Transcript Trigonometry 9e

1
Trigonometric
Functions
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1.3-1
1 Trigonometric Functions
1.1 Angles
1.2 Angle Relationships and Similar
Triangles
1.3 Trigonometric Functions
1.4 Using the Definitions of the
Trigonometric Functions
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1.3 Trigonometric Functions
Trigonometric Functions ▪ Quadrantal Angles
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Trigonometric Functions
Let (x, y) be a point other the origin on the terminal
side of an angle  in standard position. The
distance from the point to the origin is
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Trigonometric Functions
The six trigonometric functions of θ are
defined as follows:
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Example 1
FINDING FUNCTION VALUES OF AN
ANGLE
The terminal side of angle  in standard position
passes through the point (8, 15). Find the values of
the six trigonometric functions of angle .
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Example 1
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FINDING FUNCTION VALUES OF AN
ANGLE (continued)
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Example 2
FINDING FUNCTION VALUES OF AN
ANGLE
The terminal side of angle  in standard position
passes through the point (–3, –4). Find the values
of the six trigonometric functions of angle .
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Example 2
FINDING FUNCTION VALUES OF AN
ANGLE (continued)
Use the definitions of the trigonometric functions.
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Example 3
FINDING FUNCTION VALUES OF AN
ANGLE
Find the six trigonometric function values of the
angle θ in standard position, if the terminal side of θ
is defined by x + 2y = 0, x ≥ 0.
We can use any point on
the terminal side of  to
find the trigonometric
function values.
Choose x = 2.
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Example 3
FINDING FUNCTION VALUES OF AN
ANGLE (continued)
The point (2, –1) lies on the terminal side, and the
corresponding value of r is
Multiply by
to rationalize
the denominators.
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Example 4(a) FINDING FUNCTION VALUES OF
QUADRANTAL ANGLES
Find the values of the six trigonometric functions for
an angle of 90°.
The terminal side passes
through (0, 1). So x = 0, y = 1,
and r = 1.
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Example 4(b) FINDING FUNCTION VALUES OF
QUADRANTAL ANGLES
Find the values of the six
trigonometric functions for an
angle θ in standard position
with terminal side through
(–3, 0).
x = –3, y = 0, and r = 3.
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Undefined Function Values
If the terminal side of a quadrantal angle lies along
the y-axis, then the tangent and secant functions
are undefined.
If the terminal side of a quadrantal angle lies along
the x-axis, then the cotangent and cosecant
functions are undefined.
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Commonly Used Function
Values

sin 
cos 
tan 
cot 
sec 
csc 
0
0
1
0
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1
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90
1
0
undefined
0
undefined
1
180
0
1
0
undefined
1
undefined
270
1
0
undefined
0
undefined
1
360
0
1
0
undefined
1
undefined
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Using a Calculator
A calculator is degree mode
returns the correct values
for sin 90° and cos 90°.
The second screen shows
an ERROR message for tan
90° because 90° is not in
the domain of the tangent
function.
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Caution
One of the most common errors
involving calculators in trigonometry
occurs when the calculator is set for
radian measure, rather than degree
measure.
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