Math Midterm Review Solution Key

CHAPTER 5/6 - Trigonometric Functions of Angles 1. Convert -300° to radians. 2. Convert -3- to degrees. 11rr ''.;· 1f "- [c.uoo o j -�.y:-\-�] 3 .!J>O - Find the value of x in the triangle. Number 7 must be exact. 4. 1000 5. 'I..-:=.\ ooo cos. £c.£, 0 ["' =- 3+�. �] 6. 7. -- Y-,-> �-vJ :. Z-;S' fi,_ 0S ,.(,,_ :, � 85 50 ---l L....L------=-=-�--�� <3\V\�Oo-:. Sb 'd w o '6: Sr.� s,rxs>t; -:.� "' '/...-=- '5�- r o $1Y\w; +aT"'ICooo :. SS J'?> vJ -=- =- \·� J X 85" w +aY\ 3o� � w er.; . 'ot;P.> t--V.,_,___:��-u'-- � 9 �- 9, � --3--t� � 8. From the top of a 200 ft. lighthouse, the angle of depression to a ship in the ocean is 23Q. How far is the ship from the base of the lighthouse? 0 )( - +o..l"'I (o-=(- =- zoo - )(.-:::. 200 +""-V\ LP-=t � E\ . 1.. + ee. -+-J 0 � J� � e,s[_ �) 'j-:- -Z.'-S'_ '2�� ,n-� 9. To estimate the height of a mountain above a level plain, the angle of elevation to the top of the mountain is measured to be 32Q. One thousand feet closer to the mountain along the plain, it is found that the angle of elevation is 3SQ. Estimate the height of the mountain. -la.n ':1:>� � ""' h 0 J( � '32. -=- ;:;.,ooo )t +-c..,,.��- � I,, I,, (<- I 00<> � '!.2.o > \-, . -=too i< -=-- \., • (. 2Sx + z.c; = h _ � • lo 25 )( -\- b2.c;' : .-=tc:o'I<. ,c ' - \ o oo---4 l<.-: 8 3�'?>. 3 c.., 2."5'. :: . 01-S � 10. Points A and B (on the same side of a tower) are 16 m apart. The angles of elevation of the h-=- SS 3S . \ top of a tower are 38 ° and 48 ° respectively. Find the tower's height. .f�e-r L c, '° o • .h. � • 0 __...., \C, \-= >C. -; �'t'o =- !l. � 3e> - �-t-'1- ll�'t8o �)(.h (\t,p,,.)(.)��:.. '-' 1'2..'t'-\1#-t-.�I)(-:. I.Ill� l"l.\(C\1.1;. . ?�,C.. r h� 42.w t rv\ � 'f,.-::. �-:J.�':f 11. A ship leaves port at 2 p.m. and has a bearing of S 38 ° W. The ship sails at 35 knots. How many nautical miles south and how many nautical miles west will the ship have traveled by 9:30 p.m.? °1',':?('¾')� Z�z.<; ,. '"> o S St)U.-W) -;:, Z.o<.o. q n rV') S>'f'\ � J- -:. '2.IQ1,.. c-; 12. An airplane flying at 390 miles per hour has a bearing of 67 °. After flying for 5 hours, how far north and how far east has the plane traveled form its point of departure. e. 3"'Jo 1.S:: 1qso o t" cos � (,-=t -:. -- n Mh -:. I '\ '5V -::t-Co \. °l "' ""' 13. A plane is 200 miles south and 65 miles east of an airport. The pilot wants to fly directly to the airport. What bearing should be taken? 200 �+- Find the sign of the expression if the terminal point determined by 0 is in the given quadrant. 14. cos 0 cot0; QII m ( -\ t - ) w- - 15. csc 0 tan 0; QIV (- )(. ) - From the information given, find the quadrant in which 0 lies. 16. sec0> 0 and csc8 < 0 S<'Cti.l- �c. csc. (Gn'1] G 17. tan0> 0 and sec> 0 "t"CU'l Sec. f Find the missing coordinate of P, using the fact that Plies on the unit circle in the given quadrant. 20. P ( -�, c-15),_ T'{ y); Quad Ill '),. =- 'l. + � 'I�-7� .,. is \ y -=- 4-�3 ➔ S" -�J Lf ¼); Quad II (\Y·=-, 21. P (x, �..�� 2. )(. ... � .,,_ + �::. lf �:: t � � ( -f); Quad IV 22. P x, ,. � � � � � l �B=t.2.1 1,. )(2. +c-�) -=-, v.1 ,�� '1'\ Find a) the reference angle and b) the terminal point P(x, y) determined by the angle. 24 __ 3rr 4 ,....= a.') R.A. c~.31r) - - ../1 = 'f v Cos .+ - 2. 2. -- - -rz. ., C- bir) � w -s = �r )l::. cos 90 /4---1-® &4-➔ -1 ° 7 -=-s -i. -i.. _, -1 --'- 9 £D0° :::- Find the exact value of the trigonometric function. srr 27. cos 28. sin 315 ° 7rr 30. csc6 .;i-� lo.) ll..A. : 26. 96QQ '/ '=- S lY\ I _, -I -��EE 33. Find the length of an arc that subtends a central angle of 45 ° in a circle with radius 10 m. 34. Find the area of a sector with central angle 60 ° in a circle with radius 3 mi. 35. A woman is riding a bicycle whose wheels are 28 in. in diameter. If the wheels rotate at 130 revolutions per minute (rpm), find the speed at which she is traveling in mi/h. \�c:>rL-./ lc>e:>l'"'I'\,� �1T(I�),� \ rev I hr l"Vl 1-n \+-t-. I '2 1 .; I M\le �flo.Srn tn] p 5 :2-Bo +-t'-- ------- rattrt.ls= 1"4 �� 36. A boy rotates a stone in a 3 ft. long sling at the rate of 15 revolutions every 10 seconds. Find the linear and angular velocities of the stone. IO sec.. 37. y l re\/ 'S�C q.\.C r� -- 21r ra.cl ..... �V\�IA.\CU' V�lD� \ S ,e:.J • • ;l.:rrL�).f-t l1f r-(LJ)... -- 2 sin ( 2x - ;) - 1 (\lV;� Ve.lo� � q,4 nuJ...{St..c.J 't- i.e.� +-1:/,.ec.) Find the amplitude, period, vertical shift, and phase shift of the function. Then graph the function. = Amplitude:___2.___ e_ • Period:-------- -rr Phase Shift: -rr 4 Rt_6k+ • Vertical Shift: l)O'Vf'Y\ i • 2. s {.,-, 2( X:.- :- ') - J Use the graph to find the amplitude, period, and phase shift of the curve. Then write an equation that represents the curve. 38. 39. 3 2 0 -2 Amplitude: Period: -3 2, -rr' Phase Shift: Y\.,OY\.a. 5o $0 1 $-v a..=-- 2. �': '2- Period: C., '/0 Vi 3rr 7r 2 2 rr ) 5o So lo CL -: ::. " .J. � , 'S,--o c.. -::. 0 Y)_i_8_ . ��ij 4rr Phase Shift: .. ?.. cos 2 9 Equat1on:_LA __ \J .,,,________ Identify whether the graph represents y 3 Amplitude: Equation:_y_.__::_3_S_l = tan x, y = cot x, y = csc x, or y = sec x. t:x 3rr 3rr 3rr 2 2 I 43. sec.>< 3rr : 7r 7r .2 2 -rr I I I I I I I I I I I (\ I n 3rr 2 �