Transcript Angles, Degrees, and Special Triangles

Definition III: Circular Functions
Trigonometry
MATH 103
S. Rook
Overview
• Section 3.3 in the textbook:
– The six trigonometric functions and the Unit Circle
– Domain of the six trigonometric functions
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The Six Trigonometric Functions
and the Unit Circle
Unit Circle
• Recall that the Unit Circle is the special circle
with a radius of 1 and equation of x2 + y2 = 1
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Revisiting Definition I and
Definition II
• Consider a point (x, y)
on the unit circle
• By Definition I:
cos  
x x
 x
r 1
sin  
y y
 y
r 1
• By Definition II:
adj  x
cos 
 x
hyp 1
opp y
sin  
 y
hyp 1
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Circular Functions
• Therefore, (x, y) can be written as (cos θ, sin θ)
• Now consider a point (x, y) on the
circumference of the unit circle where t is the
length of the arc from (1, 0) to (x, y)
s t


• Then r  1  t
– The central angle is equivalent to the length of
the arc it cuts on the Unit Circle
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Circular Functions (Continued)
• All points encountered on the unit circle can then be
written as (cos t, sin t) where t is the distance traveled
from (1, 0)
t can be positive (counterclockwise) or negative (clockwise)
• The six trigonometric functions with respect to the Unit
Circle are:
1
y
cost  x tan t  x , x  0 csc t  y , y  0
sin t  y cott  x , y  0
y
sec t 
1
,x  0
x
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Circular Functions (Continued)
• We call these the circular functions
– The radian measure of θ is the same as the arc length from
(1, 0) to a point P on the terminal side of θ on the
circumference of the unit circle
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Circular Functions (Example)
Ex 1: Use the Unit Circle to find the six
trigonometric functions of:
a)
7
4
b)
4
3
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Circular Functions (Example)
Ex 2: Use the Unit circle to find all values of t,
0 < t < 2π where
a)
1
cos t 
2
b)
2
sin t  
2
c) tan t  1
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Circular Functions (Example)
Ex 3: If t is the positive distance from (1, 0) to
point P along the circumference of the unit
circle, sketch t on the circumference of the
unit circle and then find the value of:
a) P = (-0.9422, 0.3350); find i) cos t, ii) csc t,
and iii) cot t
b) P = (0.5231, -0.8523); find i) sin t, ii) sec t,
and iii) tan t
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Review of Functions
• Recall that a function can be thought of as a
machine which takes an input (or an
argument) value and operates on it to
produce an output value
– Ex. y = cos x: input x is placed into the cosine
function to produce output y
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Review of Functions (Example)
Ex 4: Identify i) the function, ii) the argument of
the function, and iii) the function value:
a)
b)
sin
13
6
tan

4
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Domain of the Six Trigonometric
Functions
Domain of cosine and sine
• Recall that the domain of a function is the set
of allowable input values (usually x)
• For cos t and sin t, there are no domain
restrictions
– cosine and sine are defined for every value of t
along the circumference of the unit circle
– Written in interval notation as  ,
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Domain of secant and tangent
• For sec t and tan t,
1
sec t 
cos t
and
sin t
tan t 
cos t
– Undefined when cos t = 0
• Referring to the Unit Circle, cos t  0 when t 

2
or

• Starting at 2 , cos t = 0 every additional π radians
3
2
(multiple of π)
• Therefore, sec t and tan t are undefined whenever
t

2
 k where k is an integer
and their domain



is: t | t  R, t   k where k is an integer
2


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Domain of the cosecant and
tangent
• For csc t and cot t,
1
csc t 
sin t
and
cos t
cot t 
sin t
– Undefined when sin t = 0
• Referring to the Unit Circle, sin t  0 when t  0 or 
• Starting at 0, sin t = 0 every additional π radians
(multiple of π)
• Therefore, csc t and cot t are undefined whenever
t  k wherek is an integer and their domain
is: t | t  R, t  k where k is an integer
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Domain of the Trigonometric
Functions (Example)
Ex 5: Answer the following:
a) What is the domain for the secant and the
tangent? What is the value of tan 7
2
b) What is the domain for the cosecant and
the cotangent? What is the value of csc(6 )
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Summary
• After studying these slides, you should be able to:
– Define the six trigonometric functions in terms of
circular functions
– Identify the domain of the six trigonometric functions
• Additional Practice
– See the list of suggested problems for 3.3
• Next lesson
– Arc Length and Area of a Sector (Section 3.4)
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