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Figure 2-10 shows how this can be done Draw a mirror image of Fig 2-9, with the angle q defined clockwise instead of counterclockwise Again, you get a right triangle with a hypotenuse 1 unit long, while the other two sides have lengths of sin q and cos q This triangle, like all right triangles, obeys the Pythagorean theorem As in the previous challenge, you end up with sin2 q + cos2 q = 1
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Figure 2-10 This drawing can help show that
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sin2 q + cos2 q = 1 when 3p /2 < q < 2p
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This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App A The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Approximately how many radians are there in 1 Use a calculator and round the answer off to four decimal places, assuming that p 314159 2 What is the angle in radians representing 7/8 of a circular rotation counterclockwise Express the answer in terms of p, not as a calculator-derived approximation 3 What is the angle in radians corresponding to 120 counterclockwise Express the answer in terms of p, not as a calculator-derived approximation 4 Suppose that the earth is a perfectly smooth sphere with a circumference of 40,000 kilometers (km) Based on that notion, what is the angular separation (in radians) between two points 1000 /p km apart as measured over the earth s surface along the shortest possible route 5 Sketch a graph of the function y = sin x as a dashed curve in the Cartesian xy plane Then sketch a graph of y = 2 sin x as a solid curve How do the two functions compare 6 Sketch a graph of the function y = sin x as a dashed curve in the Cartesian xy plane Then sketch a graph of y = sin 2x as a solid curve How do the two functions compare 7 The secant of an angle can never be within a certain range of values What is that range 8 The cosecant of an angle can never be within a certain range of values What is that range 9 The Pythagorean formula for the sine and cosine is sin2 q + cos2 q = 1 From this, derive the fact that sec2 q tan2 q = 1 10 Once again, consider the formula sin2 q + cos2 q = 1 From this, derive the fact that csc2 q cot2 q = 1
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CHAPTER
Polar Two-Space
The Cartesian plane isn t the only tool for graphing on a flat surface Instead of moving right-left and up-down from the origin, we can travel in a specified direction straight outward from the origin to reach a desired point The direction angle is expressed in radians with respect to a reference axis The outward distance is called the radius This scheme gives us polar two-space or the polar coordinate plane
The Variables
Figure 3-1 shows the basic polar coordinate plane The independent variable is portrayed as an angle q relative to a ray pointing to the right (or east ) That ray is the reference axis The dependent variable is portrayed as the radius r from the origin In this way, we can define points in the plane as ordered pairs of the form (q,r)
The radius In the polar plane, the radial increments are concentric circles The larger the circle, the greater the value of r In Fig 3-1, the circles aren t labeled in units We can imagine each concentric circle, working outward, as increasing by any number of units we want For example, each radial division might represent 1, 5, 10, or 100 units Whatever size increments we choose, we must make sure that they stay the same size all the way out That is, the relationship between the radius coordinate and the actual radius of the circle representing it must be linear The direction As pure mathematicians, we express polar-coordinate direction angles in radians We go counterclockwise from a reference axis pointing in the same direction as the positive x axis normally goes in the Cartesian xy plane The angular scale must be linear That is, the physical angle on the graph must be directly proportional to the value of q