Download Trig Functions of Real Numbers

Trig Functions of Real
Numbers
Lesson 2.4a
The Unit Circle




P(x, y) •
Consider a circle with
radius r = 1
θ
r=1
Wrap t onto the
circumference
Then w(t) is a function
which wraps t to a
point P(x, y)
Also, t translates to θ (radians)
t
t  s  r 
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The Unit Circle

The trig ratios for θ
can tell us x and y

Since r = 1
y
sin t   y
1
x
cos t   x
1
y
tan t 
x
P(x, y)
•
θ
r=1
View Nspire
Demo
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Example
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
4
Given t 
3
Then


 1  3 
w(t )  P  ,

 2 2 

y
What are sin, cos, tan?
What if

x
t  
Determine P(x, y)
What are the trig functions?
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Implications
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
It is now possible to take functions of
angles greater than 360 (2π) or less
than -360 (-2π)
Try these
 7 
cot 

 3 

tan 12 
cos  8 
csc8.2
Use both


Wrapping concept
Calculator (watch angle mode)
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Properties of Trig Functions
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Odd functions

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Even functions


f(-x) = - f(x)
This definition is
also applied to non
trig functions.
f(-x) = f(x)
Which of the trig functions are??
Odd
Even
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Trig Functions are Periodic
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The functions repeat themselves
The period is the smallest value, p for
which
f(x) = f(x + p)
For sin, cos, sec, csc

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The period is 2π
For tan and ctn

The period is π
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Assignment
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Lesson 2.4a
Page 166
Exercises 1 – 47 odd
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