Depression & Solving Right Triangles

Transcription

1 Math Notes Angles of Elevationl Depression & Solving Right Triangles Part 1- Angles of Elevation and Depression Angle of elevation: is the angle between the line of sight and the horizontal line (above the horizontal line). - When a person looks at something _~"'.i.lo'o.l.o.:-v.::..-..:::.0~ his or her location. Angle of depression: (below the horizontal is the angle between the line of sight and the horizontal line line). - When a person looks at something b~\0 vj location. his or her A right triangle can be formed in these problems since there is a horizontal line, a vertical line and a line of sight (diagonal line). 1

2 Things to notice: fy\1'&s\(\j L :::q a o - Lo~ cai.p. L D~ -e.\-{~ct~ion L -e\--e.\l D-\ \0(\ - L d~pv-t?s S I" Df! Example: Find the angle of elevation: A bird watcher sets up a camera on the ground and it points to the top of a tree. The tree is 10m tall and the camera is set up 15 m from the base of the tree. What is the angle of elevation? Of>P tree '~( (\c\ L e \-e. --J {,\ ~ i0(""\(e): 10 m ~ e ~ CGi\'Y\<'("iA. ~15m~ ~\j' '- L l.\ \::).e.. \ -\y\' C-\0j \ e:. - w~ \ZV\ovJ.2ft-' 0\(\{j~,Q-cli +-C\V'\ e-~ \0 ~ 0.\01-2 )$ ~ 9':: i-c\~ - \ (o,lo1-) =~~ 1.1-' a

3 Example: Angle of depression: A boat is 75 m from the base of a light house. The angle of elevation from the boat to the top of the light house is L ~5 rn ~ bota'\- What is the angle of depression? 3q, \ 0 What is the angle formed between the light house and the diagonal line of sight? 50, c,o q 0 _ 3<1. \ 'Q = 50 \10 How tall is the lighthouse (to the nearest metre)? la ~-L L o~ e.\-{ v t1\ -\--; D (\ Q'I S bur "o..(\~\0 0\- (I\-\<.("es,-\-. - \Ct~\ t-vl'tl,(lj\(; +0.(\ ~q. \ =- orr 15 -\-0-(\ 3q \ 1--,5 b /6\1.6+ i- l 5 -::: C)f P = \ to,q 5\ 3 L\!j\-. \\lo\.as~ \ ~ _{o_, _\_{Y\_ +C\ \ \.

4 Example: Find the angle of elevation A person stands 1.7 m above the ground and is looking at the top of a tree which is 27.5 m away on level ground. The tree is 18.6 m high. Find the angle of elevation. ( L~\'Io"~ I 1- -,I rn OfP ) \t- \lo\~\~ \'6.b(Y\ Loo""'. ('\j +Or 8: ~~ ~ "" C\\lu 2f 'P;..-,1j +-Od\ e=-. 0f~ d-c\j +etv\ e ~_\bi C\ ':::' b. ~\q 5 ~l ~ G ~ -~'CA(\ - \ ~ o. to Pt 5 J lo 3 L ~. \ 4

5 Example: Find the angle of elevation Sean uses a device called a transit to measure the height of a totem pole. He positions his transit 19.0 m from the totem pole and records the angle of elevation to the top of the totem pole to be 63. If Sean's transit is 1.7 m high, how tall is the totem pole? _ \ la'oi- \ hi' anj\ (.; \t\c~\i~.0- ct,,) ta n. A L \f\j lln-\- QfP So~ lofr') ~\-t.t() fo\d \,ill) 1 Iq.o :;.+nil ~d=.~ -hrc~nslt- {:-----~-=> \q.d (y\ 7 l oj,} ') )f\<'o f y( o \q,o Y' -\- Gt()103 - DfP \ q.0 i- v, '\ 102 ~ == V-~-'l-,-'-:'-""'----1 'To-km po\{ = 0rp "-t \: f ri\ 31'~ ~ \pl ~ =-l3~~ \-\--6t\\

6 Last example: Calculate a distance using the angle of depression: Natalie is rock climbing and Aaron is belaying. When Aaron pulls the rope taut to the ground, the angle of depression is 73. If Aaron is standing 8 ft from the wall, what length is of rope is off the ground? '? Jt. h'!jp The angle that the rope makes with the vertical is \1 C Find the length II' of the rope: Decide which trig ratio you will use. Le,\ \~5 \ASL \1 0 (AS Du.( Cl(\8 \( 0+ " (\~(e~ t ' \N~ "'now ~?P ) v') t\(\t,,!:tjp ~~A'H- loa- S'{l Et - opp ~p IIPlug in" the values you know. Then solve for the unknown: ~(j'{j ><S\(\\i '0j~ f5-9+ <B ~'+ 6\(\ \1 - '0 ~\\- 6 == \3+.3 -\+ l ~~

7 *There is more than one right way to solve these problems! In the last example, we could have used 73 as our angle of interest. And we could have used the cosine (Cos e ti: = adj/ hyp) to solve for the length. COs~ z: 0-.. ('\J~' (OS l'~ "~l ~~P ~jp How to Solve Problems Using Angle of Elevation or Depression: <6t~ o.tjj - Start with a diagram (right triangle) - Fill in all values that you know (into your diagram) - Decide what you are looking for (L ~ \-e.n 8 -t h?) - Choose the angle of interest and label the triangle (opp, ad], hyp) - Then choose the trig ratio to use ( remember SOH CAH TOA) Do questions p. 132 #4-6, 8, 10 Part 2 - Solving Right Triangles Solving a right triangle means: find ai/lengths of \An ~ Y'\O~f\ S I0\ -e 5 and find all Uw)\\nOVJo C\\1j' er-. Ex 1: Solve this triangle: What is unknown? LP S\~e., Q~ 3\ d\-e. f> ~ 7

8 ~~-:'....~..,' ~:.!::*:".: ;/,,'-.>." /50) :p:.i'-,,.,"'".,"' :.,.;.,.,,.,"'.,.;"'.; :~~ qoo Solving for the unknown angle: y---y---~, Q\\ I-'~ Clt\.O\ '-'\,p. ~'6 Q, _..Y'~~ Solving for the unknown sides: L P -=- \ '6 D -- t':i 0 - Y-O Lf :=. 50 -o Use trig ratio for the first unknown side For the second unknown side, use a trig ratio OR Pythagorean theorem Q ~t=\(\d. S\o.G Q 'f!..: \ Le\ ~ US(.; L Q D-S \N ~ \\V\O\D L QJ b~f> ~()\'\~j)\tla. (os &-= CL('\). \\~r Q(\ 8\e 0-\- \,,-\-Lr est- \~e W 0. {\ 1- Q:.dj LOS 40 =, acb 3\1- p opp '\J~ := \F\ NJ. S\ ~<- \-(~34n Pvz[ -----r?\ ~k Lb\A\G\ use..,,$\ (\ e.)-t(af\e 01?8~tl~...\;htO{~tn Y\'JP G\1.--\'0 1.. ~ ( 1- #t~ L.e. \'~ u~e. s.(\ e":: ~ S"" L\ 0 -= O\,? h(jp 8 3,-1 ~\ (\L\O ~3.1 ~ oyp ~ ~. L\ \ = Q?9 f' to~ LtG '" '3,'l

9 Example 2: Solve this right triangle: What's unknown? 42m You can start anywhere you like (find an angle or the missing side): Let's start with the side (FD): '- 2- Lt2 4-3\1- := c., \',\01.\ -t qlo\:: (2- r\:). \ ~5:: A"c 2-5~'d-- :: c., Find angle F: l= 'I; "'~<l\nj -Mj mho (s; (\ \ LO~ or -\.on) 0\(\ F =~2,~~.L ~ O\5~ ~'P b'2 2- o L~=-S\(\-\ (o.5~) = 36 Find angle D: L D - \~D- C'\D L\) 5qo 9

10 Solving problems with more than one right triangle: A/;~c~::-:"1~~:~c B Find the length BD: ~ \)\)\ ~<.e A 'J-.'~,.{c.t.s t)t \'(\-\-v -\-1) SO, v e. +V\: s -. u.sz ~ 0(\ -\\r\< v <~\'"\-\-~ A 'B D.._ \j.s~ 0..(\!3' -{ 0.\ \ ".\Zi~.s+~ 6 e - \4\:){\ i--(\ lanj\<"" W C\(\~.9\>p) no\\jg l\jp Find the length AC: 0\n G=2e - S\(\2b:::~eeh~P ~\ (\ '26 'A 'd-.r~.q -:: DPP \ \0 ~'0 = 0 pp \ d-~.q

11 Example: Find the angle DEF: 'D F Notice that this is one large angle formed by the angles of each right triangle. Angle DEFis the sum of the two angles. Call the two angles a and ~. Solve for these angles: Cos ca =- ~J.i '-=::- <l ::: O.3b y,(jp d-. 5 r). ~ Co~,,-\ CO, 36) \:0<- =0 0'\ 0 \ (-\~~ \ ~ 'D Et= ::: d. -\- \3 ":; (0 ~ -t ~ L\- -\-G\l\ f> =-~ ~0 --\-CA (\ ~:":- L\ := 0, tt 4- ~ \6 ~- +~h - \ l 0 \L\-Lt ) Do questions p. 131 #1-3 ~ ~ ;;t '-\~ ~~ 11