Lesson 10.2 Radian Measure and Arc Length

Transcription

1 Lesson 10.1 Defining the Circular Functions 1. Find the eact value of each epression. a. sin 0 b. cos 5 c. sin 150 d. cos 5 e. sin(0 ) f. sin(10 ) g. sin 15 h. cos 0 i. sin(0 ) j. sin 90 k. sin 70 l. sin 0. Use our calculator to find each value, approimated to four decimal places. Name each reference angle. a. sin 7 b. cos 115 c. sin(1 ) d. cos 195 e. sin 91 f. sin 0 g. cos 9 h. sin(10 ) i. cos(9 ) j. cos(8 ) k. sin l. cos(19 ). Determine whether each function whose graph is shown below is periodic or not periodic. For each periodic function, identif the period. a. b c. 8. Identif an angle that is coterminal with the given angle. Use domain 0 0. a. b. 15 c. 91 d. 1 e. 9 f. 119 g. 98 h Find sin and cos for each angle in standard position described. a. The terminal side of angle passes through the point (5, 1). b. The terminal side of angle passes through the point (, 5). c. The terminal side of angle passes through the point (1, ). CHAPTER 10 Discovering Advanced Algebra More Practice Your Skills 00 Ke Curriculum Press

2 Lesson 10. Radian Measure and Arc Length 1. Convert between radians and degrees. Give eact answers. a. 5 b. 15 c. 0 d. e. 10 f. 7 g. 1 h. 780 i. 7 j k. 10 l Find the intercepted arc length for each central angle. a. r 8 and 5 b. r 5. and.5 c. d 10 and 5 d. d and 1. Solve for. a. sin and b. cos and c. sin 1 and 70 0 d. sin 1 and 0 0 e. cos 1 and f. sin cos 1 and sin g. cos and h. 1 c os and 0. The minute hand on a watch is 85 mm long. Round our answers in a and b to the nearest tenth, and in c to the nearest thousandth. a. What is the distance the tip of the minute hand travels, in mm? b. At what speed is the tip moving, in mm/min? c. What is the angular speed of the tip in radians/min? Discovering Advanced Algebra More Practice Your Skills CHAPTER Ke Curriculum Press

3 Lesson 10. Graphing Trigonometric Functions 1. Find the period of each function in radians. a. sin b. cos c. tan d. sin 1 e. cos 1 f. tan 1 g. sin h. tan i. cos 5. Find the maimum and minimum values (if an) of each function. Also give the amplitude, if an. a. sin b. cos 8 c. tan d. sin e. cos f. tan g. sin h. 0.5 cos i..5 cos(0.5) 1.5. Write an equation for each sinusoid as a transformation of the graph of either sin or cos. More than one answer is possible. a. b. c. d.. Sketch the graph of each function for the interval 0. a. sin 1 b. sin c. tan d. cos e. tan f. cos 5. Write an equation for each sinusoid with the given characteristics. a. A cosine curve with amplitude.5, period, and phase shift b. A sine function with minimum value, maimum value 8, and one ccle starting at 0 and ending at CHAPTER 10 Discovering Advanced Algebra More Practice Your Skills 00 Ke Curriculum Press

4 Lesson 10. Inverses of Trigonometric Functions 1. Find the principal value of each epression to the nearest tenth of a degree and then to the nearest hundredth of a radian. a. sin b. cos 1 (0.95) c. cos 1 (0.015) d. sin 1 0. e. cos 1 (0.85) f. sin 1 (0.789). Find all four values of between and that satisf each equation. a. sin sin 5 b. cos cos 1 c. sin sin 1.5 d. cos cos 0.7 e. cos cos 9 5 f. sin sin 7 g. sin sin.5 h. cos cos 5.1 i. sin sin 5. Find values of approimate to three decimal places that satisf these criteria. a. Find the first two positive solutions of sin b. Find the first two positive solutions of cos c. Find the two solutions closest to zero of cos d. Find the first two positive solutions of tan.. Find each angle described to the nearest tenth of a degree. a. The largest angle of a triangle with sides of lengths 1.5 m, 0.15 m, and 17. m b. The smallest angle of a triangle with sides of lengths cm, 1 cm, and cm c. The smallest angle of a triangle in which two of the sides have lengths 1 cm and 8 cm, and in which the angle opposite the 8 cm side measures 110 Discovering Advanced Algebra More Practice Your Skills CHAPTER Ke Curriculum Press

5 Lesson 10.5 Modeling with Trigonometric Equations 1. Find all solutions for 0. Give eact values in radians. a. sin 1 b. cos 0.5 c. sin 1 d. cos 0 e. cos 0.5 f. sin 1 1 g. sin h. cos i. cos. Find all solutions for 0, rounded to the nearest hundredth. a. sin( 1.) 0.8 b. cos( 0.). c sin.7 d sin(.8) 10. Consider the graph of the function h sin (t ) 7 a. What is the vertical translation? b. What is the average value? c. What is the vertical stretch factor? d. What is the minimum value? e. What is the maimum value? f. What is the amplitude? g. What is the horizontal stretch factor? h. What is the period? i. What is the horizontal translation? j. What is the phase shift?. The number of hours of dalight on an da of the ear in Philadelphia, Pennslvania, is modeled using the equation 1. sin ( 80) 5 where represents the da number (with Januar 1 as da 1). This equation assumes a 5-da ear (not a leap ear). a. Find the number of hours of dalight in Philadelphia on da 17, the longest da of the ear (the summer solstice). b. Find the da numbers of the two das when the number of hours of dalight is closest to 1. c. Find the calendar dates for the summer solstice and for the two da numbers ou found in b. 8 CHAPTER 10 Discovering Advanced Algebra More Practice Your Skills 00 Ke Curriculum Press

6 Lesson 10. Fundamental Trigonometric Identities 1. Evaluate. Give eact values. a. tan b. cot 5 c. sec d. csc 11 e. cot f. sec g. csc h. cot i. csc. Find another function that has the same graph as each function below. (More than one answer is possible.) a. tan( ) b. sin( ) c. sin d. tan e. sec( ) f. cot() g. cos h. sec() i. csc( ). Use trigonometric identities to rewrite each epression in a simplified form containing onl sines and cosines, or as a single number. a. tan sec b. sec tan cos c. cot sin tan cos d. tan csc cos e. (csc cot )(csc cot ) f. 1 sec 1 csc (tan cos ). Determine whether each equation is an identit or not an identit. a. sin(a ) cos A b. tan A cot A c. (cos A sin A)(cos A sin A) 1 d. (sec A tan A)(sec A tan A) 1 e. csc A cot A(tan A cot A) f. (1 sin A)(1 sin A) cos A g. sec A cot A csc A h. (1 tan A)(1 tan A) sec A Discovering Advanced Algebra More Practice Your Skills CHAPTER Ke Curriculum Press

7 Lesson 10.7 Combining Trigonometric Functions 1. Use a graph or substitute values of A and B to decide whether each equation is an identit or not an identit. a. cos A 1 sin A b. sin(a B) sin A sin B c. sin( A) sin A d. cos(a ) cos A e. cos A cos A 1 f. sin A sin A g. tan A c os A cos A h. cos(a B) cos A cos B sin A sin B. Use identities from this lesson to derive an identit for sin A in terms of sin A and cos A. Show the steps ou used to derive the identit.. Rewrite each epression as a single sine or cosine. a. sin. cos.5 cos. sin.5 b. sin.8 cos.8 c. cos 0.9 cos 1.7 sin 0.9 sin 1.7 d. sin 7. cos.8 cos 7. sin.8 e. cos 0.8 sin 0.8 f. cos 0. cos.1 sin 0. sin.1. Use a sum or difference identit to find the eact value of each epression. a. sin 5 b. 1 cos 1 c. sin 1 d. cos 1 7 e. 1 sin 105 f. cos Find the eact values of sin, cos, and tan for each set of conditions. a. sin 5,0 b. cos 5 1, c. sin 1, d. cos 5, e. sec 1 0, f. csc 5, 70 CHAPTER 10 Discovering Advanced Algebra More Practice Your Skills 00 Ke Curriculum Press