Grop Members: Eploration -a: Transformed Periodic Fnctions Objectie: Gien a pre-image graph and a transformed graph of a periodic fnction, state the transformation(s). Gie the transformation applied to f () (dashed) to get the solid graph, = g().. Verball: Eqation: = g() = 4. Verball: Eqation: = g() = 6 6. Verball: Eqation: = g() = 5. Verball: Eqation: = g() = 6 6 3. Verball: Eqation: = g() = 6. Verball: Eqation: = g() = 6 6 7. What did o learn as a reslt of doing this Eploration that o did not know before? 4 / Eploration Masters Precalcls with Trigonometr: Instrctor s Resorce Book, Volme 003 Ke Crriclm Press
Grop Members: Eploration -a: Reference Angles Objectie: Learn abot measres of angles in standard position and their reference angles.. The figre shows an angle, = 5, in standard position. The reference angle, ref, is measred conterclockwise between the terminal side of and the nearest side of the horizontal ais. Show that o know what reference angle means b drawing and calclating its measre. ref 5. Sketch an angle of 30 in standard position. Sketch its reference angle and find the measre of the reference angle. 5. The figre shows = 50. Sketch the reference angle and calclate its measre. 6. Sketch an angle whose measre is between 0 and 90. What is the reference angle of this angle? 50 3. Yo shold hae drawn arrowheads on the arcs for the reference angles in Problems and. If o haen t, draw them now. Eplain wh the arc for 5 goes from the terminal side to the -ais bt the arc for 50 goes from the -ais to the terminal side. 7. The figre shows an angle of D50. Sketch the reference angle and find its measre. 50 4. Amos Take thinks the reference angle for 50 shold go to the -ais becase the terminal side is closer to it than the -ais. Tell Amos wh his conclsion does not agree with the definition of reference angle in Problem. 8. The figre in Problem 7 shows a point nits from the origin and on the terminal side of the angle. Draw a segment from this point perpendiclar to the -ais, ths forming a right triangle whose hpotense is nits long. Use what o recall from geometr to find the lengths of the two legs of the triangle. 9. What did o learn as a reslt of doing this Eploration that o did not know before? Precalcls with Trigonometr: Instrctor s Resorce Book, Volme Eploration Masters / 43 003 Ke Crriclm Press
Grop Members: Eploration -3a: Definitions of Sine and Cosine Objectie: Learn the formal definitions of sine and cosine fnctions.. The figre shows an angle of = 37 in standard position in a -coordinate sstem. Measre the angle with a protractor. Do o agree that it is 37? 5. The definitions of sine and cosine can be etended to angles that measre rotation with the aid of the reference angle. Sketch an angle of = 5. Then mark and calclate the reference angle, ref. r. The figre shows a point on the terminal side of. The - and -coordinates of the point form a right triangle whose hpotense is the distance from the origin to the point. Measre the three distances, to the nearest 0. cm. Adjacent leg, H Opposite leg, H Hpotense, r H 3. Yo recall from preios corses that the sine and cosine of an angle in a right triangle are defined: sin = = 37 opposite leg hpotense cos = adjacent leg hpotense Use the answers in Problem 3 to calclate sin 37 M cos 37 M 4. With or calclator in degree mode, find ales of sin 37 and cos 37. Do or approimate ales in Problem 3 agree with these precise ales? sin 37 H cos 37 H 6. Use or calclator to find: sin ref H cos ref H 7. The formal definitions of sine and cosine are: sin = cos = ertical coordinate radis horizontal coordinate radis Calclate sin 5 and cos 5. How are these nmbers related to the sine and cosine of the reference angle in Problem 6? How do o eplain that cos 5 is negatie? 8. State what sign the sine and cosine will hae for angles that terminate in: Qadrant I: sine cosine Qadrant II: sine cosine Qadrant III: sine cosine Qadrant IV: sine cosine 9. What did o learn as a reslt of doing this Eploration that o did not know before? 44 / Eploration Masters Precalcls with Trigonometr: Instrctor s Resorce Book, Volme 003 Ke Crriclm Press
Grop Members: Eploration -3b: -Graphs and -Graphs of Sinsoids Objectie: Show a geometric relationship between angles plotted as angles and angles plotted along the -ais. 90 80 70. The left figre shows a nit circle in a -diagram with angles marked at eer 30. Read, to two decimal places, the coordinates (, ) of the point where the ra at 60 cts the nit circle. 6. Use or obseration in Problem to plot points on the graph of =cos for each 30 from =0 to =360. Connect the points with a smooth cre.. Find cos 60 and sin 60 with or calclator. Eplain how these nmbers relate to the answers to Problem. 7. What transformation cold o appl to the graph of = sin to get the graph of = cos? 3. Plot the point (, ) =(60, sin 60 ) on the -coordinate sstem on the right at the top of this Eploration. Draw a line segment showing how this point is related to the point o plotted in Problem. 8. Eplain the difference between the wa the ale of appears on the -diagram and the wa it appears on the -diagram. 9. Wh do o think the letters and, rather than the more common letters and, are sed in the figre on the left at the top of this Eploration? 4. Withot actall calclating an more ales, plot points on the graph of =sin for each 30 from 0 to 360. Show segments connecting the appropriate points on the -diagram with points in the -diagram. 0. What did o learn as a reslt of doing this Eploration that o did not know before? 5. Connect the points in Problem 4 with a smooth cre. What geometrical figre is this cre? Precalcls with Trigonometr: Instrctor s Resorce Book, Volme Eploration Masters / 45 003 Ke Crriclm Press
Grop Members: Eploration -3c: Parent Sinsoids Objectie: Eplore the graph of the parent fnction H sin, and transform the graph.. The graph shows the fnction H sin. Plot this graph as on or grapher. Use the window shown. Trn on the grid to get the dots. Does or graph agree with this figre? 5. Write the eqation for this transformed graph. Dplicate this graph on or grapher. Eqation: 5 5 70 70. The amplitde of a periodic fnction is the ertical distance from the central ais to a high or low point. What is the amplitde of the sine fnction in Problem? Write the eqation of the transformed fnction that wold hae an amplitde of 5. 6. The dotted graph shows the reslt of three transformations. State each transformation, write the eqation of the transformed graph, and dplicate the graph on or grapher. 5 3. Plot the transformed graph as on or grapher. Does the reslting graph reall hae an amplitde of 5? 4. The solid graph shows a transformation of the sine fnction from Problem. Identif the transformation, and write the eqation for the transformed graph. Confirm that or answer is correct b plotting or eqation as 3. Verball: Eqation: 70 7. Degrees can be sed to measre rotation. What do o think is the significance of the fact that the period of the sine fnction in Problem is 360? 5 8. What did o learn as a reslt of doing this Eploration that o did not know before? 70 46 / Eploration Masters Precalcls with Trigonometr: Instrctor s Resorce Book, Volme 003 Ke Crriclm Press
Grop Members: Eploration -4a: Vales of the Si Trigonometric Fnctions Objectie: Find ales of the si trigonometric fnctions, with or withot a calclator.. Write the definitions of the si trigonometric fnctions of an angle in terms of the coordinates (, ) of a point on the terminal side and the distance r from the origin to the point. r (, ) 4. The figre shows an angle,, in standard position. The terminal side contains the point (D3, D7). Write the si trigonometric fnctions of eactl, as fractions inoling radicals if necessar. ( 3, 7) sin = cos = tan = cot = sec = csc =. Sketch 3 in standard position. Then find the si trigonometric fnctions of 3. Write the answers as decimals in ellipsis format. sin = cos = tan = cot = sec = csc = 5. The figre shows an angle of 300 in standard position. Choose a conenient point on the terminal side, determine the ales of,, and r, write them on the figre, and then find in eact form (no decimals) the si trigonometric fnctions of 300. 300 sin 3 = cos 3 = tan 3 = cot 3 = sec 3 = csc 3 = 3. Eplain wh sin 3 is positie bt tan 3 is negatie. sin 300 = cos 300 = tan 300 = cot 300 = sec 300 = csc 300 = 6. What did o learn as a reslt of doing this Eploration that o did not know before? Precalcls with Trigonometr: Instrctor s Resorce Book, Volme Eploration Masters / 47 003 Ke Crriclm Press
Grop Members: Eploration -4b: Direct Measrement of Fnction Vales Objectie: Use the definitions of sine, cosine, and tangent to calclate ales from measrements on an accrate figre.. The figre shows line segments from the origin making angles with the -ais of H 5, 30, 45, 60, and 75. Perpendiclars (dotted) are drawn from the ends of the segments to the -ais, forming right triangles. For each triangle, measre the hpotense and the two legs, to the nearest 0. cm. Write the answers on the diagram.. Use the definitions of sine, cosine, and tangent and the lengths o measred to calclate the ales of these fnctions for the fie angles. Rond the answers to two decimal places. sin cos tan 5 30 45 60 75 3. Use or grapher to make a table of ales of sine, cosine, and tangent. Write the answers, ronded to two decimal places, in this table. sin cos tan 5 30 45 60 75 4. How well do or answers in Problem, fond geometricall, compare with the answers fond nmericall in Problem 3? 5. What did o learn as a reslt of doing this Eploration that o did not know before? 48 / Eploration Masters Precalcls with Trigonometr: Instrctor s Resorce Book, Volme 003 Ke Crriclm Press
Grop Members: Eploration -5a: Measrement of Right Triangles Objectie: Gien two pieces of information abot a right triangle, find the other sides and angles.. The figre shows a right triangle with legs 4 cm and 3 cm. Do o agree that these measrements are correct? 7. Find cos A directl, sing the ale of A o stored in or calclator in Problem 3. Does the answer agree with or answer to Problem 6?. Mark the smaller acte angle as A. What nmber does tan A eqal? 8. Sketch a figre representing a right triangle with hpotense 066 ft and acte angle 8. Label the longer leg as. 3. The measre of A is eqal to the inerse tangent of the answer to Problem. This is fond on or calclator as TAN D, and means The angle whose tangent is.... Calclate the measre of A. Store this answer as A in or calclator. 9. For the triangle in Problem 8, 066 is one of the trigonometric fnctions of 8. Which fnction? 0. Calclate the length of the longer leg,. 4. Use a protractor to measre A on the figre in Problem. Does or measred answer agree with the calclated answer?. There are two was to calclate the length of the shorter leg of the triangle in Problem 8. Show that both was gie the same answer. 5. Use the Pthagorean theorem to calclate the length of the hpotense.. What did o learn as a reslt of doing this Eploration that o did not know before? 6. Use the definition of cosine to calclate cos A. Precalcls with Trigonometr: Instrctor s Resorce Book, Volme Eploration Masters / 49 003 Ke Crriclm Press
Grop Members: Eploration -5b: Accrate Right Triangle Practice Objectie: Use trigonometric fnctions to calclate nknown side and angle measres for right triangles.. The figre aboe left shows a right triangle of hpotense 0 cm and larger acte angle 68. Do o agree that these measrements are correct?. Calclate the length of the shorter leg. Show or work. 5. Calclate the measre of the smaller acte angle. Show or work. 6. Use the angle in Problem 5 to calclate the length of the hpotense. Show or work. 3. Measre the shorter leg. Does the measred ale agree with or calclated ale? 4. The figre to the right shows a right triangle with legs 6 cm and 9 cm. Do o find that these lengths are correct? 7. Calclate the hpotense again sing the Pthagorean theorem. Does it agree with or answer to Problem 6? 8. What did o learn as a reslt of doing this Eploration that o did not know before? 50 / Eploration Masters Precalcls with Trigonometr: Instrctor s Resorce Book, Volme 003 Ke Crriclm Press
Grop Members: Eploration -5c: Empire State Bilding Problem Objectie: Appl trigonometric fnctions to a right triangle problem from the real world. The Empire State Bilding in New York was the tallest bilding in the world when it was bilt in 93. To measre its height, a precalcls class finds that from a point on 5th Aene leading to the bilding, the angle of eleation to the top of the bilding is 7. The moe 307 meters closer and find that the angle of eleation is now 38.. Constrct a figre showing the street and the two points where the angles were measred. Use a scale of cm per 00 m. Constrct the eleation angles from the two points. Where the terminal sides of these angles cross is the top of the bilding. Constrct a perpendiclar from this point representing the height of the bilding. 4. B doing appropriate algebra on the two eqations in Problem 3, calclate the ales of and. 5. How well do the precise calclated ales of and agree with or measred ales of Problem?. Let be the distance from the closer point to the point where the perpendiclar meets the grond. Let be the height of the bilding. B accrate measrement on or figre, find estimates for and. 6. Look on the Internet or in a reference book to find the actal height of the Empire State Bilding. State where o fond the information. 3. Write two eqations inoling trigonometric ratios with the two known angles, the known distance, 307 m, and the nknown distances, and. 7. What did o learn as a reslt of doing this Eploration that o did not know before? Precalcls with Trigonometr: Instrctor s Resorce Book, Volme Eploration Masters / 5 003 Ke Crriclm Press
7. Vertical dilation b D 0 Eploration -a. ref =80 D5 =8 0 8 5 8. Reflection across the -ais 9. -direction 0. 0 Graphs coincide. 0. ref =50 D80 =70 50 70. 0 Graphs coincide. 0 3. Becase the angle mst be conterclockwise so that its measre will be positie 4. Becase it mst go to the nearest side of the horizontal ais 5. ref =360 D30 =50. Answers will ar. 30 50 Chapter Periodic Fnctions and Right Triangle Problems 6. ref = Eploration -a. Horizontal translation b = g() =f ( D ) ref =. Vertical dilation b factor of 3 = g() =3f () 3. Horizontal dilation b factor of = g() =f () 4. Vertical translation b D5 = g() =f () D 5 7. ref =80 +(D50 ) =30 5. Vertical translation b D5; horizontal translation b = g() =f ( D ) 6. Vertical dilation b factor of 3; horizontal translation b = g() =3f ( D ) 30 50 7. Answers will ar. Precalcls with Trigonometr: Instrctor s Resorce Book, Volme Soltions to the Eplorations / 35 003 Ke Crriclm Press
8. Dplicating the triangle aboe itself makes an angle of 60 at each erte, so the large triangle is eqianglar and therefore eqilateral. So all sides are of length, and the left (ertical) leg of the original triangle is half of, or (D becase it is below the horizontal ais). So the other (horizontal) leg is D = 3 (D 3 becase it is to the left of the ertical ais). 4. Graph. 90 80 70 3 5. Graph. 30 90 80 70 9. Answers will ar. Sinsoid Eploration -3a. =37 6. Graph.. 3. =4.6 cm; = 3.5 cm; r =5.8 cm sin 37 M0.6034 ; cos 37 M0.793 90 80 70 4. sin 37 = 0.608 ; cos 37 = 0.7986 Approimate answers are reasonabl close. 5. Graph, ref =55 55 5 7. Horizontal translation b D90 8. In the -diagram, appears as an angle in standard position. In the -diagram, it appears as the horizontal coordinate. 9. To emphasize the difference between the two was of representing an angle and its fnctions. In the -diagram, the ertical is not a fnction of the horizontal (both are fnctions of the central angle ), while in the -diagram, the ertical isa fnction of the horizontal. 6. sin ref =0.89 ; cos ref = 0.5735 7. sin 5 =0.89 ; cos 5 = D0.5735 ; 5 terminates in Qadrant II to the left of the -ais, where the -coordinates are negatie. 8. Qadrant I sine C cosine C Qadrant II sine C cosine D Qadrant III sine D cosine D Qadrant IV sine D cosine C 9. Answers will ar. Eploration -3b. (0.50, 0.87). cos 60 =0.5= the -coordinate; sin 60 = 0.8660 = the -coordinate 3. Graph. 0. Answers will ar. Eploration -3c. Yes, the graph agrees.. Amplitde = ; Y =5 sin 3. Yes 5 800 4. -dilation of 3 ; Y 3 = sin 3 5. Y 4 =8+sin 6. -translation of C60 ; -dilation of 4; -translation of D5; Y 5 = D5 +4 sin ( D 60 ) 7. represents a retrn to the starting point in a rotation. 90 80 70 8. Answers will ar. 36 / Soltions to the Eplorations Precalcls with Trigonometr: Instrctor s Resorce Book, Volme 003 Ke Crriclm Press
Eploration -4a. sin = r csc = r cos = r sec = r tan = cot =. Sketch. 3. sin cos tan 5 0.9659 0.588 0.679 30 0.8660 0.5 0.5773 45 0.707 0.707 60 0.5 0.8660.730 75 0.588 0.9659 3.730 4. The answers shold be close. 5. Answers will ar. 3 Eploration -5a. Measrements are correct.. tan A = 3 4 =0.75 3. In Qadrant II, (, ) is (negatie, positie) and r is alwas positie, so sin = r = positie positie =positie, bt tan = = positie negatie = negatie. 4. sin 3 = 0.8386 cos 3 = D0.5446 tan 3 = D.5398 sin = D 7 58 58 cos = D 3 58 58 tan = 7 3 csc = D 58 7 sec = D 58 3 cot = 3 7 csc 3 =.93 sec 3 = D.8360 cot 3 = D0.6494 3. A =tan 3 D 4 =36.8698 4. Measre of A M 37 agrees with the calclated answer. 5. Hpotense H 5 cm 6. cos A = 4 5 7. cos A H 0.8. Answers agree. 8. Draw as directed b the tet. 066 5. sin 300 = D 3 cos 300 = tan 300 = D 3 6. Answers will ar. Eploration -4b csc 300 = D 3 3 sec 300 = cot 300 = D 3 3 8 9. =cos 8 066 0. =066 ft cos 8 =94. ft. 066 ft sin 8 = 500.4566 ft (066 ft) D (94. ft) =500.4566 ft. Answers will ar... r 5 0 cm 9.7 cm.6 cm 30 0 cm 8.7 cm 5.0 cm 45 0 cm 7. cm 7. cm 60 0 cm 5.0 cm 8.7 cm 75 0 cm.6 cm 9.7 cm sin cos tan 5 0.97 0.6 0.7 30 0.87 0.50 0.57 45 0.7 0.7.00 60 0.50 0.87.74 75 0.6 0.97 3.73 Eploration -5b. Measrements are correct.. 0 cm cos 68 = 3.7460 cm 3. Measrement is correct. 4. Measrements are correct. 5. 6. tan D 6 9 =33.6900 9 cm sec 33.6900 = 0.866 cm or 6 cm csc 33.6900 = 0.866 cm 7. (6 cm) +(9 cm) = 7 cm =0.866 cm Answers agree. 8. Answers will ar. Precalcls with Trigonometr: Instrctor s Resorce Book, Volme Soltions to the Eplorations / 37 003 Ke Crriclm Press
Eploration -5c. Draw as directed b the tet.. M 580 m, M 450 m 3. tan 7 = 307 +, tan 38 = 4. B rewriting the eqations as cot 7 = 307 m + = and cot 38 =, o get 5. Answers are reasonabl close. 6. The actal height is 454 ft, or 443. m. 7. Answers will ar. Chapter 3 Applications of Trigonometric and Circlar Fnctions Eploration 3-a. Use December s temperatres for month 0. ( F) 00 50 = = 307 m cot 7 Dcot 38 307 m cot 38 cot 7 Dcot 38. -dilation of =449.7055 m M 450 m =575.5968 m M 576 m (months) 6 8 4 360 = 30 ; =cos 30 307 m + 6. H 78.3 C 6.6 cos 30( D 7 ). Actall, this shold be H 78.3 C 6.6 cos 30(t D 7), where t is time in months. 7. The fit is onl shown for the first ear. The second ear is the same. The fit is good bt not perfect. 8. Answers will ar. Eploration 3-b. 50 50 X Y 0 0 0.7 0.34 30.5 40.64 50.77 60.87 70.94 80.98 90 0 0 0 0 X Y 80 0 70 D 360 0 450 540 0 630 D 70 0 90 80 70 450 540 630 70 3. In Problem, the -dilation is 360 = 30. Here the t-dilation (if t represents time in months) is months months/degree, so = cos 30t 360 = 30 4. -translation of C7 ; H 30 cos ( D 7) 5. H 78.3 C cos 30( D 7 ) 38 / Soltions to the Eplorations Precalcls with Trigonometr: Instrctor s Resorce Book, Volme 003 Ke Crriclm Press