TrigonometrySimilar triangleshave the same shape sameanglesbutmaynot be the same size for example

3.3 9 34 123 4

3 6 18 6

Here we see that ratios of correspondingsides are the same i The magnification

4 factorg cancels

Apply this to specialtrianglesHere is on particular A

UsingthePythagorean1

a theorem we find1 145 12 12 of 92 2 3 a Tzi ca o since it is a lengtho jz

This called 45 45 go1 triangle450T l Consider the triangle1

a

4 ra we.se TI4

Another specialtriangle3002 y z Pta 22a 2

a 4 IYoo Il T 69 3WIT ta Ba O

300I jhis is called a

j z 0 60 90triangle1 604I

Intrigue551 5,3 es5y Bz

Example

38 a B 12 woo2 600 it

oriented theWe reproduce ourknown 30 60 90

Same way for A hereconvenience

So to find a note Az Zz so

a E 2 4rII 3

To find b note by r

So b

trample Find a and binz Recall the triangle

afi usoc 5b

1

These are similar so to solve fora wee note

ay trya 3

And b a 35

There are G ratios We need away

to refer to them To this end wei and an angle ofwe say a side

a triangle are adjacent if theside is one of the two sides formingthe angle i angle A and side b are adjacent

or b is adjacenta to A or A is adjacentb I s

TAc tobe Similarly cand

A are adjacentAn angle and side of a triangle areopposite if they are not adjcentSo in the aboveexample A is opposite aor a is opposite A

In the case of a righttriangle coine

angle is a right angle The side oppositethe right angle is called the hypotenuse

hypotenuse There are 2additional7

Sides called legs and twoadditional angles

ALet's choose an angle A Letssayb

n is has measure Q Of thea two legs one is oppositeA and

one is adjacent to a Here a is oppositeAand b is adjacent to A

If we had picked the other

angle B a is adjacent to B and

A b is opposite 13I h We have 6 ratios of sides

b which dependonly on thea B measure of A orb G

So with respect to we havetheir reciprocals

We call these in the picturSince opposite I

hypotenuse Inthis ratio Rthisisdepends on 0 relativeto the

anglewith measure 0

Cosa adjacentTheseratios hypotenuse hdepend ontyg

tano opposite a

adjacent b

Remember Soh CabToaWe have also their reciprocals

Ising Csce

Cos QSec 0

Iand tano

Coto

Remarte If we call the measureof the angle complementary toQ H then

Cosa sinySino cos 4tano CotyCoto tanySeco Csc 4Csce secy

ExampleSin 300 12

I cos 38 IZ

B fan 300 15 51and sin 450 42 22

Cos450 E

tan 450 7 1K

Yourcalculator knows the ratios forother triangles

For example a

If we wantto find a we form a

ratio i

ay sin 400 29 2sin400

a et 286

To avoid unnecessary errordo not round until thefinal answer Note wecould'veused cos 600 instead

Exampley Solve for a and b

a To find a form2 µ a ratio

b 2 Cos700a

So 2 a cos 700

a 2

Cos 700

re 5.848To find b we consider theratio by tan 1700

D 2 tan FooE 5.495

Note Youmightconsider instead

tf Sin 700

be a sin too2Cos OJ

Sin 700

2 Sintoj

ExampleCos too f 2tan707

Now Suppose we have informationabout the sides and want to knowthe angle

3 so

We form the ratio andrecognize it as

Sino

The calculator knows how tosolve thisarcs in E solves Sino

So o Arcs in E I 36.9

In general for OE 0 590

A resin a solves Sino a

for 0daL 1Notice that Arcs in a is an

angleA resin is seen written inother waysArcs in a as in a sin ca

But it probably shares a buttonwith is on your calculator

We can estimate 4 by 90 Oor by noting

cos 4

Again your calculator knowshow to solve this

Arccos E solves 2g cosySo 2 8 53 I

Exajifiliconsider5 of

2We see 5 is the lengthof the side opposite themarked angle and 2 is thelength adjacent to themarked angle So we

formtan a I

2

And find similarly

A retain E a 68.20

Note that while we have6 ratios only 3 are convenientWhen using a calculatorSo when you are consideringratios keep that in mind Alsouse exact values wheneverpossible