Define General Angles and Use Radian Measure
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1 1.2 a.1, a.4, a.5; P..E TEKS Define General Angles and Use Radian Measure Before You used acute angles measured in degrees. Now You will use general angles that ma be measured in radians. Wh? So ou can find the area of a curved plaing field, as in Eample 4. Ke Vocabular initial side terminal side standard position coterminal radian sector central angle In Lesson 1.1, ou worked onl with acute angles. In this lesson, ou will stud angles with measures that can be an real numbers. KEY CONCEPT Angles in Standard Position In a coordinate plane, an angle can be formed b fiing one ra, called the initial side, and rotating the other ra, called the terminal side, about the verte. An angle is in standard position if its verte is at the origin and its initial side lies on the positive -ais terminal side 908 verte 2708 initial side The measure of an angle is positive if the rotation of its terminal side is counterclockwise, and negative if the rotation is clockwise. The terminal side of an angle can make more than one complete rotation. E XAMPLE 1 Draw angles in standard position Draw an angle with the given measure in standard position. a b c Solution a. Because 2408 is 08 more than 1808, the terminal side is 08 counterclockwise past the negative -ais. b. Because 5008 is 1408 more than 08, the terminal side makes one whole revolution counterclockwise plus 1408 more. c. Because 2508 is negative, the terminal side is 508 clockwise from the positive -ais Define General Angles and Use Radian Measure 859
2 COTERMINAL ANGLES In Eample 1, the angles 5008 and 1408 are coterminal because their terminal sides coincide. An angle coterminal with a given angle can be found b adding or subtracting multiples of 08. E XAMPLE 2 Find coterminal angles Find one positive angle and one negative angle that are coterminal with (a) 2458 and (b) 958. Solution There are man such angles, depending on what multiple of 08 is added or subtracted. a b (08) GUIDED PRACTICE for Eamples 1 and 2 Draw an angle with the given measure in standard position. Then find one positive coterminal angle and one negative coterminal angle RADIAN MEASURE Angles can also be measured in radians. To define a radian, consider a circle with radius r centered at the origin as shown. One radian is the measure of an angle in standard position whose terminal side intercepts an arc of length r. Because the circumference of a circle is 2r, there are 2 radians in a full circle. Degree measure and radian measure are therefore related b the equation radians, or radians. r 1 radian r KEY CONCEPT Converting Between Degrees and Radians Degrees to radians Multipl degree measure b p radians } Radians to degrees Multipl radian measure b 1808 }. p radians 80 Chapter 1 Trigonometric Ratios and Functions
3 E XAMPLE Convert between degrees and radians READING The unit radians is often omitted. For instance, the measure 2} p radians ma be 12 written simpl as 2} p. 12 Convert (a) 1258 to radians and (b) 2 p } 12 radians to degrees. a } p radians b. 2} p } p radians21} p radians p } radians CONCEPT SUMMARY Degree and Radian Measures of Special Angles The diagram shows equivalent degree and radian measures for special angles from 08 to 08 (0 radians to 2 radians). You ma find it helpful to memorize the equivalent degree and radian measures of special angles in the first quadrant and for 908 5} p radians. All other special angles 2 are just multiples of these angles radian measure degree 08 measure GUIDED PRACTICE for Eample Convert the degree measure to radians or the radian measure to degrees p } 4 8. p } 10 SECTORS OF CIRCLES Asector is a region of a circle that is bounded b two radii and an arc of the circle. The central angle u of a sector is the angle formed b the two radii. There are simple formulas for the arc length and area of a sector when the central angle is measured in radians. KEY CONCEPT Arc Length and Area of a Sector The arc length s and area A of a sector with radius r and central angle u (measured in radians) are as follows. Arc length: s 5 ru Area: A 5 1 } 2 r 2 u r central angle u sector arc length s 1.2 Define General Angles and Use Radian Measure 81
4 E XAMPLE 4 TAKS REASONING: Multi-Step Problem SOFTBALL A softball field forms a sector with the dimensions shown. Find the length of the outfield fence and the area of the field. Solution STEP 1 Convert the measure of the central angle to radians. AVOID ERRORS You must write the measure of an angle in radians when using the formulas for the arc length and area of a sector. STEP p radians } p } 2 radians Find the arc length and the area of the sector. Arc length: s 5 ru } p ø 28 feet Area: A 5 } 1 r 2 u 5 } 1 (180) } p ø 25,400 ft2 c The length of the outfield fence is about 28 feet. The area of the field is about 25,400 square feet. GUIDED PRACTICE for Eample 4 9. WHAT IF? In Eample 4, estimate the length of the outfield fence and the area of the field if the outfield fence is 220 feet from home plate. 1.2 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Es. 11, 2, and 51 5 TAKS PRACTICE AND REASONING Es. 14, 1, 50, 5, 55, and 5 EXAMPLES 1 and on pp for Es VOCABULARY Cop and complete: An angle is in standard position if its verte is at the? and its? lies on the positive -ais. 2. WRITIN G How does the sign of an angle s measure determine its direction of rotation? VISUAL THINKING Match the angle measure with the angle p } 4 A. B. C. 82 Chapter 1 Trigonometric Ratios and Functions
5 DRAWING ANGLES Draw an angle with the given measure in standard position p } p } 1. 2p 14. TAKS REASONING Which angle measure is shown in the diagram? A B 2108 C 5708 D 908 EXAMPLES 2 and on pp for Es FINDING COTERMINAL ANGLES Find one positive angle and one negative angle that are coterminal with the given angle p } p } p p } EXAMPLE on p. 81 for Es. 2 1 CONVERTING MEASURES Convert the degree measure to radians or the radian measure to degrees p p } TAKS REASONING Which angle measure is equivalent to } 1p radians? A 08 B 908 C 7508 D p } 15 EXAMPLE 4 on p. 82 for Es. 2 8 FINDING ARC LENGTH AND AREA Find the arc length and area of a sector with the given radius r and central angle u. 2. r 5 4 in., u 5 p }. r 5 m, u 5 5p } r 5 15 cm, u r 5 12 ft, u r 5 18 m, u r 5 25 in., u ERROR ANALYSIS Describe and correct the error in finding the area of a sector with a radius of centimeters and a central angle of 408. A 5 1 } 2 () 2 (40) cm 2 HINT For Es. 9 4, set our calculator in radian mode. EVALUATING FUNCTIONS Evaluate the trigonometric function using a calculator if necessar. If possible, give an eact answer. 9. cos p } 40. sin p } tan p } 42. sec p 4. cot p } cos p } 45. sin p } 7 4. csc 4p } CHALLENGE A rotating object that passes through an angle u during time t has an angular velocit v given b the formula v 5 u } t. Find the angular velocit of the hour hand, the minute hand, and the second hand on a 12 hour clock. Give all answers in degrees per hour. 1.2 Define General Angles and Use Radian Measure 8
6 PROBLEM SOLVING EXAMPLES 1 and on pp for Es ASTRONOMY In astronom, the terminator is the da-night line on a planet that divides the planet into datime and nighttime regions. The terminator moves across the planet s surface as the planet rotates. It takes about 4 hours for Earth s terminator to move across the continental United States. Through what angle has Earth rotated during this time? Give the answer in both degrees and radians. Terminator 49. CD PLAYER When a CD plaer reads information from the outer edge of a CD, the CD spins about 200 revolutions per minute. At that speed, through what angle does a point on the CD spin in one minute? Give the answer in both degrees and radians. 50. TAKS REASONING You work ever Saturda from 9:00 A.M. to 5:00 P.M. Draw a diagram that shows the rotation completed b the hour hand of a clock during this time. Find the measure of the angle generated b the hour hand in both degrees and radians. Compare this angle with the angle generated b the minute hand from 9:00 A.M. to 5:00 P.M. EXAMPLE 4 on p. 82 for Es MULTI-STEP PROBLEM A scientist performed an eperiment to stud the effects of gravitational force on humans. In order for humans to eperience twice Earth s gravit, the were placed in a centrifuge 58 feet long and spun at a rate of about 15 revolutions per minute. a. Through how man radians did the people rotate each second? b. Find the length of the arc through which the people rotated each second. 52. MULTI-STEP PROBLEM In the shot put event at the 2004 Summer Olmpic Games, the winning shot was 21.1 meters. For a shot put to be fair, it must land within a sector having a central angle of a. If the officials drew an arc across the fair landing area marking the farthest throw, how long would the arc be? b. All fair shot puts in the 2004 Olmpics landed within a sector bounded b the arc from part (a). What is the area of this sector? 5. TAKS REASONING A spiral staircase has 15 steps. Each step is a sector with a radius of 42 inches and a central angle of } p. 8 a. What is the length of the arc formed b the outer edge of a step? b. Through what angle would ou rotate b climbing the stairs? Include a siteenth turn for stepping up on the landing. Eplain our reasoning. c. How man square inches of carpeting would ou need to cover the 15 steps? at classzone.com 84 5 WORKED-OUT SOLUTIONS on p. WS1 5 TAKS PRACTICE AND REASONING
7 54. CHALLENGE A dartboard is divided into 20 sectors. Each sector is worth a point value from 1 to 20 and has shaded regions that double or triple this value. A sector is shown below. } 4 in. }8 in. 2 1 } 8 in. }8 in. Triple Double 5 } 8 in. a. Find the areas of the entire sector, the double region, and the triple region. b. A dart ou throw randoml lands somewhere inside the sector. What is the probabilit that it lands in the double region? in the triple region? MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson 1.5; TAKS Workbook REVIEW Lesson 10.5; TAKS Workbook 55. TAKS PRACTICE Lou saves $12 per week to bu an acoustic guitar that costs $280. Which equation best represents the relationship between the amount of mone Lou still needs to save, m, and the number of weeks, n, that he has been saving? TAKS Obj. 1 A m n B m n C m 5 ( )n D m 5 ( )n 5. TAKS PRACTICE Stewart randoml selects two cards from a standard deck of 52 cards. What is the probabilit that the first card is a heart and the second card is red if he replaces the first card before selecting the second? TAKS Obj. 9 F 0.0 G 0.12 H J 0.75 QUIZ for Lessons Solve n ABC using the diagram and the given measurements. (p. 852) 1. A 5 508, a A 5 258, b B 5 708, a B 5 428, c 5 18 B a c 5. A 5 158, a 5 9. B 5 78, c 5 12 C b A Find one positive angle and one negative angle that are coterminal with the given angle. (p. 859) p Find the arc length and area of a sector with a radius of 8 inches and a central angle of u (p. 859) 12. ESCALATOR The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles has an angle of elevation of 08. The length of the escalator is 152 feet. What is the height of the escalator? (p. 852) 7p } 5 EXTRA PRACTICE for Lesson 1.2, p ONLINE QUIZ at classzone.com 85
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