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```Algebra and Trig. I
4.2 – Trigonometric Functions: The Unit Circle
The unit circle is a circle of radius 1, with its center at the origin of
a rectangular coordinate system. The equation of this unit circle is
. We can use the following formula for the length of a
circular arc,
, to find the length of the intercepted arc.
where 1 is the radius of the circle and t is the radian measure of
the central angle.
Thus the length of the intercepted arc is θ, which is also the
radian measure of the central angle. *So in a unit circle, the
radian measure of the central angle is equal to the length of
the intercepted arc.
P(x,y)
(1,0)
0
0
θ
(1,0)
θ
When θ is positive the point P is
reached by moving
counterclockwise along the unit
circle from (1,0)
When θ is negative the point P is
reached by moving clockwise
along the unit circle from (1,0)
P(x,y)
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The Six Trigonometric Functions –
The inputs of these six functions are θ and the outputs involve the
point
on the unit circle corresponding to t and the
coordinates of this point.
Trig. Functions have names that are words, rather than single
letters such as
. For example, the sine of θ is the ycoordinate of the point P on the unit circle.
The real number y depends on the letter θ and thus is a function
of t.
really means
, where sine is the name of the
function and θ, a real number, is an input.
Name
sine
cosine
tangent
Abbreviation
sin
cos
tan
Name
cosecant
secant
cotangent
Abbreviation
csc
sec
cot
Definitions of the Trigonometric Functions in Terms of
Any Circle:
If θ is a real number and
is a point on the unit circle
that corresponds to θ, and r is the radius then
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Definitions of the Trigonometric Functions in Terms of a
Unit Circle:
If θ is a real number and
is a point on the unit circle
that corresponds to θ, where r is the radius which is 1 then
Finding Values of the Trigonometric Functions
P( , )
θ
θ
0
If θ is a real number equal to the length of the
intercepted arc of an angle that measures θ
(1,0)
is the point on the
unit circle that corresponds to θ. Find the
values of the six trig. functions at t. Don’t
forget to rationalize if needed.
a)
c)
b)
d)
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f)
e)
Example –
P=
Use the figure to the left to determine the
values of the six trig. functions at
.
0
(1,0)
a)
b)
g)
h)
i)
j)
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Domain and Range of Sine and Cosine Functions –
Example – Find
Example – Find
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Trigonometric Functions at
Even and Odd Trigonometric Functions
We know that a function is even if
and odd if
. We can show that the cosine function is even and
sine is odd.
By definition, the coordinates of the points P
and Q are as follows:
P
(1, 0)
0
Q
In the above figure the x-coordinates of P and Q are the same,
thus
thus the cosine function is even.
In the above figure the y-coordinates of P and Q are negatives of
each other, thus
thus the sine function is odd
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Even and Odd Trigonometric Functions
Only cosine and secant functions are even all other are odd.
Example – Find the value of each trigonometric function:
a)
b)
Fundamental Identities
Trigonometric identities are equations that are true for all real
numbers for which the trigonometric expressions in the equations
are defined.
Reciprocal Identities
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Quotient Identities
Example – Given
and
, find the value of
each of the four remaining trigonometric functions.
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Pythagorean Identities –
Example – Given
and
using a trigonometric identity.
, find the value of
Example – Given
and
using a trigonometric identity.
, find the value of
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Periodic Functions
A function f is periodic if there exists a positive number p such
that
for all in the domain of f. The smallest
positive number p for which f is periodic is called the period of f.
Periodic Properties of the Sine and Cosine Functions
The sine and cosine functions are periodic functions and have
period 2π.
The secant and cosecant functions have period 2π.
Periodic Properties of the Tangent and Cotangent Functions
The tangent and cotangent functions are periodic functions and
have period π.
Example – Find the value of each trigonometric function.
a)
c)
b)
d)
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Repetitive Behavior of the Sine, Cosine and Tangent Functions
For any integer n and real number ,
Evaluating Trig. Functions with a Calculator
a)
b)
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