Trigonometric Functions

Transcription

1 LIALMC5_1768.QXP /6/4 1:4 AM Page 47 5 Trigonometric Functions Highwa transportation is critical to the econom of the United States. In 197 there were 115 billion miles traveled, and b the ear this increased to approimatel 5 billion miles. When an automobile travels around a curve, objects like trees, buildings, and fences situated on the curve ma obstruct a driver s vision. Trigonometr is used to determine how far inside the curve land must be cleared to provide visibilit for a safe stopping distance. (Source: Mannering, F. and W. Kilareski, Principles of Highwa Engineering and Traffic Analsis, nd Edition, John Wile & Sons, 1998.) A problem like this is presented in Eercise 61 of Section Angles 5. Trigonometric Functions 5. Evaluating Trigonometric Functions 5.4 Solving Right Triangles 47

2 LIALMC5_1768.QXP /6/4 1:4 AM Page CHAPTER 5 Trigonometric Functions 5.1 Angles Basic Terminolog Degree Measure Standard Position Coterminal Angles A B A B A B Figure 1 Terminal side Line AB Segment AB Ra AB Basic Terminolog Two distinct points A and B determine a line called line AB. The portion of the line between A and B, including points A and B themselves, is line segment AB, or simpl segment AB. The portion of line AB that starts at A and continues through B, and on past B, is called ra AB. Point A is the endpoint of the ra. (See Figure 1.) An angle is formed b rotating a ra around its endpoint. The ra in its initial position is called the initial side of the angle, while the ra in its location after the rotation is the terminal side of the angle. The endpoint of the ra is the verte of the angle. Figure shows the initial and terminal sides of an angle with verte A. If the rotation of the terminal side is counterclockwise, the angle is positive. If the rotation is clockwise, the angle is negative. Figure shows two angles, one positive and one negative. Verte A Figure Initial side C A B Positive angle Negative angle Figure An angle can be named b using the name of its verte. For eample, the angle on the right in Figure can be called angle C. Alternativel, an angle can be named using three letters, with the verte letter in the middle. Thus, the angle on the right also could be named angle ACB or angle BCA. A complete rotation of a ra gives an angle whose measure is 6. Figure 4 Degree Measure The most common unit for measuring angles is the degree. (The other common unit of measure, called the radian, is discussed in Section 6.1.) Degree measure was developed b the Bablonians, 4 ears ago. To use degree measure, we assign 6 degrees to a complete rotation of a ra.* In Figure 4, notice that the terminal side of the angle corresponds to its initial side when it makes a complete rotation. One degree, written 1, represents 1 9 of a rotation. Therefore, 9 represents of a complete rotation, and represents of a complete rotation. An angle measuring between and 9 is called an acute angle. An angle measuring eactl 9 is a right angle. An angle measuring more than 9 but less than 18 is an obtuse angle, and an angle of eactl 18 is a straight angle. See Figure 5, where we use the Greek letter (theta)** to name each angle. *The Bablonians were the first to subdivide the circumference of a circle into 6 parts. There are various theories as to wh the number 6 was chosen. One is that it is approimatel the number of das in a ear, and it has man divisors, which makes it convenient to work with. Another involves a roundabout theor dealing with the length of a Bablonian mile. **In addition to (theta), other Greek letters such as (alpha) and (beta) are sometimes used to name angles.

3 LIALMC5_1768.QXP /6/4 1:4 AM Page Angles 475 TEACHING TIP Point out that two angles can be complementar or supplementar without having a side in common. For instance, the two acute angles of a right triangle are complementar. Acute angle < < 9 Right angle = 9 Obtuse angle 9 < < 18 Figure 5 Straight angle = 18 If the sum of the measures of two positive angles is 9, the angles are called complementar. Two positive angles with measures whose sum is 18 are supplementar. EXAMPLE 1 Finding Measures of Complementar and Supplementar Angles Find the measure of each angle in Figure 6. (6m) (m) (4k) (a) (b) (6k) Figure 6 Solution (a) In Figure 6(a), since the two angles form a right angle (as indicated b the smbol), the are complementar angles. Thus, 6m m 9 9m 9 m 1. The two angles have measures of 61 6 and 1. (b) The angles in Figure 6(b) are supplementar, so 4k 6k 18 1k 18 k 18. These angle measures are and Now tr Eercises 1 and 15. O Figure 7 A = 5 Do not confuse an angle with its measure. Angle A of Figure 7 is a rotation; the measure of the rotation is 5. This measure is often epressed b saing that mangle A is 5, where mangle A is read the measure of angle A. It is convenient, however, to abbreviate mangle A 5 as A 5. Traditionall, portions of a degree have been measured with minutes and 1 seconds. One minute, written 1, is 6 of a degree. 1 1 or One second, 1, is of a minute or 6 1 The measure represents 1 degrees, 4 minutes, 8 seconds.

4 LIALMC5_1768.QXP /6/4 1:4 AM Page CHAPTER 5 Trigonometric Functions EXAMPLE Perform each calculation. (a) Calculating with Degrees, Minutes, and Seconds (b) Solution (a) Add the degrees and the minutes separatel. (b) Since , the sum is written Write 9 as Now tr Eercises and 7. Because calculators are now so prevalent, angles are commonl measured in decimal degrees. For eample, 1.48 represents , EXAMPLE Converting Between Decimal Degrees and Degrees, Minutes, and Seconds A graphing calculator performs the conversions in Eample as shown above. (a) Convert to decimal degrees. (b) Convert to degrees, minutes, and seconds. Solution (a) (b) and Add; round to the nearest thousandth Now tr Eercises and 7.

5 LIALMC5_1768.QXP /6/4 1:4 AM Page Angles 477 Verte Standard Position An angle is in standard position if its verte is at the origin and its initial side is along the positive -ais. The angles in Figures 8(a) and 8(b) are in standard position. An angle in standard position is said to lie in the quadrant in which its terminal side lies. An acute angle is in quadrant I (Figure 8(a)) and an obtuse angle is in quadrant II (Figure 8(b)). Figure 8(c) shows ranges of angle measures for each quadrant when Angles in standard position having their terminal sides along the -ais or -ais, such as angles with measures 9, 18, 7, and so on, are called quadrantal angles. Q I Terminal side Initial side Q II 18 Q II 9 < < 18 Q III 18 < < Q I < < 9 Q IV 7 < < 6 6 (a) (b) (c) Figure 8 Coterminal Angles A complete rotation of a ra results in an angle measuring 6. B continuing the rotation, angles of measure larger than 6 can be produced. The angles in Figure 9 with measures 6 and 4 have the same initial side and the same terminal side, but different amounts of rotation. Such angles are called coterminal angles; their measures differ b a multiple of 6. As shown in Figure 1, angles with measures 11 and 8 are coterminal Coterminal angles Figure 9 Figure Coterminal angles EXAMPLE 4 Finding Measures of Coterminal Angles Figure 11 Find the angles of smallest possible positive measure coterminal with each angle. (a) 98 (b) 75 Solution (a) Add or subtract 6 as man times as needed to obtain an angle with measure greater than but less than 6. Since , an angle of 188 is coterminal with an angle of 98. See Figure 11. (b) Use a rotation of See Figure 1. Figure 1 Now tr Eercises 45 and 49.

6 LIALMC5_1768.QXP /6/4 1:4 AM Page CHAPTER 5 Trigonometric Functions Sometimes it is necessar to find an epression that will generate all angles coterminal with a given angle. For eample, we can obtain an angle coterminal with 6 b adding an appropriate integer multiple of 6 to 6. Let n represent an integer; then the epression 6 n 6 represents all such coterminal angles. The table shows a few possibilities. Value of n Angle Coterminal with (the angle itself) EXAMPLE 5 Analzing the Revolutions of a CD Plaer CAV (Constant Angular Velocit) CD plaers alwas spin at the same speed. Suppose a CAV plaer makes 48 revolutions per min. Through how man degrees will a point on the edge of a CD move in sec? Solution The plaer revolves 48 times in 1 min or 6 times 8 times per sec (since 6 sec 1 min). In sec, the plaer will revolve 8 16 times. Each revolution is 6, so a point on the edge of the CD will revolve in sec. 48 Now tr Eercise Eercises (a) 6 (b) (a) (b) 1 7. (a) 45 (b) (a) 7 (b) (a) 6 (b) (a) 1 (b) Eplain the difference between a segment and a ra.. What part of a complete revolution is an angle of 45?. Concept Check What angle is its own complement? 4. Concept Check What angle is its own supplement? Find (a) the complement and (b) the supplement of each angle Find the measure of the smaller angle formed b the hands of a clock at the following times

7 LIALMC5_1768.QXP /6/4 1:4 AM Page Angles ; ; ; ; ; ; C and D n 6 45 n 6 15 n 6 7 n 6 9 n 6 15 n 6 Find the measure of each angle in Eercises See Eample (7) 16. supplementar angles with measures 1m 7 and 7m degrees 17. supplementar angles with measures 6 4 and 8 1 degrees 18. complementar angles with measures 9z 6 and z degrees Concept Check (11) Answer each question. 19. If an angle measures, how can we represent its complement?. If an angle measures, how can we represent its supplement? 1. If a positive angle has measure between and 6, how can we represent the first negative angle coterminal with it?. If a negative angle has measure between and 6, how can we represent the first positive angle coterminal with it? Perform each calculation. See Eample (4) () (5k + 5) (k + 5) Convert each angle measure to decimal degrees. Round to the nearest thousandth of a degree. See Eample Convert each angle measure to degrees, minutes, and seconds. See Eample Read about the degree smbol () in the manual for our graphing calculator. How is it used? 44. Show that 1.1 hr is the same as 1 hr, 1 min, 6 sec. Discuss the similarit between converting hours, minutes, and seconds to decimal hours and converting degrees, minutes, and seconds to decimal degrees. Find the angle of smallest positive measure coterminal with each angle. See Eample Give an epression that generates all angles coterminal with each angle. Let n represent an integer Eplain wh the answers to Eercises 56 and 57 give the same set of angles. 6. Concept Check Which two of the following are not coterminal with r? A. 6 r B. r 6 C. 6 r D. r 18

8 LIALMC5_1768.QXP /6/4 1:4 AM Page CHAPTER 5 Trigonometric Functions Angles other than those given are possible in Eercises Concept Check 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 ; 85 ; quadrant I ; 71 ; quadrant I Concept Check Locate each point in a coordinate sstem. Draw a ra from the origin through the given point. Indicate with an arrow the angle in standard position having smallest positive measure. Then find the distance r from the origin to the point, using the distance formula of Section , 7. 5, ,1 7., 74., 5 4, 4 54 ; 186 ; quadrant II ; 6 ; quadrant IV ; 16 ; quadrant III ; 8 ; quadrant II Solve each problem. See Eample Revolutions of a Turntable A turntable in a shop makes 45 revolutions per min. How man revolutions does it make per second? 76. Revolutions of a Windmill A windmill makes 9 revolutions per min. How man revolutions does it make per second? 77. Rotating Tire A tire is rotating 6 times per min. Through how man degrees does a point on the edge of 1 the tire move in sec? ; 41 ; quadrant IV (, ) (, 5) ( 5, ) 5 1 ; 519 ; quadrant III 1 (, 1) 78. Rotating Airplane Propeller An airplane propeller rotates 1 times per min. Find the number of degrees that a point on the edge of the propeller will rotate in 1 sec. 79. Rotating Pulle A pulle rotates through 75 in 1 min. How man rotations does the pulle make in an hour? 8. Surveing One student in a surveing class measures an angle as 74.5, while another student measures the same angle as 74. Find the difference between these measurements, both to the nearest minute and to the nearest hundredth of a degree (, ) rotations per hr 8. 5 or sec 4 4 (4, 4) 81. Viewing Field of a Telescope Due to Earth s rotation, celestial objects like the moon and the stars appear to move across the sk, rising in the east and setting in the west. As a result, if a telescope on Earth remains stationar while viewing a celestial object, the object will slowl move outside the viewing field of the telescope. For this reason, a motor is often attached to telescopes so that the telescope rotates at the same rate as Earth. Determine how long it should take the motor to turn the telescope through an angle of 1 min in a direction perpendicular to Earth s ais.

9 LIALMC5_1768.QXP /6/4 1:4 AM Page Trigonometric Functions Angle Measure of a Star on the American Flag Determine the measure of the angle in each point of the five-pointed star appearing on the American flag. (Hint: Inscribe the star in a circle, and use the following theorem from geometr: An angle whose verte lies on the circumference of a circle is equal to half the central angle that cuts off the same arc. See the figure.) 5. Trigonometric Functions Trigonometric Functions Quadrantal Angles Reciprocal Identities Signs and Ranges of Function Values Pthagorean Identities Quotient Identities P(, ) r Q O Figure 1 Trigonometric Functions To define the si trigonometric functions, we start with an angle in standard position, and choose an point P having coordinates, on the terminal side of angle. (The point P must not be the verte of the angle.) See Figure 1. A perpendicular from P to the -ais at point Q determines a right triangle, having vertices at O, P, and Q. We find the distance r from P, to the origin,,, using the distance formula. r (Section.1) Notice that r since distance is never negative. The si trigonometric functions of angle are sine, cosine, tangent, cotangent, secant, and cosecant. In the following definitions, we use the customar abbreviations for the names of these functions. Trigonometric Functions Let, be a point other than the origin on the terminal side of an angle in standard position. The distance from the point to the origin is r. The si trigonometric functions of are defined as follows. sin r cos r tan csc r sec r cot NOTE Although Figure 1 shows a second quadrant angle, these definitions appl to an angle. Because of the restrictions on the denominators in the definitions of tangent, cotangent, secant, and cosecant, some angles will have undefined function values.

10 LIALMC5_1768.QXP /6/4 1:4 AM Page CHAPTER 5 Trigonometric Functions 17 8 (8, 15) 15 Figure 14 = 8 = 15 r = 17 EXAMPLE 1 Finding Function Values of an Angle The terminal side of an angle in standard position passes through the point 8, 15. Find the values of the si trigonometric functions of angle. Solution Figure 14 shows angle and the triangle formed b dropping a perpendicular from the point 8, 15 to the -ais. The point 8, 15 is 8 units to the right of the -ais and 15 units above the -ais, so 8 and 15. Since r, We can now find the values of the si trigonometric functions of angle. sin r csc r r cos r 8 17 sec r 17 8 tan 15 8 cot 8 15 Now tr Eercise 7. EXAMPLE Finding Function Values of an Angle The terminal side of an angle in standard position passes through the point, 4. Find the values of the si trigonometric functions of angle. Solution As shown in Figure 15, and 4. The value of r is 4 5 (, 4) = = 4 r = 5 Then b the definitions of the trigonometric functions, sin csc 4 r cos 5 5 sec 5 5 Remember that r. tan cot Figure 15 Now tr Eercise. OP = r OP = r (, ) P (, ) P O Q Q Figure 16 We can find the si trigonometric functions using an point other than the origin on the terminal side of an angle. To see wh an point ma be used, refer to Figure 16, which shows an angle and two distinct points on its terminal side. Point P has coordinates,, and point P (read P-prime ) has coordinates,. Let r be the length of the hpotenuse of triangle OPQ, and let r be the length of the hpotenuse of triangle OPQ. Since corresponding sides of similar triangles are proportional, r r, so sin r is the same no matter which point is used to find it. A similar result holds for the other five trigonometric functions.

11 LIALMC5_1768.QXP /6/4 1:4 AM Page Trigonometric Functions 48 = = 1 r = 5 + =, Figure 17 (, 1) + =, Figure 18 We can also find the trigonometric function values of an angle if we know the equation of the line coinciding with the terminal ra. Recall from algebra that the graph of the equation (Section.) is a line that passes through the origin. If we restrict to have onl nonpositive or onl nonnegative values, we obtain as the graph a ra with endpoint at the origin. For eample, the graph of,, shown in Figure 17, is a ra that can serve as the terminal side of an angle in standard position. B choosing a point on the ra, we can find the trigonometric function values of the angle. EXAMPLE Finding Function Values of an Angle Find the si trigonometric function values of the angle in standard position, if the terminal side of is defined b,. Solution The angle is shown in Figure 18. We can use an point ecept, on the terminal side of to find the trigonometric function values. We choose and find the corresponding -value., Let. Subtract. 1 Divide b. The point, 1 lies on the terminal side, and the corresponding value of r is r 1 5. Now we use the definitions of the trigonometric functions. sin cos tan r r 1 A B Rationalize denominators. (Section R.7) csc r 5 sec r 5 cot Now tr Eercise 17. Recall that when the equation of a line is written in the form m b, the coefficient of is the slope of the line. In Eample, can be written as 1, so the slope is 1. Notice that tan 1. In general, it is true that m tan. NOTE The trigonometric function values we found in Eamples 1 are eact. If we were to use a calculator to approimate these values, the decimal results would not be acceptable if eact values were required.

12 LIALMC5_1768.QXP /6/4 1:4 AM Page CHAPTER 5 Trigonometric Functions Quadrantal Angles If the terminal side of an angle in standard position lies along the -ais, an point on this terminal side has -coordinate. Similarl, an angle with terminal side on the -ais has -coordinate for an point on the terminal side. Since the values of and appear in the denominators of some trigonometric functions, and since a fraction is undefined if its denominator is, some trigonometric function values of quadrantal angles (i.e., those with terminal side on an ais) are undefined. EXAMPLE 4 Finding Function Values of Quadrantal Angles Find the values of the si trigonometric functions for each angle. (a) an angle of 9 (b) an angle in standard position with terminal side through, Solution (a) First, we select an point on the terminal side of a 9 angle. We choose the point, 1, as shown in Figure 19. Here and 1, so r 1. Then, sin cos 9 1 tan 9 1 undefined csc sec 9 1 undefined cot 9 1. A calculator in degree mode returns the correct values for sin 9 and cos 9. The second screen shows an ERROR message for tan 9, because 9 is not in the domain of the tangent function. (, 1) 9 (, ) θ Figure 19 Figure (b) Figure shows the angle. Here,,, and r, so the trigonometric functions have the following values. sin csc undefined cos 1 sec 1 tan cot undefined Now tr Eercises 5 and 9. The conditions under which the trigonometric function values of quadrantal angles are undefined are summarized here. Undefined Function Values If the terminal side of a quadrantal angle lies along the -ais, then the tangent and secant functions are undefined. If it lies along the -ais, then the cotangent and cosecant functions are undefined.

13 LIALMC5_1768.QXP /6/4 1:4 AM Page Trigonometric Functions 485 The function values of the most commonl used quadrantal angles,, 9, 18, 7, and 6, are summarized in the following table. sin cos tan cot sec csc 1 Undefined 1 Undefined 9 1 Undefined Undefined Undefined 1 Undefined 7 1 Undefined Undefined Undefined 1 Undefined The values given in this table can be found with a calculator that has trigonometric function kes. Make sure the calculator is set in degree mode. CAUTION One of the most common errors involving calculators in trigonometr occurs when the calculator is set for radian measure, rather than degree measure. (Radian measure of angles is discussed in Chapter 6.) Be sure ou know how to set our calculator in degree mode. TEACHING TIP Students ma be tempted to associate secant with sine and cosecant with cosine. Note this common misconception. Reciprocal Identities Identities are equations that are true for all values of the variables for which all epressions are defined. 6 Identities The definitions of the trigonometric functions at the beginning of this section were written so that functions in the same column are reciprocals of each other. Since sin r and csc r, sin 1 csc and csc 1, sin provided sin. Also, cos and sec are reciprocals, as are tan and cot. In summar, we have the reciprocal identities that hold for an angle that does not lead to a denominator. Reciprocal Identities (a) sin 1 csc csc 1 sin cos 1 sec sec 1 cos tan 1 cot cot 1 tan (b) Figure 1 The screen in Figure 1(a) shows how to find csc 9, sec 18, and csc7, using the appropriate reciprocal identities and the reciprocal ke of a graphing calculator in degree mode. Be sure not to use the inverse trigonometric function kes to find the reciprocal function values. Attempting to find sec 9 b entering 1cos 9 produces an ERROR message, indicating the reciprocal is undefined. See Figure 1(b). Compare these results with the ones found in the table of quadrantal angle function values.

14 LIALMC5_1768.QXP /6/4 1:4 AM Page CHAPTER 5 Trigonometric Functions NOTE Identities can be written in different forms. For eample, sin 1 csc can be written csc 1 sin and sin csc 1. EXAMPLE 5 Find each function value. Using the Reciprocal Identities (a) cos, if sec 5 (b) sin, if Solution (a) Since cos is the reciprocal of sec, (b) sin cos sec 5. Simplif the comple fraction. (Section R.5) sin 1 csc 1 4 (Section R.7) Simplif. Multipl b csc 1 to rationalize the denominator. Now tr Eercises 45 and 47. TEACHING TIP Some students use the sentence All Students Take Calculus to remember which of the three basic functions are positive in each quadrant. A indicates all in quadrant I, S represents sine in quadrant II, T represents tangent in quadrant III, and C stands for cosine in quadrant IV. Signs and Ranges of Function Values In the definitions of the trigonometric functions, r is the distance from the origin to the point,. Distance is never negative, so r. If we choose a point, in quadrant I, then both and will be positive. Thus, the values of all si functions will be positive in quadrant I. A point, in quadrant II has and. This makes the values of sine and cosecant positive for quadrant II angles, while the other four functions take on negative values. Similar results can be obtained for the other quadrants, as summarized on the net page.

15 LIALMC5_1768.QXP /6/4 1:4 AM Page Trigonometric Functions 487 Signs of Function Values in Quadrant sin cos tan cot sec csc I II III IV <, >, r > II Sine and cosecant positive <, <, r > III Tangent and cotangent positive >, >, r > I All functions positive >, <, r > IV Cosine and secant positive EXAMPLE 6 Identifing the Quadrant of an Angle Identif the quadrant (or quadrants) of an angle that satisfies tan. sin, Solution Since sin in quadrants I and II, while tan in quadrants II and IV, both conditions are met onl in quadrant II. Now tr Eercise 57. r Figure shows an angle as it increases in measure from near toward 9. In each case, the value of r is the same. As the measure of the angle increases, increases but never eceeds r, so r. Dividing both sides b the positive number r gives r 1. In a similar wa, angles in quadrant IV suggest that 1 r, r so and 1 r 1 1 sin 1. r sin for an angle. r Similarl, 1 cos 1. The tangent of an angle is defined as. It is possible that,, or. Thus, can take an value, so tan can be an real number, as can cot. The functions sec and csc are reciprocals of the functions cos and sin, respectivel, making sec 1 or sec 1and csc 1 or csc 1. In summar, the ranges of the trigonometric functions are as follows. r Figure Ranges of Trigonometric Functions For an angle for which the indicated functions eist: 1. 1 sin 1 and 1 cos 1;. tan and cot can equal an real number;. sec 1or sec 1 and csc 1 or csc 1. (Notice that sec and csc are never between 1 and 1.)

16 LIALMC5_1768.QXP /6/4 1:4 AM Page CHAPTER 5 Trigonometric Functions EXAMPLE 7 Solution Deciding Whether a Value Is in the Range of a Trigonometric Function Decide whether each statement is possible or impossible. (a) sin 8 (b) tan (c) sec.6 (a) For an value of, 1 sin 1. Since 8 1, it is impossible to find a value of with sin 8. (b) Tangent can equal an value. Thus, tan is possible. (c) Since sec 1or sec 1, the statement sec.6 is impossible. Now tr Eercises 71 and 7. We derive three new identities from the relation- Pthagorean Identities ship r. Divide b r. Power rule for eponents (Section R.) or sin cos 1. Starting again with r and dividing through b gives or r r r r r r 1 cos sin 1 r 1 r Divide b. Power rule for eponents 1 tan sec tan, sec r tan 1 sec. On the other hand, dividing through b leads to These three identities are called the Pthagorean identities since the original equation that led to them, r, comes from the Pthagorean theorem. Pthagorean Identities 1 cot cos r, sin r csc. sin cos 1 tan 1 sec 1 cot csc Although we usuall write sin, for eample, it should be entered as sin in our calculator. To test this, verif that in degree mode, sin

17 LIALMC5_1768.QXP /6/4 1:4 AM Page Trigonometric Functions 489 As before, we have given onl one form of each identit. However, algebraic transformations produce equivalent identities. For eample, b subtracting sin from both sides of sin cos 1, we get the equivalent identit cos 1 sin. You should be able to transform these identities quickl and also recognize their equivalent forms. Looking Ahead to Calculus The reciprocal, Pthagorean, and quotient identities are used in calculus to find derivatives and integrals of trigonometric functions. A standard technique of integration called trigonometric substitution relies on the Pthagorean identities. Quotient Identities Recall that sin r and cos r. Consider the quotient of sin and cos, where cos. cos sin cos r r r r r r tan Similarl, sin cot, for sin. Thus, we have the quotient identities. Quotient Identities sin tan cos cos cot sin EXAMPLE 8 Finding Other Function Values Given One Value and the Quadrant Find sin and cos, if tan 4 and is in quadrant III. Solution Since is in quadrant III, sin and cos will both be negative. It is tempting to sa that since tan sin cos and tan 4, then sin 4 and cos. This is incorrect, however, since both sin and cos must be in the interval 1, 1. We use the Pthagorean identit tan 1 sec to find sec, and then the reciprocal identit cos 1 to find cos. tan 1 sec 4 1 sec sec sec 5 9 sec tan 4 5 sec 5 cos Choose the negative square root since sec is negative when is in quadrant III. Secant and cosine are reciprocals.

18 LIALMC5_1768.QXP /6/4 1:4 AM Page CHAPTER 5 Trigonometric Functions Since sin 1 cos sin 1 5 sin sin 16 5 sin, cos 5 Choose the negative square root. Therefore, we have sin 4 5 and cos 5. Now tr Eercise 79. NOTE Eample 8 can also be worked b drawing in standard position in quadrant III, finding r to be 5, and then using the definitions of sin and cos in terms of,, and r. 5. Eercises In Eercises 1 and 17 1, we give, in order, sine, cosine, tangent, cotangent, secant, and cosecant ; 5 5 ; 5 ; 4 ; 4 ; ; 4 5 ; 4 ; 4 ; 5 4 ; ; ; undefined; ; undefined; 1 6. ; 1; ; undefined; 1; undefined 7. ; 1 ; ; ; ; 8. 1 ; ; ; ; ; 9. ; 1; ; undefined; 1; undefined ; 5 ; 4 ; 4 ; 5 ; (5, 1) ( 1, 5) 9 Concept Check Sketch an angle in standard position such that has the smallest possible positive measure, and the given point is on the terminal side of. 1. 5, 1. 1, 5 Find the values of the si trigonometric functions for each angle in standard position having the given point on its terminal side. Rationalize denominators when applicable. See Eamples 1,, and 4.., , 5., 6. 4, 7. 1, 8., 9., 1., For an nonquadrantal angle, sin and csc will have the same sign. Eplain wh. 1. Concept Check If the terminal side of an angle is in quadrant III, what is the sign of each of the trigonometric function values of? Concept Check Suppose that the point, is in the indicated quadrant. Decide whether the given ratio is positive or negative. (Hint: Drawing a sketch ma help.) 1. II, 14. III, 15. IV, 16. IV, r r In Eercises 17, an equation of the terminal side of an angle in standard position is given with a restriction on. Sketch the smallest positive such angle, and find the values of the si trigonometric functions of. See Eample. 17., 18. 5, 19. 6,. 5, 1. Find the si trigonometric function values of the quadrantal angle 45.

19 LIALMC5_1768.QXP /6/4 1:4 AM Page Trigonometric Functions tan and cot are positive; all other function values are negative. 1. negative 14. negative 15. negative 16. negative ; 5 5 ; ; ; 54 4 ; 5 ; 5 ; 4 5 ; (, 5) ( 1, 6) 6 6 =, ; 7 7 ; 6; 1 6 ; 7; =, 1 (1, ) (5, ) 54 4 ; 4 4 ; 5 ; 5 ; ; ; ; 4 5 undefined; ; undefined; undefined. The are equal.. The are equal. 4. The are negatives of each other. 5. The are equal. 6. about.94; about decrease; increase 5 + =, + 5 =, ; 5; 5 Use the trigonometric function values of quadrantal angles given in this section to evaluate each epression. An epression such as cot 9 means cot 9, which is equal to.. sec 18 5 tan 6. 4 csc 7 cos tan 6 4 sin 18 5 cos sec 4 cot 9 cos 6 6. sin 18 cos sin 6 cos 6 If n is an integer, n 18 represents an integer multiple of 18, and n 1 9 represents an odd integer multiple of 9. Decide whether each epression is equal to, 1, 1, or is undefined. 8. cosn sinn 18. tann tann 1 9 Provide conjectures in Eercises 5.. The angles 15 and 75 are complementar. With our calculator determine sin 15 and cos 75. Make a conjecture about the sines and cosines of complementar angles, and test our hpothesis with other pairs of complementar angles. (Note: This relationship will be discussed in detail in the net section.). The angles 5 and 65 are complementar. With our calculator determine tan 5 and cot 65. Make a conjecture about the tangents and cotangents of complementar angles, and test our hpothesis with other pairs of complementar angles. (Note: This relationship will be discussed in detail in the net section.) 4. With our calculator determine sin 1 and sin1. Make a conjecture about the sines of an angle and its negative, and test our hpothesis with other angles. Also, use a geometr argument with the definition of sin to justif our hpothesis. (Note: This relationship will be discussed in detail in Section 7.1.) 5. With our calculator determine cos and cos. Make a conjecture about the cosines of an angle and its negative, and test our hpothesis with other angles. Also, use a geometr argument with the definition of cos to justif our hpothesis. (Note: This relationship will be discussed in detail in Section 7.1.) In Eercises 6 41, set our graphing calculator in parametric and degree modes. Set the window and functions (see the third screen) as shown here, and graph. A circle of radius 1 will appear on the screen. Trace to move a short distance around the circle. In the screen, the point on the circle corresponds to an angle T 5. Since r 1, cos 5 is X.96779, and sin 5 is Y This screen is a continuation of the previous one Use the right- and left-arrow kes to move to the point corresponding to. What are cos and sin? 7. For what angle T, T 9, is cos T.766? 8. For what angle T, T 9, is sin T.574? 9. For what angle T, T 9, does cos T sin T? 4. As T increases from to 9, does the cosine increase or decrease? What about the sine? 1.

20 LIALMC5_1768.QXP /6/4 1:4 AM Page CHAPTER 5 Trigonometric Functions 41. decrease; decrease 4. 1; 4. 1; The range of the cosine function is 1, 1, so cos cannot equal II 58. I 59. I or III 6. II or IV 61. ; ; 6. ; ; 6. ; ; 64. ; ; 65. ; ; 66. ; ; 67. tan 68. sin sec 7. impossible 71. impossible 7. possible 7. possible 74. possible 75. possible 76. possible 77. impossible As T increases from 9 to 18, does the cosine increase or decrease? What about the sine? 4. Concept Check What positive number a is its own reciprocal? Find a value of for which sin csc a. 4. Concept Check What negative number a is its own reciprocal? Find a value of for which cos sec a. Use the appropriate reciprocal identit to find each function value. Rationalize denominators when applicable. In Eercises 48 and 49, use a calculator. See Eample cos, if sec cot, if tan sin, if csc tan, if 48. sin, if csc cos, if sec Can a given angle satisf both sin and csc? Eplain. 51. Concept Check Eplain what is wrong with the following item that appears on a trigonometr test: Find sec, given that cos. Find the tangent of each angle. See Eample 5. cot 5 5. cot 5. cot 54. cot.4 Find a value of each variable tan 4 cot5 8 sec 6 cos5 1 Identif the quadrant or quadrants for the angle satisfing the given conditions. See Eample sin, cos tan, cot 6. cos, tan tan, cot Concept Check Give the signs of the sine, cosine, and tangent functions for each angle Concept Check Without using a calculator, decide which is greater. 67. sin or tan 68. sin or sin sin or sec Decide whether each statement is possible or impossible for an angle. See Eample sin 71. cos tan.9 7. cot sec tan sin 1 and csc 77. tan and cot Use identities to find each function value. Use a calculator in Eercises 84 and 85. See Eample tan, if sec, with in quadrant IV

21 LIALMC5_1768.QXP /6/4 1:4 AM Page Trigonometric Functions es In Eercises 87 9, we give, in order, sine, cosine, tangent, cotangent, secant, and cosecant ; 17 ; 15 8 ; 15 ; 8 ; ; 5 ; ; 4 ; ; ; 4 ; ; 5 ; 9. ; 7 ; ; ; 7 ; 5 ; ;.814; ; ; 1.87; ;.98645; ; 6.691; 1.176; false; for eample, sin cos false; sin 1 for all ft 79. sin, if cos 1, with in quadrant II 4 8. csc, if cot 1, with in quadrant IV 81. sec, if tan 7, with in quadrant III 8. cos, if csc 4, with in quadrant III 8. sin, if sec, with in quadrant IV 84. cot, if csc.58914, with in quadrant III 85. tan, if sin.49689, with in quadrant II 86. Concept Check Does there eist an angle with cos.6 and sin.8? Find all trigonometric function values for each angle. Use a calculator in Eercises 91 and 9. See Eample tan 15 with in quadrant II 88. cos with in quadrant III 8, 5, 89. tan, with in quadrant III 9. sin 5 with in quadrant I 7, 91. cot , with in quadrant IV 9. sin.16415, with in quadrant II Work each problem. csc 9. Derive the identit 1 cot b dividing r b. sin 94. Using a method similar to the one given in this section showing that cos tan, cos show that sin cot. 95. Concept Check True or false: For all angles, sin cos 1. If false, give an eample showing wh it is false. 96. Concept Check True or false: Since cot cos sin, if cot 1 with in quadrant I, then cos 1 and sin. If false, eplain wh. Use a trigonometric function ratio to solve each problem. (Source for Eercises 97 98: Parker, M., Editor, She Does Math, Mathematical Association of America, 1995.) 97. Height of a Tree A civil engineer must determine the height of the tree shown in the figure. The given angle was measured with a clinometer. She knows that sin 7.997, cos 7.4, and tan Use the pertinent trigonometric function and the measurement given in the figure to find the height of the tree to the nearest whole number. 7 5 ft This is a picture of one tpe of clinometer, called an Abne hand level and clinometer. The picture is courtes of Keuffel & Esser Co.

22 LIALMC5_1768.QXP /6/4 1:4 AM Page CHAPTER 5 Trigonometric Functions 98. (a) 1 prism diopters (b) tan (a) tan (b) tan 1. area sin 98. (Modeling) Double Vision To correct mild double vision, a small amount of prism is added to a patient s eeglasses. The amount of light shift this causes is measured in prism diopters. A patient needs 1 prism diopters horizontall and 5 prism diopters verticall. A prism that corrects for both requirements should have length r and be set at angle. See the figure. 5 r 1 (a) Use the Pthagorean theorem to find r. (b) Write an equation involving a trigonometric function of and the known prism measurements 5 and (Modeling) Distance Between the Sun and a Star Earth Suppose that a star forms an angle with respect to Earth and the sun. Let the coordinates of Earth be,, those of the star,, and those of the sun,. See the figure. Find an equation for, the distance between the sun and the star, as follows. Star θ (a) Write an equation involving a trigonometric function that relates,, and. (b) Solve our equation for. r Sun Not to scale 1. Area of a Solar Cell A solar cell converts the energ of sunlight directl into electrical energ. The amount of energ a cell produces depends on its area. Suppose a solar cell is heagonal, as shown in the figure. Epress its area in terms of sin and an side. (Hint: Consider one of the si equilateral triangles from the heagon. See the figure.) (Source: Kastner, B., Space Mathematics, NASA, 1985.) h = The straight line in the figure determines both angle (alpha) and angle (beta) with the positive -ais. Eplain wh tan tan. β α

23 LIALMC5_1768.QXP /6/4 1:4 AM Page Evaluating Trigonometric Functions Evaluating Trigonometric Functions Definitions of the Trigonometric Functions Trigonometric Function Values of Special Angles Reference Angles Special Angles as Reference Angles Finding Function Values with a Calculator Finding Angle Measures A r B (, ) C Figure Definitions of the Trigonometric Functions In Section 5. we used angles in standard position to define the trigonometric functions. There is another wa to approach them: as ratios of the lengths of the sides of right triangles. Figure shows an acute angle A in standard position. The definitions of the trigonometric function values of angle A require,, and r. As drawn in Figure, and are the lengths of the two legs of the right triangle ABC, and r is the length of the hpotenuse. The side of length is called the side opposite angle A, and the side of length is called the side adjacent to angle A. We use the lengths of these sides to replace and in the definitions of the trigonometric functions, and the length of the hpotenuse to replace r, to get the following right-triangle-based definitions. TEACHING TIP Introduce the mnemonic sohcahtoa to help students remember that sine is opposite over hpotenuse, cosine is adjacent over hpotenuse, and tangent is opposite over adjacent. These definitions will be used in applications of trigonometr in Section 5.4. Right-Triangle-Based Definitions of Trigonometric Functions For an acute angle A in standard position, sin A r cos A r tan A side opposite hpotenuse side adjacent hpotenuse side opposite side adjacent csc A r sec A r cot A hpotenuse side opposite hpotenuse side adjacent side adjacent side opposite. B 7 C 5 4 Figure 4 sin A A EXAMPLE 1 Finding Trigonometric Function Values of an Acute Angle Find the values of sin A, cos A, and tan A in the right triangle in Figure 4. Solution The length of the side opposite angle A is 7, the length of the side adjacent to angle A is 4, and the length of the hpotenuse is 5. Use the relationships given in the bo. side opposite hpotenuse 7 5 cos A side adjacent hpotenuse 4 5 tan A side opposite side adjacent 7 4 Now tr Eercise 1. NOTE Because the cosecant, secant, and cotangent ratios are the reciprocals of the sine, cosine, and tangent values, respectivel, in Eample 1 we can conclude that csc A 5 sec A 5 and cot A 4 7, 4, 7.

24 LIALMC5_1768.QXP /6/4 1:4 AM Page CHAPTER 5 Trigonometric Functions Equilateral triangle (a) right triangle (b) Figure Figure 6 Trigonometric Function Values of Special Angles Certain special angles, such as,45, and 6, occur so often in trigonometr and in more advanced mathematics that the deserve special stud. We start with an equilateral triangle, a triangle with all sides of equal length. Each angle of such a triangle measures 6. While the results we will obtain are independent of the length, for convenience we choose the length of each side to be units. See Figure 5(a). Bisecting one angle of this equilateral triangle leads to two right triangles, each of which has angles of, 6, and 9, as shown in Figure 5(b). Since the hpotenuse of one of these right triangles has length, the shortest side will have length 1. (Wh?) If represents the length of the medium side, then, Pthagorean theorem (Section 1.5) Subtract 1.. Choose the positive root. (Section 1.4) Figure 6 summarizes our results using a 6 right triangle. As shown in the figure, the side opposite the angle has length 1; that is, for the angle, Now we use the definitions of the trigonometric functions. EXAMPLE Finding Trigonometric Function Values for 6 Find the si trigonometric function values for a 6 angle. Solution cos hpotenuse, side opposite 1, side adjacent. side opposite sin hpotenuse 1 side adjacent hpotenuse Refer to Figure 6 to find the following ratios. sin 6 csc 6 side opposite tan side adjacent 1 cos 6 1 sec 6 csc 1 sec cot 1 tan 6 cot 6 Now tr Eercises 11, 1, and r = right triangle Figure 7 We find the values of the trigonometric functions for 45 b starting with a right triangle, as shown in Figure 7. This triangle is isosceles; we choose the lengths of the equal sides to be 1 unit. (As before, the results are independent of the length of the equal sides.) Since the shorter sides each have length 1, if r represents the length of the hpotenuse, then 1 1 r r r. Pthagorean theorem Choose the positive square root.

25 LIALMC5_1768.QXP /6/4 1:4 AM Page Evaluating Trigonometric Functions 497 Now we use the measures indicated on the right triangle in Figure 7. sin 45 1 csc cos 45 sec 45 1 tan cot Function values for, 45, and 6 are summarized in the table that follows. TEACHING TIP Tell students that a good wa to obtain one of these function values is to draw the appropriate famous right triangle as shown in Figure 6 or 7 and then use the right-triangle-based definition of the function value. Function Values of Special Angles sin cos tan cot sec csc Reference Angles Associated with ever nonquadrantal angle in standard position is a positive acute angle called its reference angle. A reference angle for an angle, written, is the positive acute angle made b the terminal side of angle and the -ais. Figure 8 shows several angles (each less than one complete counterclockwise revolution) in quadrants II, III, and IV, respectivel, with the reference angle also shown. In quadrant I, and are the same. If an angle is negative or has measure greater than 6, its reference angle is found b first finding its coterminal angle that is between and 6, and then using the diagrams in Figure 8. O O O in quadrant II in quadrant III Figure 8 in quadrant IV CAUTION A common error is to find the reference angle b using the terminal side of and the -ais. The reference angle is alwas found with reference to the -ais.

26 LIALMC5_1768.QXP /6/4 1:4 AM Page CHAPTER 5 Trigonometric Functions = 8 Figure EXAMPLE Finding Reference Angles Find the reference angle for each angle. (a) 18 (b) 187 Solution (a) As shown in Figure 9, the positive acute angle made b the terminal side of this angle and the -ais is For the reference angle 8. (b) First find a coterminal angle between and 6. Divide 187 b 6 to get a quotient of about.9. Begin b subtracting 6 three times (because of the in.9): The reference angle for 7 (and thus for 187 ) is See Figure. 18, Now tr Eercises 41 and = 5 Figure Special Angles as Reference Angles We can now find eact trigonometric function values of angles with reference angles of, 45, or 6. EXAMPLE 4 Finding Trigonometric Function Values of a Quadrant III Angle Find the values of the trigonometric functions for P O r Figure 1 1 = = 1 r = Solution An angle of 1 is shown in Figure 1. The reference angle is To find the trigonometric function values of 1, choose point P on the terminal side of the angle so that the distance from the origin O to P is. B the results from 6 right triangles, the coordinates of point P become, 1, with, 1, and r. Then, b the definitions of the trigonometric functions, sin 1 1 csc 1 cos 1 sec 1 tan 1 cot 1. Now tr Eercise 59. TEACHING TIP Review the signs of each trigonometric function b quadrant. As mentioned previousl, the sentence All Students Take Calculus ma help some students remember signs of the basic functions b quadrant. Notice in Eample 4 that the trigonometric function values of 1 correspond in absolute value to those of its reference angle. The signs are different for the sine, cosine, secant, and cosecant functions because 1 is a quadrant III angle. These results suggest a shortcut for finding the trigonometric function values of a nonacute angle, using the reference angle. In Eample 4, the reference angle for 1 is. Using the trigonometric function values of, and choosing the correct signs for a quadrant III angle, we obtain the results found in Eample 4.

27 LIALMC5_1768.QXP /6/4 1:4 AM Page Evaluating Trigonometric Functions 499 Similarl, we determine the values of the trigonometric functions for an nonquadrantal angle b finding the function values for its reference angle between and 9, and choosing the appropriate signs. Finding Trigonometric Function Values for An Nonquadrantal Angle 6,, Step 1 If or if then find a coterminal angle b adding or subtracting 6 as man times as needed to get an angle greater than but less than 6. Step Find the reference angle. Step Find the trigonometric function values for reference angle. Step 4 Determine the correct signs for the values found in Step. (Use the table of signs in Section 5., if necessar.) This gives the values of the trigonometric functions for angle. EXAMPLE 5 Finding Trigonometric Function Values Using Reference Angles = 6 = 4 Find the eact value of each epression. (a) cos4 (b) tan 675 Solution (a) Since an angle of 4 is coterminal with an angle of (a) 4 6 1, the reference angle is , as shown in Figure (a). Since the cosine is negative in quadrant II, cos4 cos 1 cos 6 1. = 675 = 45 (b) Figure (b) Begin b subtracting 6 to get a coterminal angle between and As shown in Figure (b), the reference angle is An angle of 15 is in quadrant IV, so the tangent will be negative, and tan 675 tan 15 tan Now tr Eercises 65 and 67. Degree mode Figure Finding Function Values with a Calculator Calculators are capable of finding trigonometric function values. For eample, the values of cos4 and tan 675, found in Eample 5, are found with a calculator as shown in Figure.

28 LIALMC5_1768.QXP /6/4 1:4 AM Page 5 5 CHAPTER 5 Trigonometric Functions TEACHING TIP Caution students not to use the sin 1, cos 1, and tan 1 kes when evaluating reciprocal functions with a calculator. CAUTION We have studied onl degree measure of angles; radian measure will be introduced in Chapter 6. When evaluating trigonometric functions of angles given in degrees, remember that the calculator must be set in degree mode. Get in the habit of alwas starting work b entering sin 9. If the displaed answer is 1, then the calculator is set for degree measure. EXAMPLE 6 Finding Function Values with a Calculator Approimate the value of each epression. (a) sin 49 1 (b) sec (c) cot (d) sin46 These screens support the results of Eample 6. We entered the angle measure in degrees and minutes for part (a). In the fifth line of the first screen, Ans 1 tells the calculator to find the reciprocal of the answer given in the previous line. Solution (a) Convert 49 1 to decimal degrees. (Section 5.1) sin 49 1 sin To eight decimal places (b) Calculators do not have secant kes. However, sec cos 1 for all angles where cos. First find cos , and then take the reciprocal to get (c) (d) cot sin sec Use the identit cot 1 tan. Now tr Eercises 75, 79, 81, and 85. Finding Angle Measures Sometimes we need to find the measure of an angle having a certain trigonometric function value. Graphing calculators have three inverse functions (denoted sin 1, cos 1, and tan 1 ) that do just that. If is an appropriate number, then sin 1, cos 1, or tan 1 give the measure of an angle whose sine, cosine, or tangent is. For the applications in this section, these functions will return values of in quadrant I. Degree mode Figure 4 EXAMPLE 7 Using an Inverse Trigonometric Function to Find an Angle Use a calculator to find an angle in the interval, 9 sin that satisfies Solution With the calculator in degree mode, we find that an angle having sine value is (While there are infinitel man such angles, the calculator gives onl this one.) We write this result as See Figure 4. sin Now tr Eercise 91.