Transcript ppt

Trigonometric Functions of Real Numbers;
Periodic Functions
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Use a unit circle to define trigonometric functions of real numbers.
Recognize the domain and range of sine and cosine functions
Use even and odd trigonometric functions
Use periodic functions
Dr .Hayk Melikyan / Departmen of Mathematics and CS
[email protected]
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The Unit Circle

A unit circle is a circle of radius 1, with its center at the
origin of a rectangular coordinate system. The equation
of this circle is x 2  y 2  r 2 .
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The Unit Circle
In a unit circle, the radian measure of the central angle is
equal to the length of the intercepted arc. Both are given
by the same real number t.
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Definition: The Value of a trigonometric function at the real number
t is its value at an angle of t radians
If t is a real number and P = (x y) is a point on the unit circle that corresponds
to t, then
Unit circle
y
Q = (0, 1)
y
sin t  y cost  x tant 
x
1
1
x
csc t 
sec t 
cot t 
y
x
y
/2
x
(1, 0)
P(x, y)
x 2 + y2 = 1
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Example: Finding Values of the Trigonometric Functions
Use the figure to find the values of the trigonometric functions
at t.
1
sin t  y 
2
3
cost  x 
2
1
1
1
y
2


tan t  
x
3
3
3
2
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3

3 3
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Example: Finding Values of the Trigonometric Functions
Use the figure to find the values of the trigonometric functions at t.
1 1 2
csct 
y 1
2
1 1
2
2
sect  


x
3
3
3
2
3 2 3

3
3
3
x
cot t   2  3
1
y
2
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The Domain and Range of the Sine
and Cosine Functions
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Text Example
Use the figure at the right to find the values of
the trigonometric functions at t = /2.
y
Solution The point P on the unit circle that
corresponds to t = /2 has coordinates (0, 1).
We use x = 0 and y = 1 to find the values of the
trigonometric functions.
P = (0, 1)
/2
(1, 0)
sin
csc

2

2
 y  1, cos


2
 x 0
x
x 2 + y2 = 1
1 1
 x 0
  1, cot    0
y 1
2 y 1
By definition, tan t = y/x and sec t = 1/x. Because x = 0, tan /2 and sec /2 are
undefined.
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The Domain and Range of the Sine and Cosine
Functions
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The domain of the sine function and the cosine
function is the set of all real numbers.

The range of these functions is the set of all real
numbers from –1 to 1, inclusive.
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Even and Odd Trigonometric Functions
The cosine and secant functions are even.
cos (-t) = cos t
sec(-t) = sec t
The sine, cosecant, tangent, and cotangent functions are odd.
sin (-t) = – sin t
csc (-t) = – csc t
tan (-t) = – tan t
cot (-t) = – cot t
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Example
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Find the exact value of sin (-45º)
2
sin(45 )   sin(45 )  
2
o
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Definition of a Periodic Function
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A function f is periodic if there exists a positive
number p such that
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f (t + p) = f (t)
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For all t in the domain of f. The smallest number p
for which f is periodic is called the period of f.
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Periodic Properties of the Sine and Cosine Functions
The sine and cosine functions are periodic functions and
have period 2.
• sin (t + 2) = sin t
• cos (t + 2) = cos t
Example
Find the exact value of cos(5)
cos(5 )  cos(4   )  cos  1
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Periodic Properties of the Tangent and Cotangent
Functions
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tan (t + ) = tan t and cot (t + ) = cot t
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The tangent and cotangent functions are
periodic functions and have period .
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Repetitive Behavior of the Sine, Cosine, and
Tangent Functions
For any integer n and real number t, is
sin (t + 2  n) = sin t,
cos (t + 2  n) = cos t, and
tan (t +  n) = tan t.
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Home Work
1.
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