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he 1958 musical Merry Andrew starred Danny Kaye as Andrew Larabee, a teacher with a flair for using unconventional methods in his classes. He uses a musical number to teach the Pythagorean theorem, singing and dancing to “The Square of the Hypotenuse”: T . . . Parallel lines don’t connect, which is just about what you might expect.Though scientific laws may change and decimals can be moved, the following is constant, and has yet to be disproved: The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides. The Pythagorean theorem, introduced in Chapter 9, is used extensively in the study of trigonometry, the foundations of which go back at least 3000 years.The word trigonometry comes from the Greek words for triangle (trigon) and measurement (metry). Today trigonometry is used in electronics, surveying, and other engineering areas, and is necessary for further courses in mathematics, such as calculus. 14.1 Angles and Their Measures 14.2 Trigonometric Functions of Angles 14.3 Trigonometric Identities 14.4 Right Triangles and Function Values 14.5 Applications of Right Triangles 14.6 The Laws of Sines and Cosines Extension Area Formulas for Triangles Collaborative Investigation Making a Point about Trigonometric Function Values Chapter 14 Test 749 750 CHAPTER 14 Trigonometry 14.1 ANGLES AND THEIR MEASURES Basic Terminology • Degree Measure • Angles in a Coordinate System A A B B A Line AB Segment AB B Ray AB Figure 1 Terminal side Vertex A Initial side Basic Terminology A line may be drawn through the two distinct points A and B. This line is called line AB. The portion of the line between A and B, including points A and B themselves, is segment AB. The portion of the line AB that starts at A and continues through B, and on past B, is called ray AB. Point A is the endpoint of the ray. See Figure 1. An angle is formed by rotating a ray around its endpoint. The ray in its initial position is called the initial side of the angle, while the ray in its location after the rotation is the terminal side of the angle. The endpoint of the ray is the vertex of the angle. See Figure 2. If the rotation of the terminal side is counterclockwise, the angle measure is positive. If the rotation is clockwise, the angle measure is negative. See Figure 3. Figure 2 A C B Positive angle Negative angle Figure 3 An angle can be named by using the name of its vertex. For example, the angle on the right in Figure 3 can be called angle C. Alternatively, an angle can be named using three letters, with the vertex letter in the middle. Thus, the angle on the right also could be named angle ACB or angle BCA. Degree Measure A complete rotation of a ray gives an angle whose measure is 360°. Figure 4 There are two systems in common use for measuring the sizes of angles. The most common unit of measure is the degree. (The other common unit of measure is called the radian.) Degree measure was developed by the Babylonians 4000 years ago. To use degree measure we assign 360 degrees to a complete rotation of a ray. In Figure 4, notice that the terminal side of the angle corresponds to its initial side when it makes a complete rotation. 1 One degree, written 1°, represents 360 of a rotation. Therefore, 90° represents 90 1 180 1 = of a complete rotation, and 180° represents 360 4 360 = 2 of a complete rotation. Angles of measure 5°, 90°, and 180° are shown in Figure 5. 90° 5° angle Figure 5 180° 14.1 ”Trigonometry, perhaps more than any other branch of mathematics, developed as the result of a continual and fertile interplay of supply and demand: the supply of applicable mathematical theories and techniques available at any given time and the demands of a single applied science, astronomy. So intimate was the relation that not until the thirteenth century was it useful to regard the two subjects as separate entities.” (From “The History of Trigonometry” by Edward S. Kennedy, in Historical Topics for the Mathematics Classroom, the Thirty-first Yearbook of N.C.T.M., 1969.) Angles and Their Measures 751 Special angles are named as shown in the following chart. Name Acute angle Angle Measure Example(s) Between 0° and 90° 60° Right angle 82° Exactly 90° 90° Obtuse angle Between 90° and 180° 138° 97° Straight angle Exactly 180° 180° If the sum of the measures of two angles is 90°, the angles are called complementary. Two angles with measures whose sum is 180° are supplementary. EXAMPLE 1 Finding Complement and Supplement Give the complement and the supplement of 50°. SOLUTION The complement of 50° is 90° - 50° = 40°. The supplement of 50° is 180° - 50° = 130°. Do not confuse an angle with its measure. The angle itself consists of the vertex together with the initial and terminal sides. The measure of the angle is the size of the rotation angle from the initial to the terminal side (commonly expressed in degrees). For example, if angle A has a 35° rotation angle, we say that m1angle A2 is 35°, where m1angle A2 is read “the measure of angle A.”We abbreviate m1angle A2 = 35° as simply angle A = 35°, or just A = 35°. Traditionally, portions of a degree have been measured with minutes and sec1 onds. One minute, written 1 œ , is 60 of a degree. 1¿ ⴝ 1 ° or 60¿ ⴝ 1° 60 1 One second, 1 fl , is 60 of a minute. 1– ⴝ 1 ¿ 1 ° ⴝ or 60– ⴝ 1¿ 60 3600 The measure 12° 42¿ 38– represents 12 degrees, 42 minutes, 38 seconds. 752 CHAPTER 14 Trigonometry EXAMPLE 2 Calculating with Degree Measure Perform each calculation. (a) 51° 29¿ + 32° 46¿ 51°29'32°46' SOLUTION (a) Add the degrees and the minutes separately. 䉴 DMS 84°15'0'' 90°0'73°12' (b) 90° - 73° 12¿ 51° 29¿ + 32° 46¿ 83° 75¿ 䉴 DMS 16°48'0'' The calculations explained in Example 2 can be done with a graphing calculator capable of working with degrees, minutes, and seconds. (b) 90° - 73° 12¿ Simplify. 83° + 1° 15¿ 84° 15¿ Write 90° as 89° 60 ¿. 75 ¿ = 60 ¿ + 15 ¿ = 1° 15 ¿ 89° 60¿ - 73° 12¿ 16° 48¿ Angles can be measured in decimal degrees. For example, 12.4238° represents 12.4238° = 12 EXAMPLE 3 4238 ° . 10,000 Converting between Decimal Degrees and Degrees, Minutes, Seconds (a) Convert 74° 8¿ 14– to decimal degrees. Round to the nearest thousandth of a degree. (b) Convert 34.817° to degrees, minutes and seconds. Round to the nearest second. SOLUTION 74°8'14'' 74.13722222 8 ° 14 ° + 60 3600 L 74° + 0.1333° + 0.0039° = 74.137° (a) 74° 8¿ 14– = 74° + 74°8'14'' 74.137 34.817 䉴 DMS 34°49'1.2'' The conversions in Example 3 can be done on some graphing calculators. The second displayed result was obtained by setting the calculator to show only three places after the decimal point. (b) 34.817° = = = = = = L 34° + 0.817° 34° + 10.8172(60 ¿2 34° + 49.02¿ 34° + 49¿ + 0.02¿ 34° + 49¿ + 10.022160–2 34° + 49¿ + 1.2– 34° 49¿ 1– 1¿ = 1 ° 60 and 1– = 1 ° 3600 Add. Round to three decimal places. 1° = 60 ¿ 1 ¿ = 60– Angles in a Coordinate System An angle u (the Greek letter theta)* is in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side lies along the positive x-axis. The two angles shown in Figures 6(a) and (b) on the next page are in standard position. An angle in standard position is said to lie in the quadrant in which its terminal side lies. For example, an acute angle is in quadrant I and an obtuse angle is in quadrant II. *The letters of the Greek alphabet are identified in a margin note on page 758. 14.1 Angles and Their Measures 753 Figure 6(c) shows ranges of angle measures for each quadrant when 0° 6 u 6 360°. Angles in standard position having their terminal sides along the x-axis or y-axis, such as angles with measures 90°, 180°, 270°, and so on, are called quadrantal angles. 90° y y QI Q II Q II 90° < < 180° Terminal side Vertex 0 x 0 Initial side 0° 360° 180° x QI 0° < < 90° Q III Q IV 180° < < 270° 270° < < 360° 270° (a) (b) (c) Figure 6 A complete rotation of a ray results in an angle of measure 360°. If the rotation is continued, angles of measure larger than 360° can be produced. The angles in Figure 7(a) have measures 60° and 420°. These two angles have the same initial side and the same terminal side, but different amounts of rotation. Angles that have the same initial side and the same terminal side are called coterminal angles. As shown in Figure 7(b), angles with measures 110° and 830° are coterminal. y y 60°° x 0 (a) x 0 Coterminal angles y 830° 110°° 420°° Coterminal angles (b) Figure 7 188° 908° x 0 EXAMPLE 4 Finding Measures of Coterminal Angles Find the angle of least possible positive measure coterminal with each angle. (a) 908° Figure 8 (b) - 75° SOLUTION (a) Add or subtract 360° from 908° as many times as needed to get an angle with measure greater than 0° but less than 360°. Because y 908° ⴚ 2 0 285° # 360° = 908° ⴚ 720° = 188°, x –75° an angle of 188° is coterminal with an angle of 908°. See Figure 8. (b) See Figure 9. Use a rotation of Figure 9 360° + 1- 75°2 = 285°. 754 CHAPTER 14 Trigonometry Sometimes it is necessary to find an expression that will generate all angles coterminal with a given angle. For example, because any angle coterminal with 60° can be obtained by adding an appropriate integer multiple of 360° to 60°, we can let n represent any integer, and the expression 60° ⴙ n # 360° will represent all such coterminal angles. Table 1 shows a few possibilities. Table 1 Value of n Angle Coterminal with 60° 2 60° + 2 # 360° = 780° 1 60° + 1 # 360° = 420° 0 60° + 0 # 360° = 60° (the angle itself) -1 60° + 1- 12 # 360° = ⴚ300° 14.1 EXERCISES Give (a) the complement and (b) the supplement of each angle. 1. 30° 2. 60° 3. 45° 4. 55° 5. 89° 6. 2° 7. If an angle measures x degrees, how can we represent its complement? 8. If an angle measures x degrees, how can we represent its supplement? Perform each calculation. 9. 62° 18 ¿ + 21° 41 ¿ 10. 75° 15¿ + 83° 32 ¿ 11. 71° 58 ¿ + 47° 29 ¿ 12. 90° - 73° 48¿ 13. 90° - 51° 28 ¿ 14. 180° - 124° 51¿ 15. 90° - 72° 58 ¿ 11– 16. 90° - 36° 18¿ 47– Convert each angle measure to decimal degrees. Use a calculator, and round to the nearest thousandth of a degree. Convert each angle measure to degrees, minutes, and seconds. Use a calculator, and round to the nearest second. 23. 31.4296° 24. 59.0854° 25. 89.9004° 26. 102.3771° 27. 178.5994° 28. 122.6853° Find the angle of least positive measure coterminal with each angle. 29. - 40° 30. - 98° 31. - 125° 32. - 203° 33. 539° 34. 699° 35. 850° 36. 1000° Give an expression that generates all angles coterminal with the given angle. Let n represent any integer. 37. 30° 38. 45° 39. 60° 40. 90° Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle. 17. 20° 54 ¿ 18. 38° 42¿ 19. 91° 35 ¿ 54– 20. 34° 51¿ 35– 41. 75° 42. 89° 43. 174° 44. 234° 21. 274° 18 ¿ 59– 22. 165° 51¿ 9– 45. 300° 46. 512° 47. - 61° 48. - 159° 14.2 Trigonometric Functions of Angles 755 14.2 TRIGONOMETRIC FUNCTIONS OF ANGLES Trigonometric Functions • Undefined Function Values y Trigonometric Functions P(x, y) ⎩ ⎪ ⎧ ⎪ ⎪ y⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ r ⎪ ⎪ ⎪ θ ⎪ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ⎧ Q x x O Figure 10 The study of trigonometry covers the six trigonometric functions defined in this section. To define these six basic functions, start with an angle u in standard position. Choose any point P having coordinates 1x, y2 on the terminal side of angle u. (The point P must not be the vertex of the angle.) See Figure 10. A perpendicular from P to the x-axis at point Q determines a triangle having vertices at O, P, and Q. The distance r from P1x, y2 to the origin, 10, 02, can be found from the distance formula. r = 21x - 022 + 1y - 022 Distance formula r = 2x 2 + y 2 Notice that r>0, because this distance is never negative. The six trigonometric functions of angle u are called sine, cosine, tangent, cotangent, secant, and cosecant. In the following definitions, we use the customary abbreviations for the names of these functions: sin, cos, tan, cot, sec, csc. Definitions of the Trigonometric Functions Let 1x, y2 be a point other than the origin on the terminal side of an angle u in standard position. The distance from the point to the origin is r = 2x 2 + y 2. The six trigonometric functions of u are defined as follows. y y x sin U ⴝ cos U ⴝ tan U ⴝ 1x ⴝ 02 r r x csc U ⴝ r y 1y ⴝ 02 sec U ⴝ r x 1x ⴝ 02 cot U ⴝ x y 1y ⴝ 02 Although Figure 10 shows a second quadrant angle, these definitions apply to any angle u. Due to the restrictions on the denominators in the definitions of tangent, cotangent, secant, and cosecant, quadrantal angles will have some undefined function values. y EXAMPLE 1 (8, 15) 17 15 The terminal side of an angle u in standard position passes through the point 18, 152. Find the values of the six trigonometric functions of angle u. x= 8 y = 15 r = 17 θ 0 8 Figure 11 Finding Function Values of an Angle x SOLUTION Figure 11 shows angle u and the triangle formed by dropping a perpendicular from the point 18, 152 to the x-axis. The point 18, 152 is 8 units to the right of the y-axis and 15 units above the x-axis, so that x = 8 and y = 15. r = 2x 2 + y 2 r = 282 + 15 2 Let x 8 and y 15. r = 264 + 225 Square 8 and 15. r = 2289 Add. r = 17 Find the square root. 756 CHAPTER 14 Trigonometry The values of the six trigonometric functions of angle u can now be found with the definitions given in the box, where x = 8, y = 15, and r = 17. y 15 = r 17 r 17 csc u = = y 15 sin u = EXAMPLE 2 x 8 = r 17 r 17 sec u = = x 8 cos u = y 15 = x 8 x 8 cot u = = y 15 tan u = Finding Function Values of an Angle The terminal side of an angle u in standard position passes through the point 1- 3, - 42. Find the values of the six trigonometric functions of u. SOLUTION As shown in Figure 12, x = - 3 and y = - 4. Find the value of r. y θ –3 sin u = (–3, –4) Figure 12 y OP = r (x′, y′) P′ OP′ = r′ P θ O Q Q′ x Remember that r 7 O. Then use the definitions of the trigonometric functions. -4 4 = 5 5 5 5 csc u = = -4 4 5 (x, y) r = 225 r = 5 x 0 –4 r = 21- 322 + 1- 422 x = –3 y = –4 r = 5 -3 3 = 5 5 5 5 sec u = = -3 3 cos u = -4 4 = -3 3 -3 3 cot u = = -4 4 tan u = The six trigonometric functions can be found from any point on the terminal side of the angle other than the origin. To see why any point may be used, refer to Figure 13, which shows an angle u and two distinct points on its terminal side. Point P has coordinates 1x, y2 and point P¿ (read “P-prime”) has coordinates 1x¿, y¿2. Let r be the length of the hypotenuse of triangle OPQ, and let r¿ be the length of the hypotenuse of triangle OP¿Q¿. Because corresponding sides of similar triangles are in proportion, y y¿ , = r r¿ Figure 13 y and thus sin u = r is the same no matter which point is used to find it. Similar results hold for the other five functions. Undefined Function Values If the terminal side of an angle in standard position lies along the y-axis, any point on this terminal side has x-coordinate 0. Similarly, an angle with terminal side on the x-axis has y-coordinate 0 for any point on the terminal side. Because the values of x and y appear in the denominators of some of the trigonometric functions, and because a fraction is undefined if its denominator is 0, some of the trigonometric function values of quadrantal angles will be undefined. EXAMPLE 3 Finding Function Values and Undefined Function Values Find values of the trigonometric functions for each angle. Identify any that are undefined. (a) an angle of 90° (b) an angle in standard position with terminal side through 1- 3, 02 14.2 (0, 1) 90° x (a) y 1 = 1 1 1 csc 90° = = 1 1 0 = 0 1 1 sec 90° = 1undefined2 0 sin 90° = 1 1undefined2 0 0 cot 90° = = 0 1 cos 90° = tan 90° = (b) Figure 14(b) shows the angle. Here, x = - 3, y = 0, and r = 3, so the trigonometric functions have the following values. θ x 0 (–3, 0) 757 SOLUTION (a) First, select any point on the terminal side of a 90° angle. We select the point 10, 12, as shown in Figure 14(a). Here x = 0 and y = 1. Verify that r = 1. Then, use the definitions of the trigonometric functions. y 0 Trigonometric Functions of Angles (b) Figure 14 0 = 0 3 3 csc u = 1undefined2 0 sin u = -3 = -1 3 3 sec u = = -1 -3 cos u = 0 = 0 -3 -3 cot u = 1undefined2 0 tan u = Undefined Function Values If the terminal side of a quadrantal angle lies along the y-axis, the tangent and secant functions are undefined. If it lies along the x-axis, the cotangent and cosecant functions are undefined. Because the most commonly used quadrantal angles are 0°, 90°, 180°, 270° and 360°, the values of the functions of these angles are summarized in Table 2. Table 2 U Function Values of Quadrantal Angles sin U cos U tan U cot U sec U csc U 0° 0 1 0 Undefined 1 Undefined 90° 1 0 Undefined 0 Undefined 1 180° 0 -1 0 Undefined -1 Undefined 270° -1 0 Undefined 0 Undefined -1 360° 0 1 0 Undefined 1 Undefined 14.2 EXERCISES In Exercises 1– 4, sketch an angle u in standard position such that u has the least possible positive measure, and the given point is on the terminal side of u. 1. 1 - 3, 42 2. 1- 4, - 32 3. 15, - 122 4. 1- 12, - 52 Find the values of the trigonometric functions for the angles in standard position having the following points on their terminal sides. Identify any that are undefined. Rationalize denominators when applicable. 5. 1 - 3, 42 6. 1- 4, - 32 7. 10, 22 8. 1- 4, 02 9. 11, 232 10. 1- 223, - 22 11. 13, 52 12. 1- 2, 72 13. 1- 8, 02 758 CHAPTER 14 Trigonometry 14. 10, 92 23. IV, 15. For any nonquadrantal angle u, sin u and csc u will have the same sign. Explain why this is so. x r 24. IV, y r 25. IV, y x 26. IV, Use the appropriate definition to determine each function value. If it is undefined, say so. 16. If cot u is undefined, what is the value of tan u? 17. How is the value of r interpreted geometrically in the definitions of the sine, cosine, secant, and cosecant functions? 27. cos 90° 28. sin 90° 29. tan 90° 30. cot 90° 31. sec 90° 32. csc 90° 18. If the terminal side of an angle u is in quadrant III, what is the sign of each of the trigonometric function values of u? 33. sin 180° 34. sin 270° 35. tan 180° 36. cot 270° Suppose that the point 1x, y2 is in the indicated quadrant. Decide whether the given ratio is positive or negative. (Hint: It may be helpful to draw a sketch.) 37. sin1- 270°2 38. cos1- 270°2 39. tan 0° 40. sec1- 180°2 y 19. II, r 41. cos 180° 42. cot 0° 20. II, x r y 21. III, r 22. III, x y x r 14.3 TRIGONOMETRIC IDENTITIES Reciprocal Identities • Signs of Function Values In Quadrants • Pythagorean Identities • Quotient Identities Reciprocal Identities The Greek Alphabet a alpha b beta g gamma d delta P epsilon z zeta h eta u theta i iota k kappa l lambda m mu n nu j xi o omicron p pi r rho s sigma t tau y upsilon f phi x chi c psi v omega The definitions of the trigonometric functions on page 755 were written so that functions directly above and below one another are reciprocals of each other. Because y sin u = r and csc u = yr , sin u = 1 csc u and csc u = 1 . sin u Also, cos u and sec u are reciprocals, as are tan u and cot u. The reciprocal identities hold for any angle u that does not lead to a zero denominator. Reciprocal Identities sin U ⴝ 1 csc U cos U ⴝ 1 sec U tan U ⴝ 1 cot U csc U ⴝ 1 sin U sec U ⴝ 1 cos U cot U ⴝ 1 tan U Identities are equations that are true for all meaningful values of the variable. When studying identities, be aware that various forms exist. For example, sin U ⴝ 1 csc U can also be written csc U ⴝ 1 sin U and You should become familiar with all forms of these identities. 1sin U21csc U2 ⴝ 1.