Tangent and Right Triangles - Math

[Pages:7]Tangent and Right Triangles

The tangent function is the function defined as

sin() tan() =

cos()

The implied domain of the tangent function is every number except for

those

which

have

cos()

=

0:

the

numbers

.

.

.

,

-

2

,

2

,

3 2

,

5 2

,

.

.

.

Graph of tangent

Period of tangent

Tangent is a periodic function with period , meaning that

tan( + ) = tan()

This follows from Lemma 10 of the previous chapter which stated that

cos( + ) = - cos() and sin( + ) = - sin()

Therefore,

sin( + ) - sin() sin()

tan( + ) =

=

=

= tan()

cos( + ) - cos() cos()

236

Tangent is an odd function

Recall that an

function is a function f (x) that has the property that

even

f ( x) = f (x) for every value of x. Examples of even functions include x2,

x4, x6, and cos(x).

An function is a function g(x) that has the property g( x) = g(x) odd

for every value of x. Examples of odd functions include x3, x5, x7, and sin(x).

We can add tangent to our list of odd functions. To see why tangent is

odd, we'll use that sine is odd and cosine is even:

tan(

)

=

sin( cos(

) )

=

sin() cos()

=

tan()

* * * * * * /*4*L * C*o?S*/ * * *

TrigonoTmriegtornyofmoretrrigyhftortriraignhgtletsriangles

Suppose that we have a right triangle. That is, a triangle one of whose

angles equalSsup.poWseetchaaltl wthee hsaidvee oaf rtihgehttrtirainagnlgelet.haTt hias toipsp, oasitteriatnheglerigohnte of whose

angle

the

ahnygploetseneuqsuealosf angle the

t2h. e

Wtrieancgallle.the side of of the triangle.

the

triangle

that

is

opposite

the

right

hypotenuse

kypoten u se.

If we focus our attention on a second angle of the right triangle, an angle

that we'll call 8, then we can label the remaining two sides as either being

opposite fromIf w8,eofrocaudsjaocuerntatoten8.tioWne'olnl caalsletchoendlenagntghles of tthhee trhigrehet tsridiaensgolef , an angle

the triangltehahtypw,eo'lplpc, aallnd,atdhj.en we can label the remaining two sides as either being

from , or

to . We'll call the lengths of the three sides of

opposite

adjacent

the triangle hyp, opp, and adj. See the picture on the following page.

237

We

call

the

side

of

the

atrnigalnegsleeqtuhaalts

is .

opWpeoscitaell

tthhee

rsigidhet

of

the

triangle

that

is

opposite

the

right

se of the triangle. angle the hypotenuse of the triangle.

hypotnus

oppOte.

GkcE. nt

aci

ttention on a sTehcoenfdollaonwgiIlnefgwopfertofhopecousrsitgiohoutnrtrraeitaltanetgnelsteito,hnaenolenanngagtlhseescoofntdheansigdlesooff tthhee right ttriangle, an angle

hen we can lsahboewl nthaebroevtmheaattionwitnheg'elltmwceaoallssui8dr,etshoeafsntheweitehanecrgalnbeelianbugeslintghethreemtraiginoinnogmtewtoricsifduensctaisonesither being

z adjacent to s6i.neW, ceo'lslincaeol,lpaptnhodesittleaenfgrotehmnst.6o,f othr eadthjarceenstidtoes6o. fWe'll

pp, and adj.

the triangle hyp, opp, and adj.

call

the

lengths

of

the

three sides

(co),

o(fe)

Proposition (13). In a right triangle as shown above, the following equa-

tions hold: opp

sin() = hyp

adj cos() =

hyp

s(e)

opp tan() =

adj

Proof: We triangle with

begin with the its side lengths

triangle labelled

on as

the top right of this page, instructed above: hyp CfoOr

(taeh)erighhyt-

potenuse, opp for the side opposite , and adj for the remaining side, which

is adjacent to . The triangle below on the left is the first triangle scaled by

the

number

1 hyp

.

That

is,

all

lengths

have

been

divided

by

the

number

hyp,

but the angles remain unchanged. What we have then is a new right triangle

whose

hypotenuse

has

length

hyp hyp

=

1.

e three formulas

Then we have the three formulas

opp

=--

hyp

adj cos(6) = --

hyp

tansi(n6()6)==ad-- hojyppp

adj cos(6) = --

hyp

SIi (e))

tan(6) =

216

kyp

216

a4j

kyp

The picture on the right shows sin() and cos(). Notice that the triangles

from

the

left

and

right

are

the

same,

so

sin()

=

opp hyp

and

cos()

=

adj hyp

.

238

Last, notice that from the definition of tangent we have

sin()

tan() =

=

cos()

opp hyp adj

opp =

adj

hyp

Problem. Find sin(8), eos(8), and tan(8) for the angle 8 shown belo Problem. Find sin(), cos(), and tan() for the angle shown below.

5

5

3

Lj.

It

3

Solution. The hypoSteonluusteioins .thTehseidheytphoattenisusoeppisotshitee stihdee rthigahtt ias nogplpe.osIittehathse right angle length 5. Of the twolreenmgtahin5i.ngOsfidthees,ttwhoe roenmeatihnaintgissiodpeps,osthitee oonf e thhaast liesnogpthposite of 8 ha 4, and the one that i4s, aadnjdactehnet otnoethhaast lisenagdtjhac3e.ntTthoer0efhoarse,length 3. Therefore,

opp 4 sin() = =

hyp 5

4 -- O]3J3 -- sin(0)

--

adj 3 cos() = =

adj 3

cos(8) --

--

hyp 5

--

opp 4 tan() = =

tan(0) --

4 --

adj 3

--

*************

239

Exercises

For #1-7, use the definition of tangent, that tan() = sin() , to identify the

cos()

given value. You can use the chart on page 227 for help.

1.) tan

3

2.) tan

4

3.) tan

6

4.) tan(0)

5.) tan

6

6.) tan

4

7.) tan

3

Suppose that is a real number, that 0 , and that cos( ) = 1. Use Lemmas 7-12 from the previous chapter, the de2finition of tangent, an3d

the periods of sine, cosine, and tangent to find the following values.

8.) sin( )

9.) tan( )

10.) sin( + )

2

11.) cos( )

2

12.) cos( + )

13.) sin( + )

14.) cos( ) 15.) sin( ) 16.) cos( + 2) 17.) sin( + 2) 18.) tan( + )

240

Match the numbered piecewise defined functions with their lettered graphs below.

sin(x) if x [0, ); and

19.) f (x) =

tan(x)

if

x

(-

2

,

0).

20.) g(x) =

tan(x)

if

x

(0,

2

);

and

cos(x) if x (-, 0].

21.) h(x) =

tan(x)

if

x

[0,

2

);

and

cos(x) if x (-, 0).

sin(x) if x (0, ); and

22.) p(x) =

tan(x)

if

x

(-

2

,

0].

A.)

B.)

/2

C.)

241

Exercises ExEEeErxxcxeeierrsrcceciisssiseeesss Find sin(6), cos(6), and tan(6) for the angles 6 given below. You might Exercises hthaFavhthteianFavhthtbtitdhhsteihaaoehtnFFUahtvahvelinatdhsosateibteiFoavsionlnwavneeeteinitistdtdensg(sbn.ooio16iPltdniegsin.attnons)Ys)g(borbtb,ioio6nlohsinoenenbbeatnt)iopbcgu(g(loynb,eo66tolleoiigb(teeaannf))mgscs6ullyidb,b,(latolina)s6abeenbieib,sicgucowb)dlylnys(beol,olhsnbe6cigeeliyisdstuwluolyt)dnda(l(eh1,estsse6u6gnhidh3iu(iwtdw)s)ndant6aahes,,tihnwggvt)ihiinwtl,ntotadePaaageehhiattgtnitnnyhahnfithtltdPdhaagnetieoeta(hnshhny6tdanedlhltatbPPeg)tene(aa.hgle6tntynyPosnnlehaPaofgeig)gtttn(y(onn.ghhniry6t6tgntreh(l(hoafgath)a)6toah..ggrthtabb)r)hneahofo.fg,ye.oagoerrtofltcrnrheoelohureoaaaerreetstdtsennnhhihao(tnagaeenewrhtntnlge)ohheei4gm,taatsreetth.lnnehh)aoahoeegmg6ennstrreoaloleoegdeergmP6mlrtseslfeioeetmyvisgmann66etttfindogohnvi6tgng(ttoaetfhofidibgnhgvivi).nnfeioeeftvidldbfhinrnnoeoneledewnettdralbbhhon.ntleeteewbthgelhlheonoetT.Ylelehlwwgeeoholntnaw.lYe.oheuegngnfo.onttggYoYrmhauhgletfYeoothmisohomsauguioffdohuisotemmafgatiofdghmaiissaeigtgfiiivisddhhngseieetidthdndete

1.)1.2)311...1))).)

4.)4.)442..46))..))

12

1212112212

10

10 10

110010

2.) 2.)2.2)242...2))).)

8 8 8 888

17 1717117717

15 15 1515

15 15

3.)

3.)3.2)353...3))).)

7

7 7 777

25 25 2255

2525

5.) 5.)5.)552..75))..))

8 8 8 888

5 5 5 555

2 2 2 222

6.) 6.)6.)662..86))..))

3 3 3 333

218 218 218218

218 218

242

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download