand to the base of the tree to be 45 feet. How tall is the tree Assume that your eyes are flve feet above the ground. We ll begin with a sketch. It s simplest to call the height of the tree x 5 feet and flnd x. Figure tan n 3 X V3 x 45 x 45V3 similar triangles Figure The tree is 45 3 5 feet tall or approximately feet tall. EXAMPLE a Find all x such that tan x -1. b Find all x such that sin x 1 2. c Find all x on the interval 0 2ä such that cos x -1 2. SOLUTIONS a There are several approaches to solving tan x -1. One approach is to use the unit circle and interpret tan x as the slope of OP. There are two points on the unit circle for which the slope of OP is 1. They are the points of intersection of the line v u and the unit circle. The points correspond to x 3 and x 4. Therefore x 3p 2nn or x f 2nn where n is any integer. 642 CHAPTER 20 Trigonometry Circles and Triangles b We begin by looking for one x-value that will satisfy sin x 1 2. In our heads dance visions of triangles we know and love. We draw one of our all-time favorites and see that x fits the bill. 6 Draw P 6 on the unit circle. Now use the unit circle to find all points P with -coordinate 1 2. Using symmetry we see that Pi P 6 and P2 P 5x 6 Therefore if sin x 1 2 then x x 6 2nn or x 5 6 2 n where n is an integer. c We begin by looking for one x such that cos x -1 2. In fact to begin with let s not even worry about the negative sign. For what x is cos x 1 2 Again don t turn to strangers turn to a triangle you know and love. Triangles We Know and Love and the Information They Give Us 643 We see that cos n 3 1 2. Let s put that on the unit circle. We label it Q because it is not .