1.4 - Phoenix Union High School District

Download Report

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
Copyright © 2009 Pearson Addison-Wesley
1.4-2
1.4 Using the Definitions of the
Trigonometric Functions
Reciprocal Identities ▪ Signs and Ranges of Function Values ▪
Pythagorean Identities ▪ Quotient Identities
Copyright © 2009 Pearson Addison-Wesley
1.1-3
1.4-3
Reciprocal Identities
For all angles θ for which both functions are
defined,
Copyright © 2009 Pearson Addison-Wesley
1.1-4
1.4-4
Example 1(a)
USING THE RECIPROCAL IDENTITIES
Since cos θ is the reciprocal of sec θ,
Copyright © 2009 Pearson Addison-Wesley
1.1-5
1.4-5
Example 1(b)
USING THE RECIPROCAL IDENTITIES
Since sin θ is the reciprocal of csc θ,
Rationalize the
denominator.
Copyright © 2009 Pearson Addison-Wesley
1.1-6
1.4-6
Signs of Function Values
 in
Quadrant sin  cos 
tan 
cot 
sec 
csc 
I
+
+
+
+
+
+
II
+




+
III


+
+


IV

+


+

Copyright © 2009 Pearson Addison-Wesley
1.4-7
Ranges of Function Values
Copyright © 2009 Pearson Addison-Wesley
1.4-8
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.
Copyright © 2009 Pearson Addison-Wesley
1.1-9
1.4-9
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.
Copyright © 2009 Pearson Addison-Wesley
1.1-10
1.4-10
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.
Copyright © 2009 Pearson Addison-Wesley
1.1-11
1.4-11
Ranges of Trigonometric
Functions
Copyright © 2009 Pearson Addison-Wesley
1.1-12
1.4-12
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
Copyright © 2009 Pearson Addison-Wesley
1.1-13
1.4-13
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,
Copyright © 2009 Pearson Addison-Wesley
1.1-14
1.4-14
Example 5
FINDING ALL FUNCTION VALUES
GIVEN ONE VALUE AND THE
QUADRANT (continued)
Remember to
rationalize the
denominator.
Copyright © 2009 Pearson Addison-Wesley
1.1-15
1.4-15
Example 5
Copyright © 2009 Pearson Addison-Wesley
FINDING ALL FUNCTION VALUES
GIVEN ONE VALUE AND THE
QUADRANT (continued)
1.1-16
1.4-16
Pythagorean Identities
For all angles θ for which the function values are
defined,
Copyright © 2009 Pearson Addison-Wesley
1.1-17
1.4-17
Quotient Identities
For all angles θ for which the denominators are
not zero,
Copyright © 2009 Pearson Addison-Wesley
1.1-18
1.4-18
Example 6
FINDING OTHER FUNCTION VALUES
GIVEN ONE VALUE AND THE
QUADRANT
Choose the positive
square root since sin θ >0.
Copyright © 2009 Pearson Addison-Wesley
1.1-19
1.4-19
Example 6
FINDING OTHER FUNCTION VALUES
GIVEN ONE VALUE AND THE
QUADRANT (continued)
To find tan θ, use the quotient identity
Copyright © 2009 Pearson Addison-Wesley
1.1-20
1.4-20
Caution
In problems like those in Examples 5
and 6, be careful to choose the
correct sign when square roots are
taken.
Copyright © 2009 Pearson Addison-Wesley
1.1-21
1.4-21
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.
Copyright © 2009 Pearson Addison-Wesley
1.1-22
1.4-22
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].
Copyright © 2009 Pearson Addison-Wesley
1.1-23
1.4-23
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.
Copyright © 2009 Pearson Addison-Wesley
1.1-24
1.4-24
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.
Copyright © 2009 Pearson Addison-Wesley
1.1-25
1.4-25
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.
Copyright © 2009 Pearson Addison-Wesley
1.1-26
1.4-26