Transcript Angles, Degrees, and Special Triangles

Definition I: Trigonometric
Functions
Trigonometry
MATH 103
S. Rook
Overview
• Section 1.3 in the textbook:
– The Six Trigonometric Functions
– Algebraic Signs of Trigonometric Functions
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The Six Trigonometric Functions
The Six Trigonometric Functions
• We begin our actual study of Trigonometry
with six common functions/ratios related to
an angle θ
• MEMORIZE the following definition of the six
trigonometric functions – they are VERY
IMPORTANT!!!
• Given an angle θ in standard position and a
point (x, y) on the terminal side of θ, then the
six trigonometric functions of θ are:
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The Six Trigonometric Functions
(Continued)
Function
Abbreviation
Definition
y
r
x
The cosine of θ
cos θ
r
y
The tangent of θ
tan θ
,x  0
x
x
The cotangent of θ
cot θ
,y0
y
r
The secant of θ
sec θ
,x  0
x
r
The cosecant of θ
csc θ
,y0
y
Where r  x 2  y 2 and x and y retain their
signs from (x, y)
The sine of θ
sin θ
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The Six Trigonometric Functions
(Continued)
Function
Abbreviation
Definition
y
r
x
The cosine of θ
cos θ
r
y
The tangent of θ
tan θ
,x  0
x
x
The cotangent of θ
cot θ
,y0
y
r
The secant of θ
sec θ
,x  0
x
r
The cosecant of θ
csc θ
,y0
y
Where r  x 2  y 2 and x and y retain their
signs from (x, y)
The sine of θ
sin θ
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The Six Trigonometric Functions
(Example)
Ex 1: For each point on the terminal side of θ, i)
draw θ in standard position and ii) find all six
trigonometric functions of θ
a) (-3, -4)
b)

2, 2

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Algebraic Signs of Trigonometric
Functions
Algebraic Signs of Trigonometric
Functions
• The sign of the six trigonometric functions
depends on which quadrant θ terminates in:
r is the distance from the origin to (x, y) so it is
ALWAYS positive
– The signs of x and y depend on which quadrant
(x, y) lies
– Remember the shorthand notation involving “the
element of” symbol:
• i.e.   QIV means theta is a standard angle which
terminates in Q IV
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Algebraic Signs of Trigonometric
Functions (Continued)
Functions
sin  
y
r
cos  
x
r
y
tan  
x
θ Є QI
θ Є QII
θ Є QIII
θ Є QIV
and
csc 
r
y
+
+
–
–
and
sec  
r
x
+
–
–
+
and
cot 
x
y
+
–
+
–
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Algebraic Signs of Trigonometric
Functions (Example)
Ex 2: Find the remaining trigonometric
functions of θ if:
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a) cos   ,  QIV
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b) cot  2,  QII
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Summary
• After studying these slides, you should be able to:
– Give the definition of ANY trigonometric function by
referencing x, y, and r
– Find any or all of the six trigonometric functions given a
point (x, y)
– Find any or all of the remaining trigonometric functions
given the value for a trigonometric function and the
quadrant in which θ terminates
• Additional Practice
– See the list of suggested problems for 1.3
• Next lesson
– Introduction to Identities (Section 1.4)
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