Trigonometry nd Construtive Geometry Trining prolems for M2 2018 term 1 Ted Szylowie tedszy@gmil.om 1 Leling geometril figures 1. Prtie writing Greek letters. αβγδɛθλµπψ 2. Lel the sides, ngles nd verties of these tringles using the lssil method, in ounterlokwise order. 3. Use primes to lel the figure in logil wy. 4. These tringles re ongruent. Lel them using the lssil method. Use primes. Write down reltionships etween ngles nd sides. 1
2 Congruene nd similrity 5. These tringles re similr. Lel them using the lssil method. Write down reltionships etween ngles nd etween sides. Wht is the zoom ftor? Is it igger or smller thn 1? 2 Congruene nd similrity 6. Copy these segments using ruler nd ompss. Don t erse your onstrution lines nd rs. Lel your work B () () B () B 7. Use ruler nd ompss to opy the ngle t. Don t erse your rs or onstrution lines. Lel your work. () () () 8. Mke ongruent opy of this tringle y SSS. Use ruler nd ompss. Don t erse your onstrution lines. Lel your work 9. Use ruler nd ompss to mke ongruent opy of this tringle y SS. Lel your work. Explin whih sides nd ngle you hve opied.
Congruene nd similrity 3 10. Use ruler nd ompss to mke ongruent opy of this tringle y S. Lel your work. Explin whih side nd ngles you hve opied. 11. Use ruler nd ompss to onstrut ounterexmple for. Construt two tringles where is true, ut not onguent. Use ruler nd ompss. Lel your tringles nd write down ll the reltionships. Is the zoom ftor igger or smller thn 1? 12. Give ounterexmple for SS, SS. Show tht hving SS true leds to two solutions, one ongruent, the other not ongruent. Use ruler nd ompss. Don t erse your onstrution lines. 13. Prove the prllelogrm re formul re se ˆ height y doing these steps: () Construt prllelogrm y ruler nd ompss. () Cut the prllelogrm into two tringles. () Copy the two tringles into seprte figures using ruler nd ompss. Lel them using lssil lelling nd primes. (d) Use SSS to prove tht the two tringles re ongruent. Explin why eh step is true. (e) Write onlusion. 14. Do the sme prllelogrm proof s in prolem 13 ut using SS. 15. Prove the prlellogrm re formul using S. Follow the steps of prolem 13. 16. Prove the prllelogrm re formul using S. Follow prolem 13. 17. Use to mke smller similr opy of this tringle. Do it with ruler nd ompss. Lel your work. Write down the reltionships etween sides nd ngles. 18. Use to mke lrger similr opy of this tringle. Use ruler nd ompss. Lel your work. Write down reltionships etween sides nd ngles.
4 90 tringles 19. Construt 90 perpendiulr lines going through point. Use ruler nd ompss. Don t erse onstrution lines nd rs. () () () 20. Here is right (90 ) tringle. We usully lel the vertex with the right ngle s C nd the longest side s. Lel the tringle nd onstrut the ltitude line h t C using ruler nd ompss. C B 21. Construt 90 tringle using ruler nd ompss. Let C e the 90 vertex. lso onstrut the ltitude line t C. Lel the verties, ngles nd sides of your figure. Don t erse your rs or onstrution lines. 3 90 tringles 22. Fill in the missing ngle. 23. Fill in the missing ngle nd el the sides with proper trigonometri nmes: hypotenuse, djent side, opposite side.
90 tringles 5 24. Here is 90 tringle. Wht is speil out side? Mke list of things. 25. Here is 90 tringle. Wht is speil out side? Mke list of things. 26. Here is 90 tringle. Wht is speil out side? Mke list of things. 27. Find sin, os, tn in terms of sides,,.
6 90 tringles 28. Find s, se, ot in terms of sides,,. 29. Consider the following tringles where 1 nd ă 1. 1 1 () Whih is igger, tn or tn 1? () Whih is igger, ot or ot 1? Explin why!
Speil ngles 7 30. Consider the following tringles where 1 nd ă 1. 1 1 () Whih is igger, sin or sin 1? () Whih is igger, os or os 1? Explin why! 31. Lel the missing ngle, lel ll sides with proper nmes nd find ll trigonometri rtios sin, os, tn, s, se, ot. in terms of sides,,. 32. Why do we hve six of these trigonometri rtios, sin, os, tn, s, se nd ot? Why re they importnt? Wht is so speil out them? 33. Explin how we were le to lulte the distne to the str 61 Cygni y using trigonometry. 34. Drw 90 tringle, lel it, nd find reltionship etween sin, os nd tn. 35. Drw 90 tringle nd lel it. Use Pythgors s lw to find reltionship etween sin, os nd 1. 36. Drw 90 tringle nd lel it. Use Pythgors s lw to find reltionship etween tn, se nd 1. 37. Drw 90 tringle nd lel it. Use Pythgors s lw to find reltionship etween ot, s nd 1. 4 Speil ngles 38. Chnge these ngles from rdins into degrees. Figure it out y drwing little irles.
8 The unit irle digrm () π{2. () π{6. () 7π. (d) 5π{8. (e) 3π{4. (f) 5π{12. (g) 2π{3. (h) 3π{2. 39. Chnge these ngles from degrees into rdins. Figure it out y drwing little irles. () 75. () 300. () 285. (d) 120. (e) 225. (f) 720. (g) 195. (h) 22.5. 40. Sketh π{6 π{3 π{2 tringle. Mke the hypotenuse 1. Lel ll the ngles nd the lengths of the sides. 41. Sketh 45 45 90 tringle. Mke the hypotenuse 1. Lel ll the ngles nd the lengths of the sides. 42. Chek tht psin q 2 ` pos q 2 1 for 30. 43. Chek tht pse q 2 ptn q 2 1 for π{4. 44. Chek tht ps q 2 pse q 2 1 for 60. 5 The unit irle digrm lwys rememer: the unit irle hs rdius 1, even if tht is not shown expliitly in the digrm. 45. Wht does unit men in the term unit irle? 46. Construt (with ruler nd ompss): xes, unit ile, ngle. 47. Sketh the sine nd osine lines nd lel them. 1 1 48. Construt with ruler nd ompss: xes, the unit irle, ngle, osine line nd sine line. 49. Wht is the ehvior of sin nd os s goes to 0 nd 90? ngle sin os Ñ 0 Ñ 90
The unit irle digrm 9 50. Prove tht the length u is the sme s sin. u 51. Prove tht length u is os. u 52. Interpret the mening of pos q 2 ` psin q 2 1 using the unit irle digrm. 53. Sketh the tn nd se lines nd lel them. 1 1 54. Construt using ruler nd ompss: xes, unit irle, ngle, tngent line nd sent line. 55. Wht is the ehvior of tn nd se s goes to 0 nd 90? ngle tn se Ñ 0 Ñ 90
10 The unit irle digrm 56. Prove tht length u is tn. Use similr tringles. u 57. Prove tht length u is se. Use similr tringles. u 58. Interpret the mening of pse q 2 ptn q 2 1 using the unit irle digrm. 59. Sketh the osent nd otngent lines nd lel them. 1 1 60. Use ruler nd ompss to onstrut xes, the unit irle, ngle, osent line nd otngent line. 61. Wht is the ehvior of s nd ot s goes to 0 nd 90? ngle s ot Ñ 0 Ñ 90
The unit irle digrm 11 62. Prove tht u is s. Use similr tringles. u 63. Prove tht u is ot. Use similr tringles. u 64. Interpret the mening of using the unit irle digrm. ps q 2 pot q 2 1 65. Orgnize the six trigonometri funtions ording to their reltionship to the unit irle. Inside the irle: Outside the irle: Prtly in nd prtly out:
12 Trigonometri identities 66. Fill in this tle of trig funtion ehviors. It s esy if you keep the unit irle digrm in mind, nd use your imgintion. sin Ñ 0 os Ñ? os Ñ? sin Ñ 1 se Ñ 1 s Ñ? s Ñ? se Ñ 8 tn Ñ 0 ot Ñ? ot Ñ? tn Ñ 8 6 Trigonometri identities 67. Wht do we men y trig identity? Give some exmples. 68. pply Pythgors s lw to tringle BC. Do you get nything new? ny new identities or reltionships etween trig funtions? B C 69. pply Pythgors s lw to tringle BC. Do you get nything new? B C
Trigonometri identities 13 70. pply Pythgors s lw to tringle BC. Do you get nything new? C B 71. Given tht the line through the irle is enter line. Wht re the ngles nd 1? 1 72. Wht re the ngles, 1, 2, 3? The line through the irle is enter line. 1 2 3 73. Wht is the reltionship etween ngles nd 1?
14 Trigonometri identities 1 74. Wht re the reltionships etween ngles, 1, 2, 3? 1 2 3 75. Use SSS ongruene to prove tht if tringle hs two equl sides, then it hs two equl ngles. 76. Prove the very eutiful irle-ngle theorem: 1 2. 1
Trigonometri identities 15 77. nd now wht hppens when the ngle 1 is on the other side? Prove reltionship etween nd 1 using the sme ides s you used in prolem 76 nd in prolem 75. 1