Question

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9) Find the value of the expression. a. cos arctan -- b. tan(arcsin(x)) = 10) From a point on a cliff 85 feet above water level an observer can see a ship. The angle of depression to the ship is 40. How far is the ship from the base of the cliff? sec? x 11) Verify the identity: -tan’x = tan x cotx 12) Find all solutions algebraically in the interval [0, 2T): sec? - 3 tan x = 5 13) Find the exact value of sin (105°). (Use the fact that 105° = 60° + 45°.) 7 14) Given cos a = - and tan 0 <0, find sin 20. 9 15) Given a triangle with B = 61', c = 15, and b = 17, sketch the triangle then solve it. If there are 2 solutions, find both.

Answer 1

© © cos | arctan (-1)] [ are cus ( 3 ) 2 cas [arcas 2 Cos 2 20 3 V13 ces forcesco ③ tan (are sin(x)) = fan foretam & d) 2 Vrx2 A 40 (10 let AB is the cliff and c is the ship. Distance of a from the base 85 feet of the cliff is aff/ Let) here, angle of depression 24° But, LDAC2 LACB [Alternate omgle] BL X LACB 24° In SABC, & AB BC 2tanyo » 85 z tango x » 22 85 ~ 1216 tanyo The required distormee is 1216 feet

secx cota tan²x 3 tanx seex - tarix stamalsus - taux Stanx (1) tittamaa seen] stomx (Proved) (12 seex - 3tanx 25 It tans - 3 tanx 25 tana-3 tamat 1-520 tanta-3 tama-420 tanta - 4 tamattoma-420 tamaltanx-4) + 1 (tana-4) 20 » (tama-4) (tanx + 1) zo tamatizo on, tana-4 20 tamaa -l tanazu : x 2X-74, 27 - 11/14 27 xztanlu) 9xz75.960 FI 4 and ax2 180°+75196 2135,3 15° 2255.960 Therefore, the required solution are 75.969,135255,969, 315º 2 3 4 4

sin (105) 2 Sin (60° +45°) 2 sinfo.cosur & cos6oo sin45° Σ V3 2 13th 242 Cosoz - 7 and tam o lo then sinoso "sing a vi- custo [i sino) ] 2 2 49 81 2 32 S 52 9 is in 2022 sino caso 2 214271-7) 5612 2 81 15 BA son Law of sine, we get 61° b SMB C 2 sinA Glza sinc b Now, SinB 2 C sine А b=17

→ 17 15 Z 1 B2610, C215, 6217 sinblo Sinc y sincz 15x single 17 cz sinis 15sindo 17 250.510 sints is r. Az 180-(B+C) 180-(61°+50.519) 268.49° 2 Again, a Z b SinB SINA S az bsinA SinB 2 17 Sin 68.490 sin 610 2 18.1 Here is one solution. . The required solution A =68.49°, C = 68.499, a z18)