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
