4.2
TRIGONOMETRIC FUNCTIONS: THE UNIT CIRCLE
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What You Should Learn
• Identify a unit circle and describe its relationship
to real numbers.
• Evaluate trigonometric functions using the unit
circle.
• Use the domain and period to evaluate sine and
cosine functions.
• Use a calculator to evaluate trigonometric
functions.
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The Unit Circle
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The Unit Circle
The unit circle
x2 + y2 = 1
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The Unit Circle
Each real number t also corresponds to a central angle 
(in standard position) whose radian measure is t.
The real number t is the (directional) length of the arc
intercepted by the angle  , given in radians.
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The Trigonometric Functions
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The Trigonometric Functions
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The Trigonometric Functions
The unit circle has been divided into eight equal arcs,
corresponding to t-values of
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The Trigonometric Functions
The unit circle has been divided into 12 equal arcs,
corresponding to t-values of
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Example 1 – Evaluating Trigonometric Functions
Evaluate the six trigonometric functions at each real
number.
a.
b.
c. t = 0 d. t = 
Solution:
a.
corresponds to the point (x, y) =
.
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Example 1 – Solution
cont’d
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Example 1 – Solution
b.
corresponds to the point (x, y) =
cont’d
.
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Example 1 – Solution
cont’d
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Example 1 – Solution
cont’d
c. t = 0 corresponds to the point (x, y) = (1, 0).
sin 0 = y = 0
cos 0 = x = 1
tan 0 =
=
=0
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Example 1 – Solution
cont’d
is undefined.
is undefined.
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Example 1 – Solution
cont’d
d. t =  corresponds to the point (x, y) = (–1, 0).
sin  = y = 0
cos  = x = –1
tan  =
=
=0
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Example 1 – Solution
cont’d
is undefined.
is undefined.
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Domain and Period of Sine and Cosine
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Domain and Period of Sine and Cosine
The domain of the sine and cosine functions is the set of all
real numbers.
–1 
y 1
–1  Sin t  1
–1 
x 1
–1  cos t  1
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Domain and Period of Sine and Cosine
sin (t + 2 n) = sin t
cos (t + 2 n) = cos t
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Domain and Period of Sine and Cosine
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Example 3 – Using the Period to Evaluate the Sine and Cosine
a. Because
b. Because
, you have
, you have
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Example 3 – Using the Period to Evaluate the Sine and Cosine
c. For
odd.
,
cont’d
because the sine function is
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