Trigonometric Ratios 8.2

The term "trigonometry" derives from the Greek ^' trigonometrid'), 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 i n many branches of pm-e mathematics as well as of applied mathematics and, consequently, much of science. Ancient Egyptians used trigonometry to reset land boundaries after the Nde 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. Al l Trigonometric functions are used for right triangles only.

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

^Each^octlo 1^ based , < \ S ^ ^ B

^ 01 ^ i v m angle 1 y ) ^ ^

TRIGONOMETRIC RATIO A B B R E V I A I T O N

Sine of A

Cosine of A

Tangent of A

Opposite Sicl6

Sin Cos

J

^ D E F I N I T I O N RATIO a.

What happens if the angle changes to B?__ ^Aji^xoLn-V s ide SuM^j-chJ

Can the angle be C?_

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

A mnemonic to help memorize this - SOHCAHTOA. V ^ S-Sine 0-Opposite leg H-H>'potenuse C-Cosine A-Adjacent leg H-H}'potenuse T-Tangent 0-Opposite leg A-Adjacent leg

Sm .Y° --Opposite leg Sm .Y° --Hypotenuse

Cos x° Adjacent leg Hypotenuse

Tau .v° Opposite leg Adjacent leg

E X A M P L E i : Determine the trigonometric ratios. Express each ratr6 as a traction.

c o s A = _lHi3_ tan A = g/}2-

sinB -_ lVL3_ cos B = 5 / [ 3

E X A M P L E 2: Deterroine the trigonometric ratios. Express each ratio as a fraction. SDM-CA14-TDA ^ ] / ;Owv : . o W | .

s in S = = 37

^ t a n S = 3/M ^ ,15

E X A M P L E 2: Determine the trigonometric ratios. Express eadxjatio as a fraction.

s m A = hYpo" "IT" tan C = '^YPP 2 J ( "

1

Verify the (ain C)^ + (cos cy = 1

2X 2

Example 3 - Use your trig table/calculator to determine the following ratios^ (calculators must be

m degree mode) _^ ^O-KoS f f t r W O r d "

c o s 8 o ° = A^^yp ^--fe^C'2)0) =

t a n 4 0 - _ d M L U n ( q o ) -Example 4 - Use youiCtrig table)to determine the measures of the angles based on the trig ratios:

sin A = 0.9848

cos B = 0.7771

tan C = 2.8