7-1. Basic Trigonometric Identities

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1 7- BJECTIVE Identif and use reciprocal identities, quotient identities, Pthagorean identities, smmetr identities, and opposite-angle identities. Basic Trigonometric Identities PTICS Man sunglasses have polarized lenses that reduce the intensit of light. When unpolarized light passes through a polarized lens, the intensit of the light is cut in half. If the light then passes through another polarized lens with its ais at an angle of to the first, the intensit of the light is again diminished. Real World p plic atio n Unpolarized light Lens is Lens is The intensit of the emerging light can be found b using the formula I I I 0 0, csc where I 0 is the intensit of the light incoming to the second polarized lens, I is the intensit of the emerging light, and is the angle between the aes of polarization. Simplif this epression and determine the intensit of light emerging from a polarized lens with its ais at a 0 angle to the original. This problem will be solved in Eample 5. In algebra, variables and constants usuall represent real numbers. The values of trigonometric functions are also real numbers. Therefore, the language and operations of algebra also appl to trigonometr. lgebraic epressions involve the operations of addition, subtraction, multiplication, division, and eponentiation. These operations are used to form trigonometric epressions. Each epression below is a trigonometric epression. cos sin a cos a sec tan statement of equalit between two epressions that is true for all values of the variable(s) for which the epressions are defined is called an identit. For eample, ( )( ) is an algebraic identit. n identit involving trigonometric epressions is called a trigonometric identit. If ou can show that a specific value of the variable in an equation makes the equation false, then ou have produced a countereample. It onl takes one countereample to prove that an equation is not an identit. Lesson 7- Basic Trigonometric Identities

2 Eample Prove that sin cos tan is not a trigonometric identit b producing a countereample. Suppose. sin cos tan sin cos tan Replace with. Since evaluating each side of the equation for the same value of produces an inequalit, the equation is not an identit. lthough producing a countereample can show that an equation is not an identit, proving that an equation is an identit generall takes more work. Proving that an equation is an identit requires showing that the equalit holds for all values of the variable where each epression is defined. Several fundamental trigonometric identities can be verified using geometr. Recall from Lesson 5- that the trigonometric functions can be defined using the unit circle. From the unit circle, sin, or and csc. That is, sin. Identities derived in this manner are cs c called reciprocal identities. (, ) Reciprocal Identities The following trigonometric identities hold for all values of where each epression is defined. sin cos cs c se c csc sec si n co s tan cot co t ta n sin Returning to the unit circle, we can sa that tan. This is an c os eample of a quotient identit. Quotient Identities The following trigonometric identities hold for all values of where each epression is defined. sin c os tan c os cot sin Chapter 7 Trigonometric Identities and Equations

3 Since the triangle in the unit circle on the previous page is a right triangle, we ma appl the Pthagorean Theorem:, or sin cos. ther identities can be derived from this one. sin cos sin c os c os cos cos Divide each side b cos. tan sec Quotient and reciprocal identities Likewise, the identit cot csc can be derived b dividing each side of the equation sin cos b sin. These are the Pthagorean identities. Pthagorean Identities The following trigonometric identities hold for all values of where each epression is defined. sin cos tan sec cot csc You can use the identities to help find the values of trigonometric functions. Eample Use the given information to find the trigonometric value. a. If sec, find cos. cos Choose an identit that involves cos and sec. se c or Substitute for sec and evaluate. b. If csc, find tan. Since there are no identities relating csc and tan, we must use two identities, one relating csc and cot and another relating cot and tan. csc cot Pthagorean identit cot Substitute for csc. 6 cot cot 7 cot Take the square root of each side. Now find tan. tan Reciprocal identit co t 7, or about. 7 Lesson 7- Basic Trigonometric Identities

4 To determine the sign of a function value, ou need to know the quadrant in which the angle terminates. The signs of function values in different quadrants are related according to the smmetries of the unit circle. Since we can determine the values of tan, cot, sec, and csc in terms of sin and/or cos with the reciprocal and quotient identities, we onl need to investigate sin and cos. Relationship Case between Diagram Conclusion angles and B The angles differ b a multiple of 60. B 60k or B 60 k 60k (a, b) Since and 60k are coterminal, the share the same value of sine and cosine. The angles differ b an odd multiple of 80. B 80 (k ) or B 80 (k ) 80 (k ) (a, b) (a, b) Since and 80 (k ) have terminal sides in diagonall opposite quadrants, the values of both sine and cosine change sign. The sum of the angles is a multiple of 60. B 60k or B 60k 60k (a, b) (a, b) Since and 60k lie in verticall adjacent quadrants, the sine values are opposite but the cosine values are the same. The sum of the angles is an odd multiple of 80. B 80 (k ) or B 80 (k ) 80 (k ) (a, b) (a, b) Since and 80 (k ) lie in horizontall adjacent quadrants, the sine values are the same but the cosine values are opposite. These general rules for sine and cosine are called smmetr identities. Smmetr Identities Case : Case : Case : Case : The following trigonometric identities hold for an integer k and all values of. sin ( 60k ) sin cos ( 60k ) cos sin ( 80 (k )) sin cos ( 80 (k )) cos sin (60k ) sin cos (60k ) cos sin (80 (k ) ) sin cos (80 (k ) ) cos To use the smmetr identities with radian measure, replace 80 with and 60 with. Chapter 7 Trigonometric Identities and Equations

5 Eample Epress each value as a trigonometric function of an angle in Quadrant I. a. sin 600 Relate 600 to an angle in Quadrant I (80 ) 600 and 60 differ b an odd multiple of 80. sin 600 sin (60 (80 )) Case, with 60 and k sin 60 b. sin 9 The sum of 9 and, which is 0 or 5, is an odd multiple of. 9 5 sin 9 sin 5 Case, with and k sin c. cos (0 ) The sum of 0 and 50 is a multiple of cos (0 ) cos (60 50 ) Case, with 50 and k cos 50 d. tan and differ b a multiple of () 6 6 Case, with and k 6 tan 7 6 sin 7 6 cos 7 6 sin () 6 cos () 6 sin 6 or tan Quotient identit cos 6 6 Rewrite using a quotient identit since the smmetr identities are in terms of sine and cosine. Lesson 7- Basic Trigonometric Identities 5

6 Case of the Smmetr Identities can be written as the opposite-angle identities when k 0. pposite- ngle Identities The following trigonometric identities hold for all values of. sin () sin cos () cos The basic trigonometric identities can be used to simplif trigonometric epressions. Simplifing a trigonometric epression means that the epression is written using the fewest trigonometric functions possible and as algebraicall simplified as possible. This ma mean writing the epression as a numerical value. Eamples Simplif sin sin cot. sin sin cot sin ( cot ) sin csc sin sin sin csc Factor. Pthagorean identit: cot csc Reciprocal identit Reciprocal identit Eample 5 PTICS Refer to the application at the beginning of the lesson. Real World p plic atio n I0 a. Simplif the formula I I 0. cs c b. Use the simplified formula to determine the intensit of light that passes through a second polarizing lens with ais at 0 to the original. I0 a. I I 0 cs c I I 0 I 0 sin I I 0 ( sin ) I I 0 cos Reciprocal identit Factor. sin cos b. I I 0 cos 0 I I 0 I I 0 The light has three-fourths the intensit it had before passing through the second polarizing lens. 6 Chapter 7 Trigonometric Identities and Equations

7 C HECK FR U NDERSTNDING Communicating Mathematics Read and stud the lesson to answer each question.. Find a countereample to show that the equation sin cos is not an identit.. Eplain wh the Pthagorean and opposite-angle identities are so named.. Write two reciprocal identities, one quotient identit, and one Pthagorean identit, each of which involves cot.. Prove that tan () tan using the quotient and opposite-angle identities. 5. You Decide Claude and Rosalinda are debating whether an equation from their homework assignment is an identit. Claude sas that since he has tried ten specific values for the variable and all of them worked, it must be an identit. Rosalinda eplained that specific values could onl be used as countereamples to prove that an equation is not an identit. Who is correct? Eplain our answer. Guided Practice Prove that each equation is not a trigonometric identit b producing a countereample. 6. sin cos tan 7. sec csc Use the given information to determine the eact trigonometric value. 8. cos, 0 90 ; sec 9. cot 5, ; tan 0. sin 5, ; cos. tan, ; sec 7 Epress each value as a trigonometric function of an angle in Quadrant I.. cos 7. csc (0 ) Simplif each epression.. c sc cot 5. cos csc tan 6. cos cot sin 7. Phsics When there is a current in a wire in a magnetic field, a force acts on the wire. The strength of the magnetic field can be determined using the formula B F c sc, where F is the force on the wire, I is the current in the wire, is the I length of the wire, and is the angle the wire makes with the magnetic field. Phsics tets often write the formula as F IB sin. Show that the two formulas are equivalent. Practice E XERCISES Prove that each equation is not a trigonometric identit b producing a countereample. 8. sin cos cot 9. s ec sin 0. sec tan c os csc. sin cos.sin tan cos. tan cot Lesson 7- Basic Trigonometric Identities 7

8 . Find a value of for which cos cos cos. Use the given information to determine the eact trigonometric value. B 5. sin 5, 0 90 ; csc 6. tan, 0 ; cot 7. sin, 0 ; cos 8. cos, ; sin 9. csc, ;cot 0. sec 5, ; tan. sin, ; tan. tan, ; cos. sec 7 5, ; sin. cos 8, ; tan 5. cot, ; sin 6. cot 8, ; csc 7. If is a second quadrant angle, and cos sec, find tan. sin cos Epress each value as a trigonometric function of an angle in Quadrant I. 8. sin cos tan 9. csc 0 5. sec (90 ). cot (660 ) Simplif each epression. C. s ec co 5. tan c os sin ( ) 6. c os ( ) 7. (sin cos ) (sin cos ) 8. sin cos sec cot 9. cos tan sin cot 50. ( cos )(csc cot ) 5. cot cos cos cot sin sin 5. cos cos 5. cos cos sin sin pplications and Problem Solving Real World p plic atio n 5. ptics Refer to the equation derived in Eample 5. What angle should the aes of two polarizing lenses make in order to block all light from passing through? 55. Critical Thinking Use the unit circle definitions of sine and cosine to provide a geometric interpretation of the opposite-angle identities. 8 Chapter 7 Trigonometric Identities and Equations

9 56. Dermatolog It has been shown that skin cancer is related to sun eposure. The rate W at which a person s skin absorbs energ from the sun depends on the energ S, in watts per square meter, provided b the sun, the surface area eposed to the sun, the abilit of the bod to absorb energ, and the angle between the sun s ras and a line perpendicular to the bod. The abilit of an object to absorb energ is related to a factor called the emissivit, e, of the object. The emissivit can be calculated using the formula e W sec. S a. Solve this equation for W. Write our answer using onl sin or cos. b. Find W if e 0.80, 0, 0.75 m, and S 000 W/m. 57. Phsics skier of mass m descends a -degree hill at a constant speed. When Newton s Laws are applied to the situation, the following sstem of equations is produced. F N mg cos 0 mg sin k F N 0 where g is the acceleration due to gravit, F N is the normal force eerted on the skier, and k is the coefficient of friction. Use the sstem to define k as a function of. 58. Geometr Show that the area of a regular polgon of n sides, each of length a, is given b na cot 8 0 n. 59. Critical Thinking The circle at the right is a unit circle with its center at the origin. B and CD are tangent to the circle. State the segments whose measures represent the ratios sin, cos, tan, sec, cot, and csc. Justif our answers. E B F C D Mied Review 60. Find Cos. (Lesson 6-8) 6. Graph cos 6. (Lesson 6-5) 6. Phsics pendulum 0 centimeters long swings 0 on each side of its vertical position. Find the length of the arc formed b the tip of the pendulum as it swings. (Lesson 6-) 6. ngle C of BC is a right angle. Solve the triangle if 0 and c 5. (Lesson 5-) 6. Find all the rational roots of the equation 8 0. (Lesson -) 65. Solve 7 0 b completing the square. (Lesson -) 66. Determine whether f() 5 is continuous or discontinuous at 5. (Lesson -5) Etra Practice See p. 8. Lesson 7- Basic Trigonometric Identities 9

10 67. Solve the sstem of equations algebraicall. (Lesson -) z z 0 5 z 68. Write the slope-intercept form of the equation of the line that passes through points at (5, ) and (, ). (Lesson -) 69. ST/CT Practice Triangle BC is inscribed in circle, and CD is tangent to circle at point C. If mbcd 0, find m. 60 B 50 C 0 D 0 E 0 B D C CREER CHICES Do maps fascinate ou? Do ou like drawing, working with computers, and geograph? You ma want to consider a career in cartograph. s a cartographer, ou would make maps, charts, and drawings. Cartograph has changed a great deal with modern technolog. Computers and satellites have become powerful new tools in making maps. s a cartographer, ou ma work with manual drafting tools as well as computer software designed for making maps. The image at the right shows how a cartographer uses a three-dimensional landscape to create a two-dimensional topographic map. There are several areas of specialization in the field of cartograph. Some of these include making maps from political boundaries and natural features, making maps from aerial photographs, and correcting original maps. Cartographer CREER VERVIEW Degree Preferred: bachelor s degree in engineering or a phsical science Related Courses: mathematics, geograph, computer science, mechanical drawing utlook: slower than average through 006 For more information on careers in cartograph, visit: 0 Chapter 7 Trigonometric Identities and Equations

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