Transcript 1.4 - Phoenix Union High School District

1
Trigonometric
Functions
Copyright © 2009 Pearson Addison-Wesley
1.4-1
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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1.4-2
1.4 Using the Definitions of the
Trigonometric Functions
Reciprocal Identities ▪ Signs and Ranges of Function Values ▪
Pythagorean Identities ▪ Quotient Identities
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Reciprocal Identities
For all angles θ for which both functions are
defined,
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Example 1(a)
USING THE RECIPROCAL IDENTITIES
Since cos θ is the reciprocal of sec θ,
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Example 1(b)
USING THE RECIPROCAL IDENTITIES
Since sin θ is the reciprocal of csc θ,
Rationalize the
denominator.
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Signs of Function Values
 in
Quadrant sin  cos 
tan 
cot 
sec 
csc 
I
+
+
+
+
+
+
II
+




+
III


+
+


IV

+


+

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Ranges of Function Values
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Example 2
DETERMINING SIGNS OF FUNCTIONS
OF NONQUADRANTAL ANGLES
Determine the signs of the trigonometric functions of
an angle in standard position with the given measure.
(a) 87°
The angle lies in quadrant I, so all of its
trigonometric function values are positive.
(b) 300°
The angle lies in quadrant IV, so the cosine and
secant are positive, while the sine, cosecant,
tangent, and cotangent are negative.
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Example 2
DETERMINING SIGNS OF FUNCTIONS
OF NONQUADRANTAL ANGLES (cont.)
Determine the signs of the trigonometric functions of
an angle in standard position with the given measure.
(c) –200°
The angle lies in quadrant II, so the sine and
cosecant are positive, while the cosine, secant,
tangent, and cotangent are negative.
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Example 3
IDENTIFYING THE QUADRANT OF AN
ANGLE
Identify the quadrant (or quadrants) of any angle 
that satisfies the given conditions.
(a) sin  > 0, tan  < 0.
Since sin  > 0 in quadrants I and II, and tan  < 0 in
quadrants II and IV, both conditions are met only in
quadrant II.
(b) cos  > 0, sec  < 0
The cosine and secant functions are both negative
in quadrants II and III, so  could be in either of
these two quadrants.
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Ranges of Trigonometric
Functions
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Example 4
DECIDING WHETHER A VALUE IS IN
THE RANGE OF A TRIGONOMETRIC
FUNCTION
Decide whether each statement is possible or
impossible.
(a) sin θ = 2.5
Impossible
(b) tan θ = 110.47
Possible
(c) sec θ = .6
Impossible
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Example 5
FINDING ALL FUNCTION VALUES
GIVEN ONE VALUE AND THE
QUADRANT
Suppose that angle  is in quadrant II and
Find the values of the other five trigonometric
functions.
Choose any point on the terminal side of angle .
Let r = 3. Then y = 2.
Since  is in quadrant II,
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Example 5
FINDING ALL FUNCTION VALUES
GIVEN ONE VALUE AND THE
QUADRANT (continued)
Remember to
rationalize the
denominator.
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Example 5
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FINDING ALL FUNCTION VALUES
GIVEN ONE VALUE AND THE
QUADRANT (continued)
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Pythagorean Identities
For all angles θ for which the function values are
defined,
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Quotient Identities
For all angles θ for which the denominators are
not zero,
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Example 6
FINDING OTHER FUNCTION VALUES
GIVEN ONE VALUE AND THE
QUADRANT
Choose the positive
square root since sin θ >0.
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Example 6
FINDING OTHER FUNCTION VALUES
GIVEN ONE VALUE AND THE
QUADRANT (continued)
To find tan θ, use the quotient identity
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Caution
In problems like those in Examples 5
and 6, be careful to choose the
correct sign when square roots are
taken.
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Example 7
FINDING OTHER FUNCTION VALUES
GIVEN ONE VALUE AND THE
QUADRANT
Find sin θ and cos θ, given that
quadrant III.
and θ is in
Since θ is in quadrant III, both sin θ and cos θ are
negative.
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Example 7
FINDING OTHER FUNCTION VALUES
GIVEN ONE VALUE AND THE
QUADRANT (continued)
Caution
It is incorrect to say that sin θ = –4
and cos θ = –3, since both sin θ and
cos θ must be in the interval [–1, 1].
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Example 7
FINDING OTHER FUNCTION VALUES
GIVEN ONE VALUE AND THE
QUADRANT (continued)
Use the identity
to find sec θ. Then
use the reciprocal identity to find cos θ.
Choose the negative
square root since sec θ <0
for θ in quadrant III.
Secant and cosine are
reciprocals.
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Example 7
FINDING OTHER FUNCTION VALUES
GIVEN ONE VALUE AND THE
QUADRANT (continued)
Choose the negative
square root since sin θ <0
for θ in quadrant III.
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Example 7
FINDING OTHER FUNCTION VALUES
GIVEN ONE VALUE AND THE
QUADRANT (continued)
This example can also be
worked by drawing θ is
standard position in quadrant
III, finding r to be 5, and then
using the definitions of sin θ
and cos θ in terms of x, y,
and r.
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