( ) ( ) ( ) Odd-Numbered Answers to Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios ( ) MATH 1330 Precalculus

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1 Odd-Numbered Answers to Eercise Set.: Special Right Triangles and Trigonometric Ratios. angles. 80. largest, smallest 7. (a) 9. (a). (a) , so + ( ) 8 8 8, so + 8. (a) 8, so ( 8 ) ( 8 ) (a), so + ( ) (a) a b The altitude divides the equilateral triangle to two congruent 0 o -0 o -90 o triangles. Therefore, a b. Sce a+ b 0, then a+ a 0, so a. Hence a b. (c) + c 0 + c 00 c 7 c 7 c. (a) 9 9 9, so , 7., 7., 9.,., 0 MATH 0 Precalculus 79

2 Odd-Numbered Answers to Eercise Set.: Special Right Triangles and Trigonometric Ratios. (a) ( BC) ( BC) ( BC) BC ( A) ( B) ( A) ( B) 8 8 ( A) ( B) 7. ( θ), ( θ) 7. ecant 9. cogent. e. (a) + + ( α) csc( α) ( α) sec( α) ( α) cot ( α). (a) ( α) csc( α) ( α) sec( α) ( α) cot ( α) csc (c) ( β) ( β) ( β) sec( β) ( β) cot ( β) ( θ) csc( θ) ( θ) sec( θ) ( θ) cot ( θ) Contued on the net page (c) ( β) csc( β) ( β) sec( β) ( β) cot ( β) 770 Universit of Houston Department of Mathematics

3 Odd-Numbered Answers to Eercise Set.: Special Right Triangles and Trigonometric Ratios 9. (a) o 0 o 0 o that ( 0 ) value of ( 0 ) both eercises. In fact, the is alwas, and the trigonometric ratio of an given number is const. csc sec cot (c) 0 csc 0 0 sec 0 0 cot 0 (d) ( 0 ) csc( 0 ) 0 sec 0 ( 0 ) cot ( 0 ). Answers var, but some import observations are below. In addition to the observations above, the followg formulas, known as the cofunction identities, are illustrated parts (d), and will be covered more detail Chapter. ( θ) ( θ) eg.. ( 0 ) ( 0 ) ( 0 ) ( 0 ) ( ) ( ) 90, ( θ) ( θ) eg.. sec( 0 ) csc( 0 ) sec( 0 ) csc( 0 ) sec( ) csc( ) sec csc 90, ( θ) ( θ) eg.. ( 0 ) cot ( 0 ) ( 0 ) cot ( 0 ) ( ) cot( ) cot 90, First, notice that when comparg Eercises 9 and 0: The answers to part are identical, for each respective trigonometric function. So, for eample, ( ) both eamples. Similarl, the answers to part (c) are identical. So, for eample, ( 0 ) regardless of the triangle beg used to compute the ratio. Along the same les, the answers to part (d) both eercises are identical. Notice, for eample, MATH 0 Precalculus 77

4 Odd-Numbered Answers to Eercise Set.: Radians, Arc Length, and the Area of a Sector. θ. θ 0. θ 7. (a) There are slightl more than radians one revolution, as shown the figure below. There are radians a complete revolution. Justification: The arc length this case is the circumference C of the circle, so s C r. Therefore, s r θ. r r Comparison to part (a):.8, and it can be seen from part (a) that there are slightl more than radians one revolution. 9. (a) 0..7 (c). (a). (a) radians radians radians radian radians 0 radians radians (c) (c) s ft 7. s d 9. s ft. θ ; cm s (Notice that s cm the solution for Eercise.).. θ ; s ft (Notice that s ft the solution for Eercise.) s 7. s 0 cm r m.8 m 8 r ft.97 ft r. 0 r m.8 m P + cm 0.9 cm A cm. (a) 0 (c) 0. A 0 ft 7. (a) 80 (c) 0 9. (a) s cm d A A ft θ ; A cm (Notice that the solution for Eercise 9.) A cm 77 Universit of Houston Department of Mathematics

5 Odd-Numbered Answers to Eercise Set.: Radians, Arc Length, and the Area of a Sector 9. ; 0 ft θ A (Notice that solution for Eercise.). r 7 cm.. r θ 7 A 0 ft the Three methods of solution are shown below: Method : (shown the tet) The lear speed v can be found ug the d formula v, where d is the disce traveled t b the pot on the CD time t. 900 Sce the rate of turn is 900 /sec, we sec note that θ 900 and t sec. 7. (a) Two methods of solution are shown below. Method : (shown the tet) The angular speed ω can be found ug the θ formula ω, where θ is radians. Sce the t 900 rate of turn is 900 /sec, we note that sec θ 900 and t sec. First, convert 900 to radians: θ θ radians Then ω radians/sec t sec Method : Take the given rate of turn, 900 /sec, equivalent to 900, and use unit conversion ratios to sec change this rate of turn from degrees/sec to radians/sec, as shown below: 900 radians 900 radians sec 80 sec 80 radians/sec To fd d, use the formula d rθ (equivalent to the formula for arc length, s rθ ). We must first convert θ 900 to radians: θ d rθ cm 0 cm. Then Fall, use the formula for lear speed: d 0 cm v 0 cm/sec t sec Method : Another formula for the lear speed v is v rω, where ω is the angular speed. (This can be derived from previous formulas, ce d rθ θ v r rω.) t t t It is given that r cm, and it is known from part (a) that w radians/sec, so: v rw ( cm)( radians/sec) 0 cm/sec. (Remember that units are not written for radians, with the eception of angular speed.) Method : Take the given rate of turn, 900 / sec, equivalent to 900, and use unit conversion ratios to sec change this rate of turn from degrees/sec to cm/sec, as shown below. Contued on the net page MATH 0 Precalculus 77

6 Odd-Numbered Answers to Eercise Set.: Radians, Arc Length, and the Area of a Sector (Note that one revolution, the circumference of the circle with a cm radius is C r cm, hence the unit conversion ratio cm. revolution 900 radians revolution cm sec 80 radians revolution 900 sec 0 cm/sec radians revolution cm 80 radians revolution (c) This problem can be solved one of three was: Method : The lear speed v can be found ug the d formula v, where d is the disce traveled t b the pot on the CD time t. 900 Sce the rate of turn is 900 /sec, we sec note that θ 900 and t sec. To fd d, use the formula d rθ (equivalent to the formula for arc length, s rθ ). We must first convert θ 900 to radians: θ The CD has a radius of cm, so the desired radius of the pot halfwa between the center of the CD and its outer edge is r ( ) cm. d rθ cm cm. Then Fall, use the formula for lear speed: d cm v cm/sec t sec Method : Another formula for the lear speed v is v rω, where ω is the angular speed. (This can be derived from previous formulas, ce d rθ θ v r rω.) t t t The CD has a radius of cm, so the desired radius of the pot halfwa between the center of the CD and its outer edge is r ( ) cm. It is known from part (a) that w radians/sec, so: v rw ( cm)( radians/sec) cm/sec. (Remember that units are not written for radians, with the eception of angular speed.) Method : Take the given rate of turn, 900 / sec, equivalent to 900, and use unit conversion ratios to sec change this rate of turn from degrees/sec to cm/sec, as shown below. (Note that one revolution, the circumference of the circle with a cm radius is C r cm, hence the unit conversion ratio cm. revolution 900 radians revolution cm sec 80 radians revolution 900 sec cm/sec radians revolution cm 80 radians revolution 9. Sce the diameter is ches, the radius, r. Take the given rate of turn, 0 miles/hr, equivalent to 0 miles, and use unit conversion ratios to change hr this rate of turn from miles/hr to rev/m, as shown below. Contued on the net page 77 Universit of Houston Department of Mathematics

7 Odd-Numbered Answers to Eercise Set.: Radians, Arc Length, and the Area of a Sector (Note that one revolution, the circumference of the circle with a radius is C r, hence the unit conversion ratio revolution. 0 miles 80 ft hr revolution hr mile ft 0 m 0 miles 80 ft hr revolution hr mile ft 0 m 8.7 rev/m, or 8.7 rpm 7. Numbers 7 and 8 show various methods for solvg these tpes of problems. One method is shown below for each question. (a) Take the given rate of turn, revolutions per second, equivalent to revolutions, and sec use unit conversion ratios to change this rate of turn from revolutions/sec to radians/sec, as shown below. revolutions radians second revolution revolutions radians second revolution 8 radians/sec v rω ( ) 0 8 radians/sec, so v 80 /sec (c) Use the lear speed from part, v 80 /sec, and then use unit conversion ratios to change this from /sec to miles/hr, as shown below: 80 ft mile 0 sec m sec 80 ft 0 sec 0 m 80 ft mile 0 sec 0 m sec 80 ft m hr.8 miles per hour (mph) 7. Hour Hand: r Rate of turn: revolution hours (Note that one revolution, the circumference of the circle with a radius is C r 8, hence the unit conversion ratio revolution. 8 hr revolution 8 0 m 0 m hr revolution hr revolution 8 0 m 0 m hr revolution 9 Mute Hand: r Rate of turn: revolution 0 m (Note that one revolution, the circumference of the circle with a radius is C r 0, hence the unit conversion ratio revolution 0. revolution 0 0 m 0 m revolution revolution 0 m 0 m 0 Second Hand: r Rate of turn: revolution m Contued on the net page 0 revolution MATH 0 Precalculus 77

8 Odd-Numbered Answers to Eercise Set.: Radians, Arc Length, and the Area of a Sector (Note that one revolution, the circumference of the circle with a radius is C r, hence the unit conversion ratio revolution. revolution 0 m m revolution revolution 0 m m 0 revolution 77 Universit of Houston Department of Mathematics

9 Odd-Numbered Answers to Eercise Set.: Unit Circle Trigonometr. (a) (c) 0. (a) 90 (c) 00. (a) (c) 0 7. (a) 7 0 MATH 0 Precalculus 777

10 Odd-Numbered Answers to Eercise Set.: Unit Circle Trigonometr The reference angle is 0, shown blue below. 0 0 (c) (c) The reference angle is 0, shown blue below (a) 0, 0, 770 0, 0, (a) 8,,,, 9. (a) The reference angle is, shown blue below.. 0, 90, 80, ,,, 7. (a) The reference angle is 0, shown blue below. 0 7 The reference angle is, shown blue below. 0 Contued the net column Contued on the net page 778 Universit of Houston Department of Mathematics

11 Odd-Numbered Answers to Eercise Set.: Unit Circle Trigonometr (c) The reference angle is, shown blue below (a) The reference angle is 0, shown blue below The reference angle is, shown blue below (c) The reference angle is 0, shown blue below II. III. IV MATH 0 Precalculus 779

12 Odd-Numbered Answers to Eercise Set.: Unit Circle Trigonometr. 7. < 9. >. ( θ) csc( θ) 780 ( θ) sec( θ) ( θ) cot ( θ). ( θ) csc( θ) ( θ) sec( θ) ( θ) cot ( θ) ( θ) sec( θ) ( θ) cot ( θ). ( θ) csc( θ) 7. The termal side of the angle tersects the unit circle at the pot ( 0, ). ( ) ( ) ( ) ( ) ( ) ( ) 90 csc sec 90 is undefed 90 is undefed cot The termal side of the angle tersects the unit circle at the pot (, 0 ). ( ) ( ) ( ) ( ) 0 csc is undefed sec 0 cot is undefed. The termal side of the angle tersects the unit 0,. circle at the pot csc 0 sec is undefed is undefed cot 0. (a) ( 00 ) ( 0 ) ( ) ( ). (a) ( 0 ) ( 0 ) 7. (a) sec sec csc csc cot 0 cot The termal side of the angle tersects the unit circle at the pot,. csc sec cot. The termal side of the angle tersects the unit circle at the pot,. 0 csc 0 0 sec 0 0 cot 0 Universit of Houston Department of Mathematics

13 Odd-Numbered Answers to Eercise Set.: Unit Circle Trigonometr. The termal side of the angle tersects the unit circle at the pot,. csc sec cot. (a) 0 0 o 0 o 0 o 0 o Diagram Diagram (a) 9. (a) 7. (a) 7. (a) 7. (a) (a) Undefed 79. (a) 8. (a) 8. (a) (a).08.0 (c) (a) (a).7.00 (d) (e) The methods parts (d) all ield the same results. MATH 0 Precalculus 78

14 Odd-Numbered Answers to Eercise Set.: Trigonometric Epressions and Identities. ( θ) ( θ) 78 + ( θ) ( θ) ( θ) ( θ) + θ θ θ + sec. ( θ) ; ( θ). ( θ ) ( given) ( θ ) 9 csc ( θ ) sec( θ ) ( θ ) cot ( θ ) 0 ; 9 7. cot ( θ) ; ( θ) csc ( θ) ; ( θ). ( θ) csc( θ) ( θ) sec( θ) ( θ ) ( given) cot ( θ ). ( θ) ( θ) csc csc + 9 ( θ) 7 ( θ) ( θ) ( θ) 9. ( θ) ( θ). ( θ) + 7 ( θ) ( θ) 7 ( θ). ( θ) ( θ) sec sec. ( θ) ( θ) cot + cot 7. θ 9. sec ( ). sec( ). ( ). ( θ ) 7. ( θ ) 9.. sec( ).. ( ) 7. ( ) 9. ( ). Notes for -9: The smbol means, therefore. Q.E.D is an abbreviation for the Lat term, Quod Erat Demonstrandum Lat for which was to be demonstrated and is frequentl written when a proof is complete.. sec Left-Hand Side sec ( ) Right-Hand Side ( ) sec Q.E.D. Universit of Houston Department of Mathematics

15 Odd-Numbered Answers to Eercise Set.: Trigonometric Epressions and Identities. ( θ) ( θ) Left-Hand Side sec( θ) csc( θ) ( θ) + cot ( θ) ( θ) ( θ) ( θ) ( θ) + ( θ) ( θ) ( θ) ( θ) ( θ) + ( θ) ( θ) ( θ) ( θ) ( θ) ( θ) ( θ) ( θ) ( θ) Right-Hand Side 7. ( ) Left-Hand Side sec sec ( θ) csc( θ) ( θ) + cot ( θ) ( θ) csc( θ) ( θ) + cot ( θ) Q.E.D. Right-Hand Side ( ) Q.E.D. 9. cot cot Left-Hand Side Right-Hand Side cot cot ( ) ( ) cot cot cot Q.E.D.. cot csc ( ) sec ( ) Left-Hand Side Right-Hand Side cot csc sec sec csc cot sec csc Q.E.D. MATH 0 Precalculus 78

16 Odd-Numbered Answers to Eercise Set.: Trigonometric Epressions and Identities. Left-Hand Side Right-Hand Side +. ( θ) + ( θ) sec ( θ) sec ( θ) Left-Hand Side ( θ) + ( θ) ( θ) ( θ) ( θ) sec ( θ) ( θ) sec ( θ) sec ( θ) sec ( θ) sec Q.E.D. + Right-Hand Side sec ( θ) sec ( θ) ( θ) + ( θ) ( θ) ( θ) sec sec Q.E.D. 7. Two methods of proof are shown. (Either method is sufficient.) Method : Work with the left-hand side and show that it is equal to the right-hand side. + Left-Hand Side + + cot csc + Contued on the net page csc cot Right-Hand Side csc cot csc cot Q.E.D. 78 Universit of Houston Department of Mathematics

17 Odd-Numbered Answers to Eercise Set.: Trigonometric Epressions and Identities Method : Work with the right-hand side and show that it is equal to the left-hand side. + Left-Hand Side + csc cot Right-Hand Side csc cot csc cot Q.E.D Two methods of proof are shown. (Either method is sufficient.) Method : Work with the left-hand side and show that it is equal to the right-hand side. + Left-Hand Side Right-Hand Side Q.E.D. Contued on the net page MATH 0 Precalculus 78

18 Odd-Numbered Answers to Eercise Set.: Trigonometric Epressions and Identities Method : Work with the right-hand side and show that it is equal to the left-hand side. + Left-Hand Side + Right-Hand Side α 7 Note: The triangle ma not be drawn to scale ( θ) csc( θ) ( θ ) sec( θ ) ( given) ( θ) cot ( θ) θ + Q.E.D. 7. θ 7. α 9 θ α Note: The triangle ma not be drawn to scale. Note: The triangle ma not be drawn to scale. ( θ ) ( given) ( θ ) csc ( θ) sec( θ) ( θ ) cot ( θ ) ( θ) csc( θ) ( θ ) sec( θ ) ( θ ) ( given) cot ( θ ) 77. ( θ) csc( θ) ( θ) sec( θ) ( θ) cot ( θ) 78 Universit of Houston Department of Mathematics

19 Odd-Numbered Answers to Eercise Set.: Trigonometric Epressions and Identities 0 csc ( θ) ( θ) ( θ) sec( θ) ( θ) cot ( θ) MATH 0 Precalculus 787