Trigonometry - The Six Trigonometric Ratios

Identifying the six trigonometric ratios

1. THE SIX
TRIGONOMETRIC
RATIOS
MS. REYBETH D. RACELIS, LPT

2. OBJECTIVES
•Illustrates the six
trigonometric ratios
•Find the trigonometric ratios
of special angles.
•Use trigonometric ratios to
solve problems involving right
triangles.

3. ARITHMETIC
•isconcernedmainlywiththestudy
of structure, operations and
applications ofwholenumbersand
positiverationalnumberswhether
intheformoffraction,decimalor
percent.

4. ALGEBRA
•isconcernedwiththestudyof
thevariables represented by
letters and the operations
relatingthesevariables.

5. Advanced Definition of Algebra
Algebra(from Arabic al-jabrmeaning
"reunion of broken parts") is the
branch of mathematics concerning the
study of the rules of operations and
relations, and the constructions and
concepts arising from them,
including terms, polynomials,
equations and algebraic structures.

6. FATHER OF ALGEBRA
MuhammadibnMusaal-Khwarizmi
wasa9th-centuryPersianMuslim
mathematician andastronomer.Heis
knownasthe"fatherofalgebra",a
wordderivedfromthetitleofhis
book,Kitabal-Jabr.

7. AL-KHWARIZMI, THE FATHER OF ALGEBRA
•AbuJa'farMuhammadibnMusaal-Khwarizmilivedin
Baghdad,around780to850CE(orAD).Hewasoneof
thefirsttowriteaboutalgebra(usingwords,not
letters).
•Around825hewrotethebook"HisabAl-jabrw’al-
muqabala", from which we get the
word algebra (meaning 'restoration ofbroken
parts').Thisbookincludedmanywordproblems,
especiallydealingwithinheritance.
•Al-Khwarizmi helped establish widespread use
ofHindu-Arabicnumbers:
•1,2,3,...

8. •TheHindu-Arabicsystemwasmucheasiertousewhen
performingmathematical operations, sinceitisa
base-10system.EvertriedtomultiplyusingRoman
numerals...?

9. FATHER OF ALGEBRA
Diophantus of Alexandra
-was a Greek mathematician whom many
have considered as the “Father of
Algebra”. He lived during the 3
rd
century
A.D. and wrote the treatise Arithmetica.
His fame lies in representing unknown by
“symbols”.

10. GEOMETRY
-isabranchofmathematicsthatstudies
the sizes,shapes,positions angles and
dimensionsofthings.

11. FATHER OF GEOMETRY
Euclidentershistoryasoneofthe
greatestofallmathematicians andheis
oftenreferred toasthefatherof
geometry.Thestandardgeometrymostof
uslearnedinschooliscalledEuclidian
Geometry.

12. TRIGONOMETRY
•Abranchofmathematics that
studies triangles and the
relationship between the
lengthsoftheirsidesandthe
anglesbetweenthosesides.

13. •The word trigonometrycomes from
the Greek words for triangle
(trigōnon) and measure
(metron).

14. FATHER OF TRIGONOMETRY
Hipparchus of Nicaea
HipparchuswasaGreekastronomerwholivedbetween
190-120B.C.Heisconsidered thefather of
trigonometry, abranchofmathematics whichstudies
theanglesofsidesoftriangles.

15. CONCEPTS:
•TRIANGLE–apolygonwith3vertices,3
sideswhicharelinesegments.
•TRIGONOMETRIC RATIO–aratiothat
describesarelationshipbetweenaside
andanangleofatriangle

16. •ADJACENTSIDE–thesidenexttothe
referenceangleinarighttriangle.
•OPPOSITE SIDE–thesideacrossthe
referenceangleinarighttriangle.
•HYPOTENUSE–thelongestsideofaright
triangle.
-alwaysoppositetothe90°
angleinarighttriangle.

17. •RIGHTTRIANGLE–inwhichoneangle
isarightangle.
•SOH-CAH-TOA–amnemonicusedfor
rememberingtheequation.
•MNEMONIC–atoolusedtohelp
rememberfactorlargeamountof
information.

18. Sine, Cosine and Tangent
The main functions in trigonometry are Sine, Cosine
and Tangent
They are simply one side of a right -angled triangle
divided by another.
For any angle "θ":
(Sine, Cosine and
Tangent are
often abbreviated
tosin, cos and tan.)

19. OTHER FUNCTIONS (COTANGENT, SECANT,
COSECANT)
Similar to Sine, Cosine and Tangent, there are
three othertrigonometric functions which are
made by dividing one side by another:

20. PYTHAGOREAN THEOREM
•statesthatthesquareofthe
hypotenuseofarighttriangle
isequaltothesumofthe
squaresoftheothertwosides.
c²=a²+b²

21. •Pythagoras ofSamos is
oftenreferredtoasthefirst
pure mathematician. Greek
philosopher,mathematician, and
discovered the Pythagorean
theorem.

22. EXERCISES
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