Trigonometric Ratios

The term "trigonometry" derives from the Greek (!'thgonometrj'a"), meaning "triangle measuring".

Trigonometry specifically deals with the relationships between the sides and the angles of triangles, that is, the trigonometric functions, and with calculations based on these functions. Trigonometry has important applications in many branches of pure mathematics as well as of applied mathematics and, consequently, much of science. Ancient Egj^tians used trigonometry to reset land boundaries after the Nile river flooded each year, and the Babylonians used i t to measure distances to nearby stars. Trigonometry is used in engineering, cartography, medical imaging, and many other fields.

I n this chapter we study the relationship between the ratios of sides of r ight triangles. These ratios are called Trigonometric Ratios. All Trigonometric functions are used for right triangles only.

The three ratios we use i n Geometry are SINE, COSINE, & TANGENT.

^• l̂Each roMo iS based. .op|x:)sr-[-e

TRIGONOMETRIC RATIO

Sine of A

•„^' Cosine of A

Tangent ofA

ABBREVIATION

Si

Cos

an

DEFINITION

_ oppQ^tfe Side I , 6b

RATIO PL c

b c

*** o p p o s i t e Sidt. ognd 4ne

What happens if the angle changes to B? a d ^ q m l V Sfd^, [1̂ )111̂ 1 S>a5l' - c h /

Can the angle be C? MOj HeVer W | 6 -tVie'^O" ATl j Ie .

The right angle is never used in trigonometry, as the hypotenuse doesn't change.

A mnemonic to help memorize this JsOHCAHTOAl S-Sine 0-Opposite leg H-Hypotenuse

1

C-Cosine A-Adjacent leg H-Hypotenuse T-Tangent 0-Opposite leg A-Adjacent leg

Sni .Y° = Opposite leg Hypotenuse

Cos x° = Adjacent leg Hypotenuse

Tan .v° = Opposite leg Tan .v° = Adjacent leg

E X A M P L E Determine the trigonometric ratios. Express each ratio as a fraction.

c o s A = _ ! H i 3 _ ^ . ^ 2 3 l t a n A = ^ / l 2 _ , H 1 Ipl

s i n B = ] ! V i 3 _ « , ^ i ^ l cos B = 5/13 g .3S4(p t a n B = l i /S__« Z.HfiOO

S i n ' hypo

Cos- ^

E X A M P L E 2- Determine t h ^ g i ^ o n ^ ^ c ^ r a ^ . Express each ratio as a fraction.

s i n S = J / S . * ^ . ^OOO c o s E = ^ / 5 _ ^ ' bOOO t a n S = i / y L ^ . 1 5 0 O

! the t n E X A M P L E 2: Determine £^e1arigonometric ratios. Express eaobNCatio as a fraction.

^ t a n C = _ZI l l 2X V5

^ I f f \ \B zx

Verify the (sin C)2 + (cos C)2 = 1 ^

-1- + ^ - I

Example 3 - Use your trig table/calculator to determine the following ratios^ (calculators must be in degree mode) , _ kr\0^t^ V^OS sin 10° = « n cos 80° =

tan 40» = »ft3'^ I