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Chapter 4
Pre-Calculus
OHHS
4.3 The Circular Functions
• Solve Trig Functions of Any Angle
• Solve Trig Functions of Real
Numbers
• Understand Periodic Functions
• Analyze the 16-point Unit Circle
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Anatomy of an Angle
Initial Side
Vertex
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Angle Rotation
Positive Angle
Counter
Clockwise
Negative Angle
Clockwise
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Standard Position
y
Vertex at
(0,0)
Initial Side
x
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Quadrantal Angles
• Terminal side of a standard position
angle is on an axis.
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Coterminal Angles
• Angles with the same
initial and terminal sides,
but with different
rotations.
• All of these are
coterminal angles.
45º
405º
-315º
765º
-675º
-1035º
1485º
-1395º
How did I find all these
angles?
45º+360ºn
where n is an integer.
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Example
• Find a positive angle and a negative
angle that are coterminal with
• (a) 30
• 30+360 = 390 30-360=-330
• (b) 
4
•

9
 2 
4
4

7
 2 
4
4
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Now You Try
P. 381, #1
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1st Quadrant Trig
P(x,y)
r
θ
x
r
csc θ =
y
y
y
• sin θ =
r
x
• cos θ =
r
y
• tan θ =
x
x
r
sec θ =
cot θ =
y
x
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Example
Let θ be the acute angle in standard position whose
terminal side contains the point (5, 3). Find the six
trigonometric functions of θ.
3
3 34
sin θ =
=
P(5,3)
34
34
34
θ
5
34
csc θ =
3
3
5
5 34
cos θ =
=
34
34
3
tan θ =
5
34
sec θ =
5
5
cot θ =
3
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Now You Try
• P. 381, #5
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Example
Let θ be the acute angle in standard position whose
terminal side contains the point (-5, 3). Find the six
trigonometric functions of θ.
3
3 34
P(-5,3)
sin θ =
=
34
34
34
3
θ
-5
34
csc θ =
3
-5
-5 34
cos θ =
=
34
34
3
tan θ = 5
 34
sec θ =
5
-5
cot θ =
3
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Now You Try
• P. 381, #11
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Reference Angle
• The angle formed between the terminal
side and the nearest part of the x-axis.
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Example
• Find the exact values of the six
trigonometric functions of 315
Reference
315
Angle
1
45
45
-1
2
1  2
sin  

2
2
1
2
cos  

2
2
tan   1
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Now You Try
• P. 381, #25
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Example
• Find the coordinates on the unit circle
where θ = 210º
sin 210º =
θ = 210º
x
-1 
2
y
3
2
30º
tan 210º =
1
sec 210º =

3 1


, 2

2


2 3

3
3
3
cos 210º =
2
csc 210º =
3  3
2
1

2
cot 210º =
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Now You Try
• P. 381, #29
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Trig Functions of
Quadrantal Angles
cot(180) =
x 1

y 0
undefined
180
(-1, 0)
sin(180) =
r = x 2 + y2
2
0
2
r = (-1) + 0 = 1
y 0

1
r
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Now You Try
• P. 381, #41
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Using One Trig Ratio to Find
the Others
3
• If sin θ = and tan θ < 0, find cos θ.
7
• In which quadrant is this true? Q2
 40
cos  
7
3
7
θ
 40
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Using One Trig Ratio to Find
the Others
• If sec θ = 3 and sin θ > 0, find tan θ.
• In which quadrant is this true? Q1
8
tan 
2 2
1
3
82 2
1
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Now You Try
• P. 381, #43
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The Unit Circle
• A circle with radius = 1
1
• What is its circumference?
• 2
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The Unit Circle
Wrapping
Function
π
2
3π
4
π
4

0
5π
4
0
π
4
Circumference
2
π
2
3π
4
3π
2

or
2
7π
4
5π
4
3π
2
7π
4
2
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Using the Unit Circle to Find Trig
Ratios
• The terminal
side of any
angle t
intersects the
unit circle at
(cos t, sin t)
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Trigonometric Functions on the
Unit Circle
sin t = y
1
csc t =
y
cos t = x
1
sec t =
x
y
tan t =
x
x
cot t =
y
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Unit Circle Example
• Find tan 
3π
4

π
2
y
tan  =
x
π
4
0
tan  =
-1
(-1, 0)
0
5π
4
3π
2
tan  = 0
7π
4
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Periodic Functions
• A function is periodic if there is a
positive number c such that
f(t + c) = f(t)
for all values of t in the domain of f.
• The smallest such number c is called
the period of the function.
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Your Turn
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Home Work
• P. 381,
• #2, 6, 8, 18, 20, 22, 26, 30,
36, 40, 44, 48, 50, 52,
61-66, 68, 70
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