Transcript circular functions. - Phoenix Union High School District

3
Radian
Measure and
Circular
Functions
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3.3-1
Radian Measure and Circular
3 Functions
3.1 Radian Measure
3.2 Applications of Radian Measure
3.3 The Unit Circle and Circular Functions
3.4 Linear and Angular Speed
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3.3-2
3.3 The Unit Circle and Circular
Functions
Circular Functions ▪ Finding Values of Circular Functions ▪
Determining a Number with a Given Circular Function Value ▪
Applying Circular Functions
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Circular Functions
A unit circle has its center at the origin and a
radius of 1 unit.
The trigonometric functions of
angle θ in radians are found by
choosing a point (x, y) on the
unit circle can be rewritten as
functions of the arc length s.
When interpreted this way, they
are called circular functions.
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Circular Functions
For any real number s represented by a
directed arc on the unit circle,
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The Unit Circle
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The Unit Circle
 The unit circle is symmetric with respect to the
x-axis, the y-axis, and the origin.
If a point (a, b) lies on the unit circle, so do
(a,–b), (–a, b) and (–a, –b).
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The Unit Circle
 For a point on the unit circle, its reference arc
is the shortest arc from the point itself to the
nearest point on the x-axis.
For example, the quadrant I real number
is associated with the point
on the
unit circle.
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The Unit Circle
Since sin s = y and cos s = x, we can replace x
and y in the equation of the unit circle
to obtain the Pythagorean identity
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Domains of Circular Functions
Sine and Cosine Functions:
Tangent and Secant Functions:
Cotangent and Cosecant Functions:
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Evaluating A Circular Function
Circular function values of real numbers are
obtained in the same manner as
trigonometric function values of angles
measured in radians.
This applies both to methods of finding
exact values (such as reference angle
analysis) and to calculator approximations.
Calculators must be in radian mode
when finding circular function values.
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Example 1
FINDING EXACT CIRCULAR FUNCTION
VALUES
Find the exact values of
Evaluating a circular function
at the real number
is
equivalent to evaluating it at
radians.
An angle of
intersects the
circle at the point (0, –1).
Since sin s = y, cos s = x, and
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Example 2(a) FINDING EXACT CIRCULAR FUNCTION
VALUES
Use the figure to find the exact values of
The real number
corresponds to the
unit circle point
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Example 2(b) FINDING EXACT CIRCULAR FUNCTION
VALUES
Use the figure and the definition of tangent to find
the exact value of
Moving around the unit
circle
units in the
negative direction
yields the same
ending point as
moving around the
circle units in the
positive direction.
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Example 2(b) FINDING EXACT CIRCULAR FUNCTION
VALUES
corresponds to
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Example 2(c) FINDING EXACT CIRCULAR FUNCTION
VALUES
Use reference angles and degree/radian
conversion to find the exact value of
An angle of
corresponds to an angle of 120°.
In standard position, 120° lies in quadrant II with a
reference angle of 60°, so
Cosine is negative
in quadrant II.
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Example 3
APPROXIMATING CIRCULAR
FUNCTION VALUES
Find a calculator approximation for each circular
function value.
(a) cos 1.85 ≈ –.2756
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(b) cos .5149 ≈ .8703
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Example 3
APPROXIMATING CIRCULAR
FUNCTION VALUES (continued)
Find a calculator approximation for each circular
function value.
(c) cot 1.3209 ≈ .2552
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(d) sec –2.9234 ≈ –1.0243
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Caution
A common error in trigonometry is
using a calculator in degree mode
when radian mode should be used.
Remember, if you are finding a
circular function value of a real
number, the calculator must be in
radian mode.
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Example 4(a) FINDING A NUMBER GIVEN ITS
CIRCULAR FUNCTION VALUE
Approximate the value of s in the interval
if cos s = .9685.
Use the inverse cosine function of a calculator.
, so in the given interval, s ≈ .2517.
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Example 4(b) FINDING A NUMBER GIVEN ITS
CIRCULAR FUNCTION VALUE
Find the exact value of s in the interval
if tan s = 1.
Recall that
negative.
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, and in quadrant III, tan s is
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Example 5
MODELING THE ANGLE OF ELEVATION
OF THE SUN
The angle of elevation of the sun in the sky at any
latitude L is calculated with the formula
where
corresponds to sunrise and
occurs if the sun is directly overhead. ω is the
number of radians that Earth has rotated through
since noon, when ω = 0. D is the declination of the
sun, which varies because Earth is tilted on its axis.
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Example 5
MODELING THE ANGLE OF ELEVATION
OF THE SUN (continued)
Sacramento, CA has latitude L = 38.5° or .6720
radian. Find the angle of elevation of the sun θ at 3
P.M. on February 29, 2008, where at that time,
D ≈ –.1425 and ω ≈ .7854.
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Example 5
MODELING THE ANGLE OF ELEVATION
OF THE SUN (continued)
The angle of elevation of the sun is about .4773
radian or 27.3°.
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