Chapter 7 Trigonometric Identities and Equations 7-1 Basic Trigonometric Identities Pages

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1 Trigonometric Identities and Equations 7- Basic Trigonometric Identities Pages Sample answer: tan, cot, cot tan cos cot, cot csc 5. Rosalinda is correct; there may be other values for which the equation is not true. 7. Sample answer: Pythagorean identities are derived by applying the Pythagorean Theorem to a right triangle. The opposite angle identities are so named because A is the opposite of A. (A) 4. tan(a) cos(a) A cos A A cos A tana 6. Sample answer: cos 3 3. csc B F csc I BI F csc BI F csc F BI csc F BI 9. Sample answer: 45. Sample answer: sec 6. csc 8. Sample answer: Sample answer: 30. Sample answer: 0 Glencoe/McGraw-Hill 74 Advanced Mathematical Concepts

2 3. Sample answer: Sample answer: cos 8 4. csc cot csc cos Let (, y) be the point where the terminal side of A intersects the unit circle when A is in standard position. When A is reflected about the -ais to obtain A, the y-coordinate is multiplied by, but the -coordinate is unchanged. So, (A) y A and cos (A) cos A tan 5 4. sec csc 46. tan 48. cos csc a. W eas cos 56b W Glencoe/McGraw-Hill 75 Advanced Mathematical Concepts

3 57. k tan a tan, so n n h a a h cot. The area of the isosceles tan a 80 n There are n such triangles, triangle is (a) cot cot. a 4 80 n h a 4 80 n so A na cot. 59. EF and cos OF ce the circle is a unit circle. CD CD tan CD OD CO CO sec CO OD y O A E B F C D OF BA BA EOF OBA, so BA. EF OA cos OF Then cot BA. Also by EF EO OB OB similar triangles,, or. EF OA EF OB Then csc OB. EF 6. y y cos ( 6 ) cm O Glencoe/McGraw-Hill 76 Advanced Mathematical Concepts

4 63. a.0, B 70, b , (, 5, 3) 64.,, 66. continuous 68. y C 7- Verifying Trigonometric Identities Pages Answers will vary. 3. Sample answer: They are the trigonometric functions with which most people are most familiar. cot 5. cos csc cos c os cos cos cos cos 7. csc cot csc cot csc cot csc cot csc cot csc cot csc cot ( cot ) cot csc cot csc cot csc cot csc cot csc cot csc cot csc cot csc cot. Sample answer: Squaring each side can turn two unequal quantities into equal quantities. For eample,, but (). 4. Answers will vary. 6. tan sec c os co s co s cos cos cos + cos cos 8. tan sec cos tan cos cos cos tan cos cos cos tan cos tan cos tan cos tan tan Glencoe/McGraw-Hill 77 Advanced Mathematical Concepts

5 9. (A cosa) A cota A A cosa cos A A cota A cosa A cota A A cosa A 0. Sample answer: 4 A cota cosa A A A cota A cota A cota. Sample answer: cos 3. tan A tan A tan A sec A csc A co s A A A cos A I cos. R I cos R I cos R I cos R I cot R csc I c os R I c os R I cos R 4. cos cot cos cos cos cos tan A tan A Glencoe/McGraw-Hill 78 Advanced Mathematical Concepts

6 tan 5. sec tan 6. sec cos cos sec tan cos cos c os sec sec tan sec tan cos cos c os sec cos cos cos + sec cos ( + cos ) sec cos 7. sec csc tan cot cos sec csc cos sec csc cos sec csc sec csc sec csc sec csc cos cos cos cos cos cos cos cos cos cos sec = sec 8. cos cos cos ( cos ) cos cos cos cos cos ( cos )( cos ) cos cos cos sec csc sec csc Glencoe/McGraw-Hill 79 Advanced Mathematical Concepts

7 9. ( A cos A) ( A cos A) sec A csc A sec A csc A sec A csc A ( A cos A) sec A csc A ( A cos A) cos A A ( A cos A) cos A A A cos A ( A cos A) ( A cos A) cos y. y cos y y y y cos y( y) sec A csc A sec A csc A y cos y cos y( y) y cos y cos y y y cos y cos y cot 3. csc csc csc csc csc (csc )(csc ) csc csc csc csc y cos y y cos y y 0. ( )(tan sec ) cos tan tan sec sec cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos. cos cos() () cos cos ( ) cos 4. cosb cot B csc B B cosb cot B B B cosb cot B B B B cosb cot B cosb cot B B B cos B B cos B cosb cot B cosb B cosb cot B cosb cot B Glencoe/McGraw-Hill 80 Advanced Mathematical Concepts

8 5. cos tan cos 6. (csc cot ) cos cos cos cos csc csc cot cot cos 7. cos cos tan cot cos s in cos c os cos cos s in c cos os cos cos cos cos cos cos cos cos cos cos ( cos )( cos ) cos cos Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts 8. cos tan sec cos tan cos cos cos cos cos cos cos cos cos cos ( cos ) cos ( cos ) ( cos )( cos ) cos cos cos cos cos cos cos cos sec cos tan cos sec cos tan cos cos cos cos cos cos cos cos cos sec cos tan sec cos tan sec cos tan sec sec cos tan cos sec cos sec cos tan cos cos sec sec cos tan cos tan sec sec cos tan sec cos tan sec cos tan

9 9. Sample answer: sec 3. Sample answer: cos Sample answer: yes 39. no 4. f () 30. Sample answer: tan 3. Sample answer: 34. Sample answer: cot 36. no 38. yes 40a. P I0 R( cos ft) I 0 R 40b. P csc ft 4. a c cos b cos a c cos cos b cos c cos a cos b cos b Then cos cos b a c cos c cos a cos c cos a a cos a cos c c g 43. y ( tan ) tan v By the Law of Sines, b a a, so b. tan a cot c 44. Sample answer: tan 46. Then A ab a A a a A a A (80 ( )) A a ( ) Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts

10 47. y ( 90 ) shirts, 40 pants 53. D , y 3 5. {6}, {4, 4}; no; 6 is paired with two elements of the range. 7-3 Sum and Difference Identities Pages Find a countereample, such as 30 and y The opposite side for 90 A is the adjacent side for A, so the right-triangle ratio for (90 A) is the same as that for cos A Find the coe, e, or tangent, respectively, of the sum or difference, then take the reciprocal. 4. cot( ) tan( ) tan tan tan tan tan tan tan tan co t cot co t cot cot cot cot cot (90 A) cos A (90 ) cos A cos(90 ) A cos A cos A 0 A cos A cos A cos A cot cot cot cot Glencoe/McGraw-Hill 83 Advanced Mathematical Concepts

11 . tan cot cos cos cos cos ( ) 0 (cos ) (cos ) 0 ( ) 3. cos n 0 t. ( y) cos cos cot cot cot cot cot cot ( y) ( y) cos y cos y ( y) ( y) ( y) cot tan y csc sec y c os y c os y co s y c os y c os y co s y cos y cos y Glencoe/McGraw-Hill 84 Advanced Mathematical Concepts

12 cos cos cos 35. cos(60 A) (30 A) cos 60 cos A 60 A 30 cos A cos 30 A cos A 3 A 0 cos 36. (A ) A A cos cos A A ( A)() (cos A)(0) A A A 3 cos A A 37. cos(80 ) cos cos 80 cos 80 cos cos 0 cos cos cos 39. (A B) (A B) (A B) tan A tan B sec A sec B A B c os A c os B co s A cos B A s in B c os A cosb co s A cos B cos A cos B cos A cos B (A B) A cos B cos A B (A B) (A B) 38. tan( 45 ) tan tan 45 tan tan 45 tan (tan )() tan tan 40. cos(a B) cos(a B) cos(a B) tan tan tan tan tan tan tan tan tan A tan B sec A sec B A B c os A c os B co s A cos B A B c os A c os B co s A cos B cos A cos B cos A cos B cos(a B) cos A cos B A B cos(a B) cos(a B) Glencoe/McGraw-Hill 85 Advanced Mathematical Concepts

13 4. sec(a B) sec(a B) sec(a B) co s A cos B A B c os A c os B sec A sec B tan A tan B 4. co s A cos B A B c os A c os B sec(a B) cos A cos B A B sec(a B) cos(a B) sec(a B) sec(a B) cos A cos B cos A cos B ( y) ( y) y ( cos y cos y) ( cos y cos y) y ( cos y) (cos y) y cos y cos y y cos y y y cos y y (cos y y) y( cos ) y ( )() ( y)() y y y 43. V L I 0 L t 44. n n n ( ) ( 60 ) n cos 30 cos 30 n 3 cos n 3 cos Glencoe/McGraw-Hill 86 Advanced Mathematical Concepts

14 45. A 46a. 46b. cos h cos h h 47. tan( ) tan( ) tan( ) tan( ) ( ) cos( ) cos cos cos cos cos cos c os cos c os cos c os cos c os cos s in c os cos tan tan tan tan Replace with to find tan( ) tan tan() tan( ()) tan tan() tan tan tan( ) tan tan 46c. cos 48a. Answers will vary. 48b. tan A tan B tan C tan A tan B tan C tan A tan B tan(80 (A B)) tan A tan B tan(80 (A B)) tan A tan B tan A tan B tan 80 tan(a B) tan 80 tan(a B) 0 tan(a B) 0 tan(a B) tan A tan B tan tan A Tan B 0 tan(a B) 0 tan(a B) tan A tan B tan(a B) tan A tan B tan(a B) tan A tan B (tan A tan B) tan(a B) tan A tan B tan(a B) tan A tan B tan(a B)( tan A tan B) tan(a B) tan A tan B tan(a B) ( tan A tan B )tan(a B) tan A tan B tan(a B) tan A tan B tan(a B) tan A tan B tan(a B) tan 80 tan(a B) tan 80 tan(a B) Glencoe/McGraw-Hill 87 Advanced Mathematical Concepts

15 49. sec cos csc cot sec cos cos cot cot sec cos cos cos sec sec sec sec y 8 ( t ) about 83 miles ft 6. { 5 or 3} , 5. k, where k is an integer 54. 8, 360, ft, 55.9 ft ,, A Chapter Mid-Chapter Quiz Page cos 4 4. tan cot sec csc cos Glencoe/McGraw-Hill 88 Advanced Mathematical Concepts

16 csc 5. sec csc sec csc sec cos cos sec sec csc csc csc 6. cot sec cos tan cos csc cos cos cos 7. tan( ) cot csc csc csc tan tan tan( ) tan tan tan tan tan( ) tan tan tan tan tan( ) tan tan tan( ) tan( ) cot tan cot tan tan tan Glencoe/McGraw-Hill 89 Advanced Mathematical Concepts

17 7-4 Double-Angle and Half-Angle Identities Pages If you are only given the value of cos, then cos cos is the best identity to use. If you are only given the value of, then cos is the best identity to use. If you are given the values of both cos and, then cos cos is just as good as the other two.. cos cos cos cos c os Letting yields cos or, c os. 3a. III or IV 3b. I or II 3c. I, II, III, or IV 5. Both answers are correct. She obtained two different representations of the same number. One way to verify this is to evaluate each epression with a calculator. To verify it algebraically, square each answer and then simplify. The same result is obtained in each case. Since each of the original answers is positive, and they have the same square, the original answers are the same number Sample answer: ,, Glencoe/McGraw-Hill 90 Advanced Mathematical Concepts

18 ,, A A A s A A sec A A sec A A co s A co s A co co s A A A cos A cos A cos A 0. tan tan tan tan tan tan. cos cos cot tan cot tan tan tan tan tan tan cot tan tan A A cos A A A 3. P I0 R I 0 R cost ,, ,, ,, , 9, ,, ,, Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts

19 csc sec csc 5 sec csc sec csc cos sec csc cos csc sec sec csc sec csc sec csc 9. cos A A cos A A cos A cos A A cos A A cos A A cos A A (cos A A)(cos A A) cos A A cos A A cos A A 3. cos cos (cos ) cos cos (cos ) cos cos (cos ) (cos cos ) (cos ) (cos )(cos ) cos (cos ) cos cos 30. ( cos ) cos cos cos cos 3. sec sec sec sec cos cos cos Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts

20 A A 33. tan cos A ( ) A cos cos A tan cos A 3 4 cos ( ) A A cos A tan cos A ( ) ( ) A A cos 3 3 A 3 4 tan cos A tan tan A A A cos A tan A 35. cos 3 4cos 3 3cos cos ( ) 4cos 3 3cos cos cos 4cos 3 3cos (cos )cos cos 4cos 3 3cos (cos )cos ( cos ) cos 4cos 3 3cos cos 3 cos cos cos 3 4cos 3 3cos 4cos 3 3cos 4cos 3 3cos cos Glencoe/McGraw-Hill 93 Advanced Mathematical Concepts

21 37. PBD is an inscribed angle that subtends the same arc as the central angle POD, so mpbd. By right triangle PA trigonometry, tan BA PA OA cos. 38. R R R v cos ( ) g cos v cos ( 45 ) g cos 45 v cos ( cos 45 cos 45 ) g cos 45 R v cos ( ) (cos ) g R R R v cos ( cos ) g v ( cos cos ) g v g ( cos ( cos ) ) v R ( cos ) g Glencoe/McGraw-Hill 94 Advanced Mathematical Concepts

22 39a. c o s L co L s c o s L cos L 39b tan( 30 ) tan( 30 ) 3 tan tan 30 3 tan tan 30 tan 3 3 tan 3 tan 3 3 tan 3 tan tan tan tan 4. Sample answer: cos cos 0 () 44. about 460 ft (7, ) Glencoe/McGraw-Hill 95 Advanced Mathematical Concepts

23 7-5 Solving Trigonometric Equations Pages A trigonometric identity is an equation that is true for all values of the variable for which each side of the equation is defined. A trigonometric equation that is not an identity is only true for certain values of the variable and , where is any integer , ,, 6 6 6,. k 3. (k ) All trigonometric functions are periodic. Adding the least common multiple of the periods of the functions that appear to any solution to the equation will always produce another solution. 4. Each type of equation may require adding, subtracting, multiplying, or dividing each side by the same number. Quadratic and trigonometric equations can often be solved by factoring. Linear and quadratic equations do not require identities. All linear and quadratic equations can be solved algebraically, whereas some trigonometric equations require a graphing calculator. A linear equation has at most one solution. A quadratic equation has at most two solutions. A trigonometric equation usually has infinitely many solutions unless the values of the variable are restricted , , k k, k, 6 6 k Glencoe/McGraw-Hill 96 Advanced Mathematical Concepts

24 , , , 45, 80, , 0, 80, , 50, 80, , ,,, , k, 6 6 k 39. k, 6 k 4. k k 45. k 47. k, k or , , , , ,,,,, ,,, ,, , k, 6 6 k k, 6 6 k 5 4. k, 3 3 k 44. k, k k, 3 3 k k, k, 6 6 k or , , Glencoe/McGraw-Hill 97 Advanced Mathematical Concepts

25 or s Sample answer: or a b. Measure the angles of incidence and refraction to determine the inde of refraction. If the inde is.4, the diamond is genuine. 60. b y y 3cos O about 8 rps 67. ( )( )( ) 69. (4, 3) 7. g() 66. undefined 68. ma: (, 7), min: (, 3) 70. (6,, 3) 7. C g() 3 O Glencoe/McGraw-Hill 98 Advanced Mathematical Concepts

26 7-6 Normal Form of a Linear Equation Pages Normal means perpendicular.. Compute cos 30 and 30. Use these as the coefficients of and y, respectively, in the normal form. The normal form 3 is y The statement is true. The given line is tangent to the circle centered at the origin with radius p. 4. Form Equation Information Displayed Slope-Intercept y m b slope and y-intercept Point-Slope y y m( ) slope and a point on the line Standard A By C 0 none Normal cos y p 0 length of the normal and the angle the normal makes with the -ais See students work for sample problems y y y y 0; ; y 0; y 3 0; 3; ; 8 a.. 3y 30 0 b..6 miles Glencoe/McGraw-Hill 99 Advanced Mathematical Concepts

27 3. y y y y y 5 0; 5; y y y y y 0; ; y 3 0; 3; y 0; ; ; 3; y y 0; ; y ; ; ; ; y ; ; y a. y O y units 34a b. tan 34c. cot 34d. cot.5 ft 33b. y.5 0 Glencoe/McGraw-Hill 00 Advanced Mathematical Concepts

28 5 35a. y b c. See students work. 35d. The line with normal form cos y p 0 makes an angle of with the positive -ais and has a normal of length p. The graph of Armando s equation is a line whose normal makes an angle of with the -ais and also has length p. Therefore, the graph of Armando s equation is the graph of the original line rotated counterclockwise about the origin. Armando is correct. See students graphs. 37. $ a. The angles of the quadrilateral are 80, 90,, and 90. Then , which simplifies to. If the lines intersect so that is an interior angle of the quadrilateral, the equation works out to be b. tan tan ( ) tan tan tan tan 38. 3, 40. If the lines intersect so that is an interior angle of the quadrilateral, the equation works out to be tan tan tan. tan tan 5 3 Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts

29 cm in. by 6.5 in. by.5 in. 45. (6, 3) 4. 6, 44. 6, E 7-7 Distance from a Point to a Line Pages The distance from a point to a line is the distance from that point to the closest point on the line. 3. In the figure, P and Q are any points on the lines. The right triangles are congruent by AAS. The corresponding congruent sides of the triangles show that the same distance is always obtained between the two lines. P. The sign should be chosen opposite the sign of C where A By C 0 is the standard form of the equation of the line. 4. The formula is valid in either case. Eamples will vary. For a vertical line, a, the formula subtracts a from the -coordinate of the point. For a horizontal line, y b, the formula subtracts b from the y-coordinate of the point. Q (0 63 ) (30 83)y ; (0 63) (83 30)y ft Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts

30 y 0; 6 y (0 33 ) (3 30 )y ; ( 0 33) (3 30 )y (68 57 ) (7 87 )y ; (68 57) (7 87)y a. Linda 8b. 48 Glencoe/McGraw-Hill 03 Advanced Mathematical Concepts

31 9..09 m ,, The radius of the circle is [(5) )] (6 ( ) or 5. Now find the distance from the center of the circle to the line. A By C d A B 5(5) ()(6) 3 d 5 ) ( 65 d 3 d 5 Since the distance from the center of the circle to the line is the same as the radius of the circle, the line can only intersect the circle in one point. That is, the line is tangent to the circle y y miles y csc ( 60 ) O about.8 s 39. (6,, 5) C Glencoe/McGraw-Hill 04 Advanced Mathematical Concepts