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Lesson 6.5 & 6.6: The Law of Sines & The Law of Cosines

The Law of Sines

When do you use the Law of Sines?

Law of Sines Formula:

Examples:

1. A satellite orbiting the earth passes directly overhead at observation stations inPhoenix and Los Angeles, 340 mi apart. At an instant when the satellite is betweenthese two stations, its angle of elevation is simultaneously observed to be 60" atPhoenix and 75' at Los Angeles. How far is the satellite from Los Angeles?

$a\e\\r\e1\n(\5- ) u@s

5

tN\en\tq

fnto.

)l

C

ttu-x.Atintqb'\ = t\\ t\(\(L\*)'* (q\ = t\u(q)

x = rqo tsxel1= j\\ e\ H

3\bq' = \a\sG)( = -1-

Case 'l:One $tde

( xtxb \rrlo' ar\$\eB.

sY t\.\Case 2:

F$tecieU

Ttroo t\de3 ioq o\e o+

\he tng\e oFQot\\ethsse iid.s. (tS A)

Case 3:\so $rdet I {ne e\r$\c \n be\ueen\\etn. (SAt)

Case 4:Tnree S\de3.

(srs)

$\!!\\ =$nls- =3s!o.bc{tPirr \$e ts$e, rrcr$n \e siAe oPPs\\\e

$rowr \i \\ a q\\e\ \a\$.

Q\ces..\Y'wrr\e \f.stoae\es

2. Solve the triangle below.

h b=-1

)s.(rS) =r\Y\ ab" ) sg-)=sw

8(}.\tb.\ . strn\$)=Srry

o.. rrrn (t\*) * 15Q S trnhb' ) 6-ttqi " .:ill.t,-lslrf!\-P)S-ttt'r

1"*\t\5Q\ ^* G5.\

A

Which of the four cases for Law of Sines is the triangle above? N NaThe Ambisuous Case (SSA)+

lf we are given a triangle with two sides and an angle, but the angle is not adjacent toeither of the two sides, there are three potential outcomes:

'oX,"tl"1il"\"!\[i'ln:\s"*T;*IULX[)z.\os to\

a\ ear eruu, q$ *\.}:*' il:;ill:H;r-*:**T" o::.No to\u(ont. (t lvra{eNe er$ h€. qret\ed 6\\\^r*re t\\e( \nea\\r\Scnee.\\ .Examples:

1. Solve triangle ABC, where 1A = 45", o=7fr, andb:7.

.""(=\tF.')=(+) rrztr'.'(q1") * S'gs(B){E-.! =trn(q\-T'fz

t "%\r'\%\ , .s

ris$S\ = t$JS.

.\^^.^(\"\ =\sfn\$B>]-* --.qr5U)

arsr(!Q-) :,1:^^=SS9lv x\$t$" )

q * \i.5B* t$ otlp#6$

How many solutions are there? Explain.\nev e $ e$r\ a$e tthL\\\0Yr be ( ft$ *t

e."vt (B\ * t t (ty-\r('\$&,q': \cr*t ".ai',i1',!\r{i"fi'.';

B = 30" or B ="\50o , hurJevev , S *\EQn be(auser\ urou\& vro\a\e \he \tian$e An$* t$Kn\he ct e\\

l*"*.*s axr$e

Non""'

\5.. 105.

2. Solve triangle ABC if < ,4

q)

\c=a\$'b

Q\ ^J a5-l' t

= 43.1o, a = 186.2, and b = 248.6.

rtu,xss:-c:u"-$( 16 1ur .r')

a=70, and b =\22.

5\r,(\l") = s\0 \"11-

E{'.(*t\ =1-\SSS1: \TLjlq'(q1t

$rn t\5\ = fro, , \,r1lrs.(\D\ -- ?.,

q\$\ (S\a\\.u

\A.1-a\f\ (\^L)trvr (B\ =

--\o^O\a\qe Lrt

rin(utf ) _\t\".1-

u!(\rr) -- iyq(rt'\)rtr".-f 9r

4S15.,=$! "Lt)!T)l\

!'.'i\'l \t,".1tt',I(1\',\")

Cr = srn (\t '\')

rrds,:-$ =\tu,aLa

( r_ s \\5.'L

3. Solve triangle ABC, where < A = 42",

\b

\ g =\'La e

How many solutions are there? Explain.

\exe bye \s to\ukrOrrt bec-au\eand \.\-'\ ta ou\ttde o( \\et\n (s\ . \"\1 g L-\ ) \l

Triangle 1 Triangle 2

1A=43.L a: L86.2 1A:43.L a: L86.2

.- ^A< B * bb.r b :248.6 < B s\\\f b = 248.6

< c r+f\.\" c :'LBl.t < C EIJL:I c: \s5.1-

t ut \e\ned'

ng \.\l

The Law of Cosines

The law of sines cannot be used directly to solve triangles if we know two sides and theangle in between them (SAS) or if we know all three sides (SSS).

The two cases in which the Law of Cosines applies are SA$ and t$3

of=f*f-lbc,cot(n)

d + Q.," - taq, got(B)

t'+\r"- a"b... (C')

t'1-

c,A

Examples:

1. A tunnel is to be built through a mountain. To estimate the length of the tunnel, asurveyor makes the measurements shown in the figure below. Use the surveyor'sdata to approximate the length of the tunnel.

c. = (rrt)'+ (trf - t(3t$taYfr *t15(tt-'\'\

d*C.=. e, t \\to.t q\

\e \eng\h s \ \e \ut'ne\ r'rou\d'

be absu\ q\v 't Q* '

2. Find the angles in the triangle below.

' r,: 12

*= t'+ \1? -t(t)(rt\ cst(N\?b * 't_!1 - \q1 car (A\a$ -r-tfr

- \t1 =l-!qt cgt(A\cattA\ = ffi[ = QQ;-\ (H) re \1.'ro

tf * t" +ta * t!%1(B) c.s% Lg\$\\ = t\ v'15 * tU c"os(C\t\\ = tq -t\ cot(C\- 11 it_

b5 = -to c-ot (C)

c"ot (C\ = *H

t = cot-\ (-E\g r \11.\"

qL s.,,\\b's

3. SolvetriangleABC,where 1A=46.5",b:t0.5, andc = 18.0.

5ar\$' 6 e \1.1\t\L = (\\.5f + (rh.r-f - t(\\ s\ (rt t\ ea t ( C\

b)=\S.5

Heron's Formula for Area (of a Triarlgle)

We can apply the Law of Cosines to finding the area of a triangle:

Example:

A businessman wishes to buy a triangular lot in a busy downtown location (see the figurebelow). The lot frontages on the three adjacent streets are 125,280, and 315 ft. Find thearea of the lot.

**$a jN = tt\.\q - -Ltt.tco$(C)-lSq.qq -1t\ \q_

tq.5\s -t-tl .t(ot(C)c0t((\ : - ?s'j5\

c Tt.l..l_C t Qt.1o

\toqsr.ss{Js\r$s\ u5 \\'

I\ne area o\ tv\an$\e A$C. \t $\\en

A=

br5"

u0hey e t = t(c +b + C,) (Une te\s$-Pex\Yr\e\erand al b ? q, t\e r$ne t\de \en$h't

o\ tlne N)

{b}L-